Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on...

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Дата:	2016
Автори:	Nazarov, F., Sodin, M.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2016
Назва видання:	Журнал математической физики, анализа, геометрии
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Цитувати:	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions / F. Nazarov, M. Sodin // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 3. — С. 205-278. — Бібліогр.: 30 назв. — англ.

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spelling	irk-123456789-1405542018-07-11T01:23:10Z Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions Nazarov, F. Sodin, M. We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains. 2016 Article Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions / F. Nazarov, M. Sodin // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 3. — С. 205-278. — Бібліогр.: 30 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.03.205 Mathematics Subject Classification 2010: 60G15 http://dspace.nbuv.gov.ua/handle/123456789/140554 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.
format	Article
author	Nazarov, F. Sodin, M.
spellingShingle	Nazarov, F. Sodin, M. Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions Журнал математической физики, анализа, геометрии
author_facet	Nazarov, F. Sodin, M.
author_sort	Nazarov, F.
title	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
title_short	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
title_full	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
title_fullStr	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
title_full_unstemmed	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
title_sort	asymptotic laws for the spatial distribution and the number of connected components of zero sets of gaussian random functions
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/140554
citation_txt	Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions / F. Nazarov, M. Sodin // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 3. — С. 205-278. — Бібліогр.: 30 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT nazarovf asymptoticlawsforthespatialdistributionandthenumberofconnectedcomponentsofzerosetsofgaussianrandomfunctions AT sodinm asymptoticlawsforthespatialdistributionandthenumberofconnectedcomponentsofzerosetsofgaussianrandomfunctions
first_indexed	2025-07-10T10:43:20Z
last_indexed	2025-07-10T10:43:20Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2016, vol. 12, No. 3, pp. 205–278 Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions F. Nazarov∗ Dept. of Math. Sciences, Kent State University, Kent OH 44242, USA E-mail: nazarov@math.kent.edu M. Sodin∗∗ School of Math. Sciences Tel Aviv University Tel Aviv 69978, Israel E-mail: sodin@post.tau.ac.il Received September 2, 2015 We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensem- bles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains. Key words: smooth Gaussian functions of several real variables, the num- ber of connected components of the zero set, ergodicity. Mathematics Subject Classification 2010: 60G15. Contents 1. Introduction and the Main Results . . . . . . . . . . . . . . . 206 2. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 ∗Supported by grants No. 2006136, 2012037 of the United States – Israel Binational Science Foundation and by U.S. National Science Foundation Grants DMS-0800243, DMS-1265623. ∗∗Supported by grants No. 2006136, 2012037 of the United States – Israel Binational Science Foundation and by grant No. 166/11 of the Israel Science Foundation of the Israel Academy of Sciences and Humanities. c© F. Nazarov and M. Sodin, 2016 F. Nazarov and M. Sodin 3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4. Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. Quantitative Versions of Bulinskaya’s Lemma . . . . . . 225 6. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7. Recovering the Function ν̄ by a Double Scaling Limit 239 8. Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 9. The Manifold Case. Proof of Theorem 3 . . . . . . . . . . . 247 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 A. Smooth Gaussian Function . . . . . . . . . . . . . . . . . . . . . 250 B. Proof of The Fomin–Grenander–Maruyama Theorem 272 C. Condition (ρ4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 In memory of Volodya Matsaev 1. Introduction and the Main Results The result we present has two main versions. The first one treats zero sets of smooth Gaussian functions on the Euclidean space Rm with translation-invariant distributions. The second version deals with parametric ensembles of smooth Gaussian functions in open domains in Rm. We also show how to translate the second version to parametric ensembles of smooth Gaussian functions on smooth manifolds without boundary. In Appendix A, all parts of the theory of smooth Gaussian functions needed for understanding this work are developed from scratch. Appendix B contains the proof of the Fomin–Grenander–Maruyama theorem in the multidimensional setting. None of the results in these Appendices is our own work. 1.1. The translation invariant case Suppose F : Rm → R is a continuous Gaussian random function with translation- invariant distribution (meaning that for every v ∈ Rm, the continuous Gaussian functions F and F (·+v) have the same distribution). Then the covariance kernel K(x, y) = E{F (x)F (y)} 206 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components of F depends only on the difference x−y and can be written in the form K(x, y) = k(x−y) where k : Rm → R is a symmetric positive definite function. By Bochner’s theorem?, k can be written as the Fourier integral k(x) = (Fρ)(x) = ∫ Rm e2πi(x·λ) dρ(λ) of some finite symmetric positive Borel measure ρ on Rm, which is called the spectral measure of F . We denote by Z(F ) = F−1{0} the (random) zero set of F . Let S be any bounded open convex set in Rm containing the origin. By S(R) we denote the set {x ∈ Rm : x/R ∈ S}. By N S (R; F ) we denote the number of the connected components of Z(F ) that are contained in S(R). When S is the unit ball B = {x : \|x\| < 1}, we will write N(R; F ) instead of N B (R;F ). We say that a finite complex-valued measure µ on Rm is Hermitian if for each Borel set E ⊂ Rm, we have µ(−E) = µ(E). By Fµ we denote the Fourier integral of the measure µ, and by spt(µ) we denote the closed support of µ. Theorem 1. Suppose that the spectral measure ρ of a continuous Gaussian translation-invariant function F satisfies the following conditions: (ρ1) ∫ Rm \|λ\|4 dρ(λ) < ∞; (ρ2) ρ has no atoms; (ρ3) ρ is not supported on a linear hyperplane. Then there exists a constant ν > 0 such that for every bounded open convex set S ⊂ Rm containing the origin, lim R→∞ N S (R; F ) volS(R) = ν almost surely and lim R→∞ E ∣∣∣NS (R; F ) volS(R) − ν ∣∣∣ = 0 . (1.1.1) Furthermore, ν > 0 provided that (ρ4) there exist a finite compactly supported Hermitian measure µ with spt(µ) ⊂ spt(ρ) and a bounded domain D ⊂ Rm such that Fµ ∣∣ ∂D < 0 and (Fµ)(u0) > 0 for some u0 ∈ D. 1.1.1. Rôle of conditions (ρ1) − (ρ3). Condition (ρ1) guarantees that F ∈ C2−(Rm) def= ⋂ α∈(0,1) C1+α(Rm). Condition (ρ3) says that the distribution of the gradient ∇F is non-degenerate. Together conditions (ρ1) and (ρ3) allow us to ?See [3, § 20] for the original proof or [17] for a clear self-contained exposition. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 207 F. Nazarov and M. Sodin think of the zero set Z(F ) as a collection of pairwise disjoint smooth hypersurfaces that partition Rm into “nodal domains”. The translation invariance allows us to consider the probability distribution measure generated by F on an appropriate space of functions as an invariant measure with respect to the action of the abelian group Rm by translations (τvg)(·) = g(· + v). Condition (ρ2) ensures that this action is ergodic, which in turn implies that the limit ν is non-random. 1.1.2. Condition (ρ4). Condition (ρ4) is essentially equivalent to the possibility to deterministically create at least one bounded connected component of the zero set Z(F ). The measures not satisfying (ρ4) have to be very degenerate. In particular, the support of any measure not satisfying (ρ4) has to be contained in a quadratic hypersurface in Rm. We prove this, as well as some other observations pertaining to condition (ρ4), in Appendix C.. On the other hand, the Fourier transform of the Lebesgue surface measure on the sphere centered at the origin is radial and sign changing. So if spt(ρ) is a sphere in Rm centered at the origin then (ρ4) is still satisfied. These observations suffice to check condition (ρ4) in most interesting exam- ples. 1.1.3. What can be said about the constant ν? Unfortunately, the proof of Theorem 1 does not provide much information about the value of the constant ν. There is a huge discrepancy between the lower bounds that can be extracted from the “barrier construction” introduced in [25] and the upper bounds obtained by computing the mean number of special points in the nodal domains or in the zero set (cf. Nastasescu’s undergraduate thesis [24]). It is worth noting that the limiting constant ν̄ equals the expectation E{ vol(G0)−1 } , where G0 is the connected component of Rm\Z(F ) containing the origin (or any other given point in Rm). The random variable vol(G0) is, perhaps, even more mysterious than N(R; F ). Our theorem shows that P{ vol(G0) < +∞} > 0, but we still do not even know how to prove that this probability is 1, not mentioning any efficient tail estimate for its distribution. 1.1.4. Further remarks about Theorem 1. Theorem 1 can be viewed as a version of the “law of large numbers” for the “connected component process” on Rm associated with the Gaussian function F . In most applications, one does not need as strong convergence as is guaranteed by Theorem 1 and just the convergence in probability (which is equivalent to the convergence in distribution for constant limits) is enough. Note also that the value of the intensity ν(F ) is completely determined by the covariance kernel k(x−y) of F , or, which is the same, by the spectral measure ρ. Our last remark concerns a non-degenerate linear change of variables. Let T : Rm → Rm be a non-degenerate linear operator and let F̃ (x) = F (Tx). Then 208 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components F̃ is also a Gaussian translation-invariant function. Moreover, for every S ⊂ Rm and R > 0, we have NTS(R;F ) = NS(R; F̃ ), whence, ENS(R; F̃ ) volS(R) = \|det T \| ENTS(R;F ) vol(TS)(R) . Thus, if the intensity ν(F ) exists, then so does ν(F̃ ), and we have the relation ν(F̃ ) = \| detT \| ν(F ) . 1.2. Parametric Gaussian ensembles Definition 1 (parametric Gaussian ensemble). A parametric Gaussian en- semble (f L ) on an open set U ⊂ Rm (or on an m-dimensional manifold X with- out boundary) is any family (f L ) of continuous Gaussian functions on U (on X, respectively) indexed by some countable set of numbers L > 1 accumulating only at +∞. Many interesting parametric Gaussian ensembles (in particular, two examples considered below in Sec. 2) arise from the following construction. Let X be a smooth compact m-dimensional manifold without boundary. Let HL be a sequence of real finite-dimensional Hilbert spaces of continuous functions on X indexed by some scaling parameter L > 1 so that limL→∞ dimHL = ∞. Since, for every x ∈ X, the point evaluation HL 3 f 7→ f(x) is a continuous linear functional on HL, there is a unique function Kx L ∈ HL such that f(x) = 〈f, Kx L〉. The function KL(x, y) = Kx L(y) is called the reproducing kernel of the space HL. Since, Kx L ∈ HL, we have Kx L(y) = 〈Kx L,Ky L〉, so KL(x, y) is symmetric. Now let { ek } be an orthonormal basis in HL. Then for every f ∈ HL, we have f = ∑ k〈f, ek〉ek in HL and, therefore, pointwise. Thus KL(x, y) = Kx L(y) = ∑ k ek(x)ek(y). Consider the continuous Gaussian function f L (x) = ∑ k ξkek(x), x ∈ X , where ξk are independent standard real Gaussian random variables. The covari- ance kernel of the Gaussian function f L equals E{ f L (x)f L (y) } = ∑ k ek(x)ek(y) = KL(x, y) , Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 209 F. Nazarov and M. Sodin so it does not depend on the choice of the orthonormal basis { ek } and coincides with the reproducing kernel of HL. It follows that the distribution of f L also does not depend on the choice of the basis and is completely determined by the space HL itself. We shall call this continuous Gaussian function f L the continuous Gaussian function generated by HL. 1.2.1. Normalization. We say that a continuous Gaussian function f on U with the covariance kernel K is normalized if E{ f(x)2 } = K(x, x) = 1 for all x ∈ U . If the random Gaussian function f is not normalized but non-degenerate (that is, E{ f(x)2 } > 0, or, what is the same, P{f(x) = 0} = 0 for every x ∈ U), we can just replace f by f̃(x) = f(x)√ K(x,x) , which will correspond to replacing the covariance kernel K(x, y) by K̃(x, y) = K(x, y)√ K(x, x) K(y, y) , (1.2.1.) without affecting the zero set Z(f) in any way. Note that if we allow f to degenerate at some points uncontrollably, then the zero set of f may contain deterministic pieces of arbitrarily complicated structure and our talk about the asymptotic behavior of the number of nodal components of f may easily become totally meaningless. Thus, • we will always assume that all continuous Gaussian functions and all para- metric Gaussian ensembles in this paper are normalized. Note that in many basic examples, including the ones we consider below in Sec. 2, the function x 7→ KL(x, x) is constant, so the normalization of K reduces to the division by that constant. 1.2.2. Scaling and translation-invariant local limits. Let U be an open set in Rm and let (f L ) be a parametric Gaussian ensemble on U . Let K L be the covariance kernel of f L . We define the scaled covariance kernel Kx,L at a point x ∈ U by Kx,L(u, v) = KL ( x + u L , x + v L ) . Note that Kx,L is the covariance kernel of the scaled Gaussian function fx,L(u) = f L ( x + u L ) , i.e., Kx,L(u, v) = E{ fx,L(u)fx,L(v) } . Note also that if x ∈ U is fixed and L →∞, the sets Ux,L = {u ∈ Rm : x + u L ∈ U} exhaust Rm. 210 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Definition 2 (translation-invariant limit). Let (f L ) be a parametric Gaussian ensemble on an open set U ⊂ Rm and let K L be the covariance kernel of f L . Let x ∈ U . We say that the scaled covariance kernels Kx,L have a translation- invariant limit if there exists a continuous function kx : Rm → R such that, for each u, v ∈ Rm, lim L→∞ Kx,L(u, v) = kx(u− v) . (1.2.2) We say that the parametric Gaussian ensemble (f L ) has a translation invariant limit at the point x if there exists a translation invariant continuous Gaussian function Fx on Rm such that, for every finite point set U ∈ Rm, the finite- dimensional Gaussian vectors fx,L\|U converge to Fx\|U in distribution. We call the function Fx the local limiting function and its spectral measure ρx the local limiting spectral measure of the parametric Gaussian ensemble (f L ) at the point x. If a parametric Gaussian ensemble (f L ) on U has a translation invariant limit Fx at some point x ∈ U , then the scaled covariance kernels Kx,L have a translation invariant limit as well and the limiting kernel kx(u− v) is the covariance kernel of Fx. On the other hand, without any additional assumptions, covariance kernels Kx,L(u, v) may have a translation invariant limit kx(u−v) that corresponds to no continuous Gaussian function F . However, within the set-up considered in this paper, these notions become equivalent. It is natural to believe that if a parametric Gaussian ensemble (f L ) on U has a translation invariant limit at every point x ∈ U , then for large L, we can count the nodal components of f L in some open set V ⊂ U by partitioning V into nice sets Vj of size larger than 1/L, choosing some points xj ∈ Vj , approximating the number of nodal components of f L in each set Vj by the number of nodal compo- nents of Fxj in (Vj)xj ,L = { v ∈ Rm : xj +L−1v ∈ Vj } , and adding all these counts up. If we are lucky enough, the nodal components of Fx may have asymptotic intensity ν̄(x) = ν(Fx) and then the total count we get will be typically close to ∑ j ν̄(xj) vol(Vj)xj ,L = Lm ∑ j ν̄(xj) volVj . If we are even luckier, the quantity ν̄(x) may depend on x in a nice enough way for the Riemann sums ∑ j ν̄(xj) volVj to converge to ∫ V ν̄ d vol. The formalization of this intuitive argument requires some accuracy, especially because the standard integral calculus nowadays is Lebesgue, not Riemann. The classical form of a convergence statement for integrals in the Lebesgue language is that of the dominated convergence theorem, whose general structure is • Given a sequence of nice objects that converge in some fairly weak and easy to check sense to some limiting object, and assuming that our pre-limiting Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 211 F. Nazarov and M. Sodin objects are uniformly controlled in some way, the limiting object is nice as well, and some integral functional of the limiting object is the limit of the integral functionals of the pre-limit objects. Our Theorem 2 will be exactly of this structure. We have already introduced in Definition 2 the modes of convergence we will be using. Now it is time to define “controllability”. 1.2.3. Uniform smoothness of covariance kernels K L . The control we want to impose will be two-fold. First, we will need to restrict the typical speed of oscillation of the continuous Gaussian functions f L . Some restriction of this type is inevitable because fast oscillating continuous Gaussian functions like the Brownian motion on R1 change sign infinitely many times near every their zero and they still have fairly decent moduli of continuity on the Hölder scale. The control we will impose will guarantee that fx,L ∈ C2−(U) = ⋂ τ∈(0,1) C1+τ (U) on every compact subset of U . For k > 1, by Ck,k(U ×U) we denote the class of functions g : U ×U → R for which all partial derivatives ∂α x ∂β y g(x, y), \|α\|, \|β\| 6 k (taken in any order) exist and are continuous?. For L > 1, a compact set Q ⊂ U , and g ∈ Ck,k(U × U), we put ‖g‖L,Q,k def= max \|α\|,\|β\|6k max x,y∈Q L−(\|α\|+\|β\|)∣∣∂α x ∂β y g(x, y) ∣∣. When L = 1, we will write ‖g‖Q,k instead of ‖g‖1,Q,k. If the covariance kernel K of a continuous Gaussian function f on U belongs to Ck,k(U×U), then the semi-norms ‖K‖L,Q,k can be computed on “the diagonal” α = β and x = y: ‖K‖L,Q,k = max\|α\|6k max x∈Q L−2\|α\|∣∣∂α x ∂α y K(x, y)\|y=x ∣∣. A näıve explanation to this fact comes from the Cauchy–Schwarz inequality com- bined with the formula ∂α x ∂β y K(x, y) = E{ ∂α x f(x) ∂β y f(y) } , which is true in the case when the derivatives on the RHS exist and are continuous random functions. The proof of this fact for the general case will be given in Appendix A.11. The uniform smoothness of the kernels KL with k > 1 is more than enough to erase any distinction between the existence of a translation invariant limit of the kernels Kx,L and the existence of a translation invariant limit at x of the parametric Gaussian ensemble (f L ). ?in which case, they do not depend on the order 212 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components 1.2.4. Local uniform non-degeneracy of the parametric Gaussian ensemble (f L ). Our second restriction will be of the opposite character. While the local uniform smoothness guarantees that the continuous Gaussian functions f L do not change too fast or in a too rough way, the condition we discuss in this section will ensure that f L cannot change too slowly or in a too predictable way in any direction. Without any such restriction, there will be nothing that would prevent long regular components to prevail and, with our methods, we will either not be able to say anything at all in such case, or will just conclude that some limit is 0, which would merely mean that the particular scaling we have chosen is a wrong one for the problem. With all this in mind, let us pass to the formal definitions. Let K ∈ C1,1(U × U) and let Cx be the matrix with the entries Cx(i, j) = ∂xi ∂yj K(x, y)\|y=x , x ∈ U . If K is the covariance kernel of some C1 Gaussian function f on U , then Cx is the covariance matrix of the Gaussian random vector ∇f(x). Assuming that detCx 6= 0, we can say that the density of the probability distribution of the random Gaussian vector ∇f(x) in Rm is given by p(ξ) = 1 (2π)m/2 √ det Cx e− 1 2 (C−1 x ξ ξ) . Since in this case C−1 x is positive definite, we have max ξ p(ξ) = p(0) = (2π)−m/2(detCx)−1/2 . Definition 3 (local uniform non-degeneracy of (f L )). We say that a para- metric Gaussian ensemble (f L ) on some open set U ⊂ Rm is locally uniformly non-degenerate if the corresponding kernels KL(x, y) are at least in C1,1(U × U) and for every compact set Q ⊂ U , lim L→∞ inf x∈Q detCx,L > 0 where Cx,L is the matrix with the entries Cx,L(i, j) = ∂ui∂vjKx,L(u, v) ∣∣ u=v=0 = L−2∂xi∂yjKL(x, y) ∣∣ y=x . As the argument above shows, if f L is C1-smooth, then our non-degeneracy condition just means that, for every compact set Q ⊂ U , there is a uniform upper bound for the densities of the distributions of all Gaussian vectors L−1∇f L (x) with x ∈ Q. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 213 F. Nazarov and M. Sodin Suppose that, for some x ∈ U , the kernels Kx,L have a translation invariant limit and the convergence holds in the semi-norm ‖ · ‖Q,1 for some compact set Q containing x in its interior. Then the matrix Cx,L converges to the matrix cx with the entries cx(i, j) = −( ∂ui∂ujkx ) (0) = 4π2 ∫ Rm λiλj dρx(λ) and we see that in this case the limiting measure ρx satisfies inf ξ∈Sm−1 ∫ Rm ∣∣λ ξ ∣∣2 dρx(λ) > 0 , which means that ρx cannot be supported on any linear hyperplane { λ : λ ξ = 0 } , i.e., condition (ρ3) is satisfied. 1.2.5. Controllability. Now we are ready to say what we mean by a locally uniformly controllable parametric Gaussian ensemble (f L ) on U . Definition 4 (locally uniform controllability). The parametric Gaussian en- semble (f L ) on an open set U ⊂ Rm is locally uniformly controllable if it is locally uniformly non-degenerate and the corresponding covariance kernels KL satisfy lim L→∞ ‖KL‖L,Q,2 < ∞ for every compact set Q ⊂ U . The above considerations combined with results presented in Appendix (see A.11 and A.12) imply that • if the kernels KL are locally uniformly controllable and if the scaled kernels Kx,L have translation-invariant limits, then the limiting spectral measure ρx satisfies assumptions (ρ1) and (ρ3) of Theorem 1. 1.2.6. Tameness Definition 5 (tame ensembles). The parametric Gaussian ensemble (f L ) on an open set U ⊂ Rm is tame if (i) it is locally uniformly controllable, and there exists a Borel subset U ′ ⊂ U of full Lebesgue measure such that, for all x ∈ U ′, (ii) the scaled kernels (Kx,L) have translation invariant limits; (iii) the limiting spectral measure ρx has no atoms. A tame parametric Gaussian ensemble (f L ) has a translation invariant limit at every point x ∈ U ′. Moreover, by Theorem 1, the point intensity ν̄(x) def= ν(Fx) associated with the ensemble (f L ) is well-defined on U ′. 214 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components 1.3. The main result Before we state our second main theorem, we will introduce one more object. Let U be an open set in Rm and let f be a continuous Gaussian function on U . We say that a (depending on the implicit probability variable ω) Borel measure n on U is a connected component counting measure of f if spt(n) ⊂ Z(f) and the n-mass of each connected component of Z(f) equals 1. Note that we do not require the dependence of n on ω to be measurable in any sense (for this reason, we do not call n a random measure), so in the statement of the next theorem we will have to use “the upper expectation” E∗ instead of the usual one E . Theorem 2. Suppose that (f L ) is a tame parametric Gaussian ensemble on an open set U ⊂ Rm. Then (i) the function x 7→ ν̄(x) is measurable and locally bounded in U ; and (ii) for every sequence of connected component counting measures n L of f L and for every compactly supported in U continuous function ϕ, we have lim L→∞ E∗ {∣∣∣ 1 Lm ∫ ϕ dn L − ∫ ϕν̄ d vol ∣∣∣ } = 0 . Note that the second statement of that theorem can be strengthened to lim L→∞ E∗ {∣∣∣ 1 Lm ∫ ϕdn L − ∫ ϕν̄ d vol ∣∣∣ q} = 0 for some q = q(m) > 1 (which tends to 1 as m →∞) without any essential change in the proof but we are not aware of any application of this stronger result for which the current version would not suffice as well. 1.4. The manifold version of Theorem 2 Theorem 2 can be transferred to parametric Gaussian ensembles on smooth manifolds without boundary. Everywhere in this section X is an m-dimensional C2-manifold without boundary (not necessarily compact) that can be covered by countably many charts, all charts being assumed open and C2-smooth, and (f L ) is a parametric Gaussian ensemble on X. We start with two definitions. Definition 6 (tame ensembles on manifolds). We say that a parametric Gaus- sian ensemble (f L ) on X is tame if, for every chart π : U → X, the parametric Gaussian ensemble (f L ◦ π) is tame on U . This definition implies that for every chart π : U → X, the parametric Gaus- sian ensemble (f L ◦ π) satisfies the assumptions of Theorem 2. So the associated point intensity ν̄π belongs to L∞loc(U). Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 215 F. Nazarov and M. Sodin Definition 7 (Volumes compatible with smooth structure). We say that a locally finite Borel positive measure vol X on X is a volume compatible with the smooth structure of X if for every chart π : U → X, the measures π∗ vol and vol X are mutually absolutely continuous and the corresponding Radon–Nikodym densities are continuous on π(U). Of course, the main example we have in mind giving this definition is that of a smooth Riemannian manifold X and the volume generated by the Riemannian metric on X. We also note that despite the manifold X may be endowed with no measure, the words “almost every x ∈ X” still have meaning because all push-forward measures π∗ vol corresponding to various charts π : U → X of X are mutually absolutely continuous wherever they can be compared to each other. At last, we can state the manifold version of Theorem 2. Theorem 3. Suppose that (f L ) is a tame parametric Gaussian ensemble on X. Then (i) there exists a locally finite Borel non-negative measure n∞ on X such that for every choice of connected component counting measures n L of f L and every function ϕ ∈ C0(X), lim L→∞ E∗ {∣∣∣ 1 Lm ∫ ϕdn L − ∫ ϕd n∞ ∣∣∣ } = 0 ; (ii) for every chart π : U → X, the measure n∞ coincides on π(U) with the push- forward π∗(ν̄π vol) where ν̄π is the point intensity associated with the parametric Gaussian ensemble f L ◦ π; (iii) if vol X is some volume measure compatible with the smooth structure of X, then n∞ is absolutely continuous with respect to vol X , and there exists a set X ′ ⊂ X of full vol X such that, for every x ∈ X ′, the quantity n(x) = ν̄π(π−1(x)) dπ∗ vol d vol X (x) is well-defined and does not depend on the choice of the chart π : U → X with x ∈ π(U). Moreover, dn∞ = nd vol X . The point of part (iii) is that, for vol X -almost all x ∈ X, it allows one to compute the Radon–Nikodym derivative dn∞ d vol X (x) using any chart containing x. In particular, nothing prevents us from choosing for each point its own individual chart. 216 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components 1.4.1. How to verify tameness? Theorem 3, as stated, has an essential shortcoming: it may be somewhat unpleasant to verify tameness of (f L ) because formally it requires one to estimate various quantities in the local coordinates given by π for every chart π : U → X, however weird or ugly. The next two observations (both of purely technical nature) allow one to substantially reduce this workload. Recall that an atlas on X is any family of charts A = { πα : Uα → X } α such that ⋃ α πα(Uα) = X. Here is our first observation: • Suppose A is an atlas on X and that, for every chart πα ∈ A, (f L ◦ πα) is tame on Uα. Then (f L ) is tame on X. The possibility to check the tameness for the charts from any atlas of our choice is quite a relief. However, one unpleasant thing still remains. It may (and often does) happen that for every point x ∈ X there is one “preferred” chart πx : Ux → X covering x such that the computations in this chart are a piece of cake in any infinitesimal neighborhood of x but not quite so even a bit away from x. In this case we would strongly prefer to compute all quantities and check all conditions at x using its preferred chart πx. However, we are still formally required to run the computations concurrently on any compact subset of any given chart using the local coordinates given by that particular chart. Our next observation takes care of this difficulty. Definition 8. We say that an atlas A of X has uniformly bounded distortions if there exists a constant A > 0 such that all partial derivatives of orders 6 2 of all coordinate functions of all transition maps between the charts of A are bounded by A. Note that this definition doesn’t require X to be uniform in any sense; rather it requires that the charts in A be small enough so that X doesn’t show any non- trivial structure within the union of each chart with all charts it intersects, and that the chart scalings be more or less consistent with each other within small regions. Our second observation says that • If the atlas A has uniformly bounded distortions, then to check the tameness of (f L ) on X, it suffices to check the relevant conditions and uniform bounds (on compact subsets of X) for the related quantities computed in the charts (Ux, πx) at the points π−1 x (x) only. These two observations may be not obvious and we will explain them more in Section 9.. Note that in our examples, we will deal with compact manifolds admitting a transitive group G of diffeomorphisms leaving the parametric Gaussian ensemble (f L ) under consideration invariant (meaning that for each g ∈ G and each L, Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 217 F. Nazarov and M. Sodin the continuous Gaussian functions fL and fL ◦ g have the same distribution). In such situation, all one needs is to find one chart π : U → X such that the atlas consisting of the charts g ◦ π, g ∈ G, has uniformly bounded distortions. Then one may fix his/her favorite point x = π(u) in that chart, and establish all the required bounds and conditions at this single point for this single chart. All passages about “almost every x” and suprema and infima over Q in all conditions can be ignored in such setup because all the related objects and quantities do not depend on x at all. 1.5. The final remarks about Theorems 2 and 3 1.5.1. Note that the particular choice of the counting measures nL plays no rôle. The reason is that, for large L, with high probability the overwhelming part of nL comes from components of arbitrarily small diameter. Such components can be viewed as single points at the macroscopic level. 1.5.2. If the manifold X is compact, we can apply the conclusion of Theo- rem 3 to ϕ ≡ 1 and to obtain the asymptotics (n∞(X) + o(1))Lm for the typical (and the mean) total number of nodal components of f L on X as L → ∞. Of course, this asymptotic law is really useful only when n∞(X) > 0. Finding an asymptotic formula (or even a decent estimate) for the variance of the total number of nodal components in such regimes remains an open problem. 1.5.3. The proof of Theorem 2 also shows that the value ν̄(x) can be recovered as a double-scaling limit. In Lemma 12 we show that, for almost every x ∈ U and for each ε > 0, we have lim R→∞ lim L→∞ P {∣∣∣N ( x,R/L; f L ) volB(R) − ν̄(x) ∣∣∣ > ε } = 0 where N ( x, R L ; f L ) = N(R, fx,L) is the number of the connected components of the zero set Z(f L ) contained in the open ball centered at x of radius R/L. 1.5.4. A few words should be said about the measurability issues. While we prove every measurability result that is necessary for the completeness of the formal exposition, when possible, we circumvent this discussion by using upper integral and upper expectation instead of the usual ones. Note that the Borel measurability of similar quantities has been discussed in detail in Rozenshein’s Master Thesis [26, Sec. 5]. 1.6. Pertinent works 1.6.1. The earliest non-trivial lower bound for the mean number of connected components is, probably, due to Malevich. In [23], she considered a C2-smooth 218 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components translation-invariant Gaussian random function F on R2 with positive covariance kernel decaying at a certain rate at infinity. She proved that EN(R; F )/R2 is bounded from below and from above by two positive constants. Her proof of the lower bound uses Slepian’s inequality and probably cannot be immediately extended to models with covariance kernels that change their signs. 1.6.2. Several years ago, Bogomolny and Schmit [5] proposed a bond percola- tion model for the description of the zero set of the translation-invariant Gaussian function F on R2 whose spectral measure is the Lebesgue measure on the unit circumference. This model completely ignores slowly decaying correlations be- tween values of the random function at different points and is very far from being rigorous. The predictions of Bogomolny and Schmit were checked by computa- tional experiments carried out by Nastasescu [24], Konrad [18], and Beliaev and Kereta [2]. The observed value of the constant ν was very close to but still notice- ably less than the Bogomolny and Schmit prediction. It would be very interesting to reveal a hidden “universality law” that provides the rigorous foundation for the work done by Bogomolny and Schmit. Note also that it is not clear whether or to what degree their approach can be extended to make reasonably accurate predic- tions about the behavior of nodal components of translation-invariant Gaussian functions corresponding to other spectral measures in R2 or in dimensions m > 2. 1.6.3. In [25], we showed that for the Gaussian ensemble of spherical har- monics of large degree L on the two-dimensional sphere, the total number N(fL) of connected components of Z(fL) satisfies P {∣∣L−2N(f L )− υ ∣∣ > ε } < C(ε)e−c(ε) dimHL , with some υ > 0. The limiting function for this ensemble is the one considered by Bogomolny and Schmit. The case of higher dimension (in a slightly different setting) was treated by Rozenshein in [26]. The exponential concentration of N(f L )/L2 is interesting since this model has slowly decaying correlations. We were unable to prove the exponential concentration for other ensembles considered here. The difficulty is caused by the small components, which do not exist when f L is an eigenfunction of the Laplacian. Even in the univariate case, the question about the exponential concentration in Theorem 1 remains open; cf. Tsirelson’s lecture notes [30]. Some lower bounds for the number of connected components of the zero set and for other similar quantities were obtained in different settings by Bourgain and Rudnick [7], Fyodorov, Lerario, Lundberg [9], Gayet and Welschinger [10, 11, 12], Lerario and Lundberg [22] using the “barrier construction” from [25]. 1.6.4. Certain versions of main results of this work were presented at the St. Petersburg Summer School in Probability and Statistical Physics (June, 2012) and appeared in the lecture notes [29]. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 219 F. Nazarov and M. Sodin 1.6.5. There have been several works of interest relying on ideas and tech- niques developed in this paper, among which those by Bourgain [6], Canzani and Sarnak [8], Kurlberg and Wigman [20] and Sarnak and Wigman [28] deserve special attention of the reader. Acknowledgments. On many occasions, Boris Tsirelson helped us by pro- viding information and references concerning Gaussian measures and measura- bility. Leonid Polterovich and Zeév Rudnick have read parts of a preliminary version of this work and made several valuable comments, which we took into ac- count. We have had encouraging discussions of this work with Andrei Okounkov, Peter Sarnak, and Jean-Yves Welschinger. Alex Barnett and Maria Nastasescu showed us the beautiful and inspiring simulations. We thank them all. 2. Examples Here, we point out two examples illustrating Theorem 3. In our examples, the manifold X has a natural Riemannian metric and a transitive group of isometric diffeomorphisms that leaves the distribution of (f L ) invariant. As discussed near the end of Sec. 1.4, this will allow us to check the conditions of Theorem 3 at just one point x ∈ X with respect to a natural local chart associated with this point and to conclude that the limiting measure n∞ on the manifold X is a constant multiple of the Riemannian volume on X. Moreover, since in our examples the kernels Kx,L(u, v) converge to k(u−v) uniformly with all derivatives on compact subsets of Rm×Rm, the uniform smoothness and non-degeneracy of the kernels Kx,L(u, v) can be derived from the corresponding properties of the limiting kernel k(u − v). Passing to the limit in our examples is an elementary exercise in Taylor calculus and complex analysis. This list of examples may be continued (see [8, 20, 26, 28]) but the two ones we included into this paper should be already enough to convey its main message, which is • Under not unreasonably unfavorable conditions, establishing the asymptotics for the number of nodal domains for parametric Gaussian ensembles is about as easy (or, if the reader prefers, as hard) as establishing the convergence of the scaled kernels and investigating the resulting limiting processes. 2.1. Trigonometric ensemble Here Hn is the subspace of L2(Tm) that consists of real-valued trigonometric polynomials Re ∑ ν∈Zm : \|ν\|∞6n cνe 2πi(ν·x) in m variables of degree 6 n in each variable. A straightforward computation shows that the corresponding normalized covariance kernel coincides with the 220 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components product of m Dirichlet’s kernels: Kn(x, y) = m∏ j=1 sin [π(2n + 1)(xj − yj)] (2n + 1) sin [π(xj − yj)] . In this case, it is natural to choose the degree n as the scaling parameter L. The scaled kernels Kx,n(u, v) = Kn(x + n−1u, x + n−1v) do not depend on the choice of the point x ∈ Tm. They extend analytically from Rm × Rm to Cm × Cm and the extensions converge uniformly on compact subsets of Cm × Cm to m∏ j=1 sin 2π(uj − vj) 2π(uj − vj) . This implies the convergence with all derivatives on all compact subsets of Rm× Rm. The limiting spectral measure ρ is the normalized Lebesgue measure on the cube [−1, 1]m ⊂ Rm. 2.2. Kostlan’s ensemble In this case, Hn is the space of the homogeneous real-valued polynomials of degree n in m + 1 variables restricted to the unit sphere Sm. The scalar product in Hn is given by 〈f, g〉 = ∑ \|J \|=n ( n J )−1 fJgJ (2.2.1) where f(X) = ∑ \|J \|=n fJXJ , g(X) = ∑ \|J \|=n gJXJ , XJ = xj0 0 xj1 1 xj2 2 . . . xjm m , and J = (j0, j1, j2, . . . , jm), \|J \| = j0 + j1 + j2 + . . . + jm, ( n J ) = n! j0!j1!j2! . . . jm! . The form of the scalar product (2.2.1) comes from the complexification: extending the homogeneous polynomials f and g to Cm+1, one can show that 〈f, g〉Hn = c(n, m) ∫ Cm+1 f(Z)g(Z)e−\|Z\| 2 d vol(Z) , i.e., 〈f, g〉Hn coincides (up to a positive factor) with the scalar product in the Fock–Bargmann space (or any other weighted L2-space of entire functions with fast decaying radial weight). Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 221 F. Nazarov and M. Sodin It is known that the complexified Kostlan ensemble is the only unitarily invari- ant Gaussian ensemble of homogeneous polynomials. On the other hand, there are many other orthogonally invariant Gaussian ensembles, all of them having been classified by Kostlan [19] (see [9, Section 2] for some details). The normalized covariance kernel of Kostlan’s ensemble equals (x · y)n. Take x = (0, . . . , 0, 1) (the “North Pole”) and consider the local chart π(u) =( u, √ 1− \|u\|2) where u runs over a small neighbourhood of the origin in Rm. Then π(u) π(v) = m∑ j=1 ujvj + ( 1− m∑ j=1 u2 j ) 1 2 ( 1− m∑ j=1 v2 j ) 1 2 = 1− 1 2 m∑ j=1 (uj − vj)2 + O (\|u\|4 + \|v\|4) as u, v → 0 . This suggests that the correct scaling in this case is L = √ n and the limiting covariance kernel is lim n→∞ ( π(n− 1 2 u) π(n− 1 2 v)) )n = lim n→∞ ( 1− (2n)−1 m∑ j=1 (uj − vj)2 + n−2O (\|u\|4 + \|v\|4) )n = exp { −1 2 m∑ j=1 (uj − vj)2 } . The justification of the local uniform convergence with all derivatives is similar to that in the previous example, and we skip it. The limiting spectral measure is the Gaussian measure on Rm with the density cme−2π2\|λ\|2 . An interesting feature of this example is a very rapid off diagonal decay of the covariance kernel. 3. Notation We denote by B(x, r) the open ball of radius r centered at x, B̄(x, r) denotes the corresponding closed ball. B(r) always denotes the open ball of radius r centered at the origin. For a closed set Γ ⊂ Rm, we denote by N(x, r; Γ) the number of the con- nected components of Γ that are contained in the open ball B(x, r), and by N∗(x, r; Γ) the number of the connected components of Γ that intersect the closed ball B̄(x, r). If Γ = Z(f) is the zero set of a continuous function f , we will abuse the notation slightly and write N(x, r; f) instead of N(x, r; Z(f)). For a bounded open convex set S and R > 0, we denote by NS(R; Γ) the number of connected components of Γ that are contained in S(R) = {u : R−1u ∈ S}. 222 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Throughout the paper, we denote by c and C various positive constants, which may depend on the dimension m and on the parameters of the Gaussian process or ensemble under consideration (the parameters in the conditions of Theorems 1 and 2) but on nothing else. The values of these constants may vary from line to line. Usually, the constants denoted by C should be thought of as large, and the constants denoted by c as small. The notation a . b means that a 6 C · b. Quite frequently, we will use the smoothness class C2−(U) (U ⊂ Rm is an open set), which we define as C2−(U) = ⋂ 0<β<1 C1+β(U) . Recall that to check that g ∈ C1+β(U) it suffices to show that g ∈ C1(U) and the first order partial derivatives ∂xig are β-Hölder functions on any closed ball B̄ ⊂ U . 4. Lemmata In this section, we present several lemmas needed for the proofs of Theorems 1 and 2. 4.1. Some integral geometry The first result is taken from [25, Claim 5.1] where it appears in a slightly different form. Lemma 1. Suppose Γ ⊂ Rm is a closed set and S ⊃ B(1) is a bounded open convex set. Then, for 0 < r < R, ∫ S(R−r) N(u, r; Γ) volB(r) d vol(u) 6 NS(R; Γ) 6 ∫ S(R+r) N∗(u, r; Γ) volB(r) d vol(u) . Note that N(u, r; Γ) is lower semicontinuous as a function of u. Proving the Lebesgue measurability of u 7→ N∗(u, r; Γ) without additional assumptions on Γ may be somewhat nontrivial. However, we will apply this lemma only in the case when the set of connected components of Γ is countable. Also, replacing the integral on the RHS by the upper Lebesgue integral will not affect the argument in any way. So, we will not dwell on this particular measurability. P r o o f. For a connected component γ of Γ, we put G∗(γ) = ⋂ y∈γ B(y, r), G∗(γ) = ⋃ y∈γ B̄(y, r) . Note that since γ is closed, G∗(γ) is open and G∗(γ) is closed. Also, for any y ∈ γ, G∗(γ) ⊂ B(y, r) ⊂ G∗(γ). Hence, Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 223 F. Nazarov and M. Sodin ∫ S(R−r) N(u, r; Γ) d vol(u) = ∫ S(R−r) ( ∑ γ : γ⊂B(u,r) 1 ) d vol(u) 6 ∫ S(R−r) ( ∑ γ : γ⊂S(R), u∈G∗(γ) 1 ) d vol(u) = ∑ γ⊂S(R) vol ( G∗(γ) ∩ S(R− r) ) 6 NS(R; Γ) volB(r) , proving the left inequality. On the other hand, ∫ S(R+r) N∗(u, r; Γ) d vol(u) = ∫ S(R+r) ( ∑ γ : u∈G∗(γ) 1 ) d vol(u) = ∑ γ vol ( G∗(γ) ∩ S(R + r) ) . Since for every connected component γ having a common point y with S(R), we have B(y, r) ⊂ G∗(γ) ∩ S(R + r), the last sum is at least NS(R; Γ) volB(r), so the right inequality holds as well. 4.2. Stability of components of the zero set under small perturbations If zero is not a critical value of a smooth function then the zero set of this function is stable under small perturbations. The following lemma, which quan- tifies this general principle, is taken from [25, Claim 4.2] where it was proven in the two-dimensional case. The proof of the general case needs no changes. Denote by V+t the open t-neighbourhood of a set V ⊂ Rm. Lemma 2. Fix α, β > 0. Let F be a C1-smooth function on an open ball B ⊂ Rm such that at every point u ∈ B, either \|F (u)\| > α, or \|∇F (u)\| > β. Then each component γ of the zero set Z(F ) with dist(γ, ∂B) > α/β is contained in an open “annulus” Aγ ⊂ γ+α/β bounded by two smooth connected hypersufaces such that F = +α on one boundary component of Aγ, and F = −α on the other one. Furthermore, the “annuli” Aγ are pairwise disjoint. As an immediate corollary, we obtain 224 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Lemma 3. Under the assumptions of the previous lemma, suppose that G ∈ C(B) with supB \|G\| < α. Then each component γ of Z(F ) with dist(γ, ∂B) > α/β generates a component γ̃ of the zero set Z(F + G) such that γ̃ ⊂ γ+α/β and different components γ1 6= γ2 of Z(F ) generate different components γ̃1 6= γ̃2 of Z(F + G). 4.3. Statistical independence of g and ∇g Quite often, we will use the following well-known fact: Lemma 4. Suppose U ⊂ Rn is an open set and g : U → R is a Gaussian C1-function on U that has constant variance. Then g(u) and its gradient ∇g(u) are independent for every u ∈ U . P r o o f. Denote by gui the partial derivative ∂uig. The covariance kernel K(u, v) = E{ g(u)g(v) } is a C1-function, and E{ gui(u)g(u) } = Kui(u, v) ∣∣ v=u . Since the function u′ 7→ K(u′, u) attains its maximal value at u′ = u and is C1- smooth, we have Kui(u, v) ∣∣ v=u = 0. Therefore, E{ gui(u)g(u) } = 0. Since g(u) and ∇g(u) are jointly Gaussian, this orthogonality implies their independence. We will be using the following corollary: Lemma 5. Suppose F : Rm → R is a Gaussian random function with translation- invariant distribution whose spectral measure ρ satisfies conditions (ρ1) and (ρ3). Then the distribution of the Gaussian vector ( F (u),∇F (u) ) does not degenerate. P r o o f of Lemma 5. By Lemma 4, F (u) and ∇F (u) are independent. Hence, it suffices to show that the distribution of ∇F (u) does not degenerate. If it degenerates, then there exists a non-zero vector v ∈ Rm such that 0 = E{ (v∇F )2 } = 4π2 ∫ Rm (v λ)2 dρ(λ) , which is impossible since, due to condition (ρ3), the spectral measure ρ cannot be supported on a linear hyperplane. 5. Quantitative Versions of Bulinskaya’s Lemma 5.1. Preliminaries The purpose of this part is to show that certain “bad events” have negligibly small probability. The particular bad events we want to get rid of are the event that the random Gaussian function and its gradient are simultaneously small at some point and the event that Z(f) has too many connected components. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 225 F. Nazarov and M. Sodin Everywhere in this part, BR ⊂ Rm is a fixed ball of large radius R > 1, S = ∂BR, U is an open neighbourhood of BR+1, and f is a continuous Gaussian function on U with the covariance kernel K. As usual, we will assume that the function f is normalized, that is, E\|f(x)\|2 = K(x, x) = 1, x ∈ U . We will impose certain bounds on the smoothness and non-degeneracy. These bounds are normalized versions of estimates used in the definition of controllability of parametric Gaussian ensembles. Namely, we will assume that (i) the kernel K is C2,2(U × U)-smooth and max \|α\|62 max x∈B̄R+1 ∣∣∂α x ∂α y K(x, y)\|y=x ∣∣ 6 M < ∞ , and that (ii) the process f is non-degenerate on U and inf x∈B̄R+1 det Cx > κ > 0 , where Cx is the covariance matrix of the Gaussian random vector ∇f(x), that is, the matrix with the entries Cx(i, j) = ∂xi∂yjK(x, y)\|y=x. • Till the end of Sec. 5, the constants M and κ remain fixed and all the con- stants that appear in the conclusions of all results proven here may depend on M and κ. As shown in Appendix A.11, the smoothness assumption (i) yields that, al- most surely, the process f is C2−(U)-smooth. We will be frequently using a quantitative version of this statement, which is also given in A.11. For a closed ball B̄ ⊂ U , denote by ‖f‖B̄,1+β the least N such that max B̄ \|f \| 6 N, max B̄ \|∇f \| 6 N, and \|∇f(x)−∇f(y)\| 6 N \|x−y\|β for x, y ∈ B̄. Then, for every β < 1 and every p < ∞, sup x∈B̄R E{‖f‖p B̄,1+β } 6 C(β, p, M) < ∞ . 5.2. The function Φ A prominent rôle in our approach will be played by the function Φ(x) = \|f(x)\|−t\|∇f(x)\|−tm , x ∈ U, t ∈ (0, 1) , 226 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components and by its spherical version ΦS(x) = \|f(x)\|−t\|∇Sf(x)\|−t(m−1) , x ∈ S = ∂BR, t ∈ (0, 1) , where ∇Sf(x) is the projection of the vector ∇f(x) to the tangent space to S at the point x ∈ S. The main feature of this function is that if f and ∇f (or ∇Sf) are very small at two points that are close to each other (in particular, if they are small at the same point), then Φ (ΦS correspondingly) is very large in a neighbourhood of these two points. At the same time, since f is normalized and ∇f is non-degenerate, • the moments E{ Φq(x) } and E{ Φq S(x) } are bounded locally uniformly on U and uniformly on S whenever we fix t < 1 < q so that tq < 1. Moreover, if t and q satisfying this restrictions are fixed, the suprema supB̄R+1 E{ Φq } and supS E { Φq S } are bounded by constants depending only on κ. 5.3. Almost surely, zero is not a critical value of f As a warm up, we prove a useful qualitative result that goes back to Bulin- skaya. Lemma 6. Almost surely, the following assertions hold: (i) zero is not a critical value of f ; (ii) there is no point z ∈ S ∩ Z(f) at which ∇Sf(z) = 0. P r o o f. In the first case, we use the function Φ. Fix a compact set Q ⊂ U and take a positive δ < dist(Q, ∂U). Consider the event ΩQ = {∃z ∈ Q : such that f(z) = 0, ∇f(z) = 0 } and take a ball B̄ ⊂ U centered at z of radius less than δ. Since the function ∇f(x) is β-Hölder with every β < 1, we have, for all x ∈ B̄, \|f(x)\| . \|x− z\|, \|∇f(x)\| . \|x− z\|β , whence Φ(x) & \|x − z\|−t(1+βm). Hence, choosing t and β so close to 1 that t(1 + βm) > m, we see that ∫ Q+δ Φ dvol > ∫ B Φdvol = +∞ . Recalling that E{Φ(x)} is uniformly bounded on Q̄+δ and using Fubini’s theorem, we conclude that the event ΩQ has zero probability. It remains to note that U can be covered by countably many compact subsets. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 227 F. Nazarov and M. Sodin Similarly, in the second case we take ΦS . As above, the expectation E{ΦS(x)} is uniformly bounded on S. Suppose that, for some z ∈ S ∩ Z(f), ∇Sf(z) = 0, that is, the gradient ∇f(z) is orthogonal to the sphere S. Then, for x ∈ S, we have \|f(x)\| . \|x− z\|, \|∇Sf(x)\| . \|x− z\|β + R−1\|x− z\| . \|x− z\|β , and, thereby, ΦS(x) & \|x − z\|−t(1+β(m−1)). Therefore, choosing t and β so close to 1 that t(1 + β(m− 1)) > m− 1, we get ∫ S Φd volS = +∞, and conclude that the event we consider has zero probability. 5.4. With probability close to one, f and ∇f cannot be simultaneously small Here, we prove a quantitative version of Lemma 6. Lemma 7. Given δ > 0, there exists τ > 0 (possibly, depending on R) such that P{ min x∈B̄R max{\|f(x)\|, \|∇f(x)\|} < τ } < δ . P r o o f. Denote by Ωτ the event {∃z ∈ B̄R : \|f(z)\|, \|∇f(z)\| < τ } and put W = 1 + ‖f‖B̄R+1,1+β . The parameter β ∈ (0, 1) will be specified later. If the event Ωτ occurs, then in the ball B = B(z, τ) with τ ∈ (0, 1), we have \|f(x)\| 6 τ + τ‖f‖B̄R+1,1+β = Wτ , and \|∇f(x)\| 6 τ + τβ‖f‖B̄R+1,1+β < Wτβ . Then, on Ωτ , Φ(x) > τ−t(1+βm)W−t(1+m) , for x ∈ B and ∫ BR+1 Φdvol > ∫ B Φd vol > cτm−t(1+βm)W−t(1+m) . 228 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Therefore, P{ Ωτ } 6 Cτ t(1+βm)−m vol(BR+1) E { W t(1+m) 1 vol(BR+1) ∫ BR+1 Φd vol } 6Cτ t(1+βm)−m vol(BR+1) ( E{ W pt(1+m) }) 1 p ( 1 vol(BR+1) ∫ BR+1 E{ Φq}d vol ) 1 q , with 1 p + 1 q = 1. The only restriction we have is β < 1 < q < 1 t . So we can take β and t so close to 1 that the exponent t(1 + βm) −m = t −m(1 − β) remains positive. This completes the proof. 5.5. General principle for estimating the number of connected com- ponents Our next aim is to estimate how many connected components of various kinds Z(f) may have. We start with “an abstract scheme”, which our estimates will be based on. Let (X,µ) be a measure space with 0 < µ(X) < ∞, and let X = ⋃ j Xj be a cover of X with bounded covering number C0 (that is, for every x ∈ X, # { j : x ∈ Xj } 6 C0). Let (Ω,P) be a probability space, and let { (Yi(ω), zi(ω)) } 16i6N(ω) be disjoint subsets of X with marked points zi ∈ Yi depending on the parameter ω ∈ Ω. Our aim is to estimate the cardinality N(ω) of the collection {Yi}. Lemma 8. Let Φ: X → R+ be a random function such that, for some q > 1, sup X E{Φq} < ∞ . Let {Wj} be non-negative random variables such that, for any p < ∞, sup j E{W p j } < ∞ . Suppose that, for every pair (i, j) with zi ∈ Xj, we have ∫ Xj∩Yi Φdµ > ρµ(Xj ∩ Yi)−σW−η j with some ρ, σ, η > 0. Then E∗{N q} 6 (C0C(ρ, σ))q µ(X)q [ sup X E{ Φq }] 1 1+σ [ sup j E{ W qη σ j }] σ 1+σ . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 229 F. Nazarov and M. Sodin P r o o f of Lemma 8. Let Nj be the number of i’s such that zi ∈ Xj . Then, for at least 1 2Nj indices i with this property, we have µ(Xj ∩Yi) 6 2 Nj µ(Xj), and therefore, ∫ Xj∩Yi Φdµ > c(ρ, σ)Nσ j µ(Xj)−σW−η j , whence, ∫ Xj Φ dµ > c(ρ, σ)N1+σ j µ(Xj)−σW−η j . Applying Hölder’s inequality with the exponents 1 + σ and 1+σ σ , we get N 6 ∑ j Nj = ∑ j Njµ(Xj) − σ 1+σ W − η 1+σ j µ(Xj) σ 1+σ W η 1+σ j 6 (∑ j N1+σ j µ(Xj)−σW−η j ) 1 1+σ (∑ j µ(Xj)W η σ j ) σ 1+σ 6 C(ρ, σ) (∫ X Φdµ ) 1 1+σ (∑ j µ(Xj)W η σ j ) σ 1+σ . Then E∗{N q } 6 C(ρ, σ)q E {(∫ X Φdµ ) q 1+σ (∑ j µ(Xj)W η σ j ) qσ 1+σ } and, applying Hölder’s inequality with the same exponents again, we obtain E∗{N q } 6 C(ρ, σ)q [ E {(∫ X Φ dµ )q}] 1 1+σ [ E {(∑ j µ(Xj)W η σ j )q}] σ 1+σ . At last, using Hölder’s inequality with the exponents q q−1 and q, we get E {(∫ X Φdµ )q} 6 µ(X)q−1 ∫ X E{ Φq } dµ 6 µ(X)q sup X E{ Φq } and E {(∑ j µ(Xj)W η σ j )q} = E {(∑ j µ(Xj) 1− 1 q µ(Xj) 1 q W η σ j )q} 6 [∑ j µ(Xj) ]q−1 [∑ j µ(Xj) E { W qη σ j }] 6 ( C0µ(X) )q sup j E{ W qη σ j } . 230 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Finally, E∗{N q} 6 C(ρ, σ)q [ µ(X)q sup X E{Φq} ] 1 1+σ [( C0µ(X) )q sup j E{ W qη σ j }] σ 1+σ 6 (C0C(ρ, σ))q µ(X)q [ sup x∈X E{ Φ(x)q }] 1 1+σ [ sup j E{ W qη σ j }] σ 1+σ , completing the proof. 5.6. Components on a sphere For the sphere S = ∂BR, we denote by N(S; f) the number of connected components of S \ Z(f). Lemma 9. There are positive constants C < ∞ and q > 1 such that E∗{Nq(S; f)} 6 CRq(m−1). P r o o f of Lemma 9. We cover the sphere S with bounded covering number by closed spherical caps Xj of Euclidean radius 1, and denote by B̄j the closed m-dimensional Euclidean balls of radius 1 having the same centers as Xj . The total number of the caps in the cover is . Rm−1. By Yi we denote the connected components of S \ Z(f). In each domain Yi we fix a point zi where the gradient ∇f(zi) is directed normally to S, that is, ∇Sf(zi) = 0. The number of i’s such that, for some j, Xj ⊂ Yi is . Rm−1. Thus, in what follows, we consider only those i’s for which Yi does not contain any Xj . In order to apply Lemma 8 with the function ΦS and with Wj = ‖f‖B̄j ,1+β we need to establish the lower bounds for the integrals ∫ Xj∩Yi ΦS d volS(x) with ΦS = \|f \|−t \|∇sf \|−t(m−1) , assuming that zi ∈ Xj . Since the sets Yi ∩ Xj and Xj \ Yi aren’t empty, the closed set ∂Yi ∩ Xj is not empty too. Denote ρi = dist(zi, ∂Yi ∩Xj) 6 2 and take a closest to zi point p ∈ ∂Yi∩Xj . By Vi we denote the spherical cap centered at zi such that p ∈ ∂Vi. Note that, by the construction, volS(Yi ∩ Xj) > volS(Vi ∩ Xj) & ρm−1 i . Since f(p) = 0, we have \|f(x)\| . ρi ‖f‖B̄j ,1+β = ρiWj , x ∈ Vi ∩Xj . Furthermore, since ∇Sf(zi) = 0, \|∇Sf(x)\| . ρβ i ‖f‖B̄j ,1+β + ρi R ‖f‖B̄j ,1+β . ρβ i Wj , x ∈ Vi ∩Xj . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 231 F. Nazarov and M. Sodin Hence, on Vi ∩Xj we have ΦS & ρ −t(1+β(m−1)) i W−tm j & (volS(Vi ∩Xj)) −t( 1 m−1 +β) W−tm j , and ∫ Yi∩Xj ΦS d volS(x) > ∫ Vi∩Xj ΦS d volS(x) & (volS(Vi ∩Xj)) 1−t( 1 m−1 +β)W−tm j . Now, fixing the parameters t and β so close to 1 that t( 1 m−1 +β) > 1, we see that the RHS is > (volS(Yi ∩ Xj)) 1−t( 1 m−1 +β)W−tm j . At last, applying Lemma 8, we complete the proof. 5.7. Regular components Definition 9. We call a connected component G of the set U \Z(f) regular, if G is compactly supported in U and vol(G) < vol(B(1)). By Nreg(BR; f) we denote the number of regular connected components G compactly contained in BR. Lemma 10. There exist constants q > 1 and C < ∞ such that E∗{N q reg(BR; f) } 6 CRqm . P r o o f. The proof of this lemma follows closely that of Lemma 9. Cover the ball B̄R by closed balls Xj of radius 1 with centers in BR keeping the covering number bounded, and put X = ⋃ j Xj . Then B̄R ⊂ X ⊂ BR+1. Denote by {Yi} the set of regular nodal domains of f that are contained in BR. In each domain Yi choose a point zi with ∇f(zi) = 0. In order to apply Lemma 8 with the function Φ = \|f \|−t\|∇f \|−tm and with Wj = ‖f‖B̄j ,1+β, we need to estimate from below the integrals ∫ Xj∩Yi Φd vol assuming that zi ∈ Xj . Since vol(Yi) < vol(Xj), we note again that ∂Yi ∩ Xj 6= ∅. Put ρi = dist(zi, ∂Yi ∩ Xj) 6 2, take a closest to zi point p ⊂ ∂Yi ∩ Xj , and denote Vi = B(zi, ρi). By the construction, vol(Yi ∩Xj) > vol(Vi ∩Xj) & ρm i . Since f(p) = 0 and ∇f(zi) = 0, we have \|f(x)\| 6 ρi ‖f‖B̄j ,1+β = ρiWj , x ∈ Vi ∩Xj , and \|∇f(x)\| . ρβ i ‖f‖B̄j ,1+β = ρβ i Wj , x ∈ Vi ∩Xj . 232 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Hence, on Vi ∩Xj , Φ & ρ −t(1+βm) i W −t(1+m) j & (vol(Vi ∩Xj))−t(β+ 1 m ) W −t(m+1) j , and ∫ Yi∩Xj Φd vol > ∫ Vi∩Xj Φd vol & (vol(Vi ∩Xj))1−t(β+ 1 m ) W −t(m+1) j . Fixing the parameters t and β so close to 1 that t(β + 1 m) > 1, we get ∫ Yj∩Xj Φdvol & (vol(Yi ∩Xj))1−t(β+ 1 m ) W −t(m+1) j . Finally, Lemma 8 ends the job. 5.8. The moment estimate for the total number of connected compo- nents If the function f is C1-smooth and 0 is not a critical value, then we can bound the number of connected components γ of Z(f) contained in BR by the number of connected components G of U \Z(f) compactly contained in BR. All we need for that is to note that each γ ⊂ BR is the outer boundary? of some G compactly supported in BR and no two different connected components γ ⊂ BR of Z(f) can serve as the outer boundary of the same connected component G of U \Z(f) simultaneously. Thus, combining the estimate of Lemma 10 with the trivial bound # { G : Ḡ ⊂ BR, vol(G) > vol(B(1)) } 6 Rm , we conclude that, for some q > 1, E∗{N(BR; f)q } . Rmq . If, in addition, f is non-degenerate on S = ∂BR in the sense that f and ∇Sf do not vanish simultaneously anywhere on S (due to Lemma 6 this event has proba- bility 1), then, arguing in a similar way, we can estimate the number of connected components of Z(f) intersecting S by the number of connected components of S \Z(f). Thus, the result of Lemma 9 can be viewed as an upper bound for the q-th moment of the number of connected components of Z(f) intersecting S. We will use these observations several times when referring to Lemmas 10 and 9 as if they were about the connected components of Z(f) rather than about those of U \ Z(f) and S \ Z(f). ?i.e., the part of the boundary of G that bounds the unbounded connected component of Rm \G as well Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 233 F. Nazarov and M. Sodin 6. Proof of Theorem 1 6.1. Preliminaries We need several basic notions from the ergodic theory. Suppose ( Ω, S,P) is a probability space on which Rm acts by measure-preserving transformations τv, v ∈ Rm. This means that for each v ∈ Rm, τv : Ω → Ω is a S-measurable transformation, τu ◦ τv = τu+v, τ−v = τ−1 v , and for each v ∈ Rm and each A ∈ S, we have P(τvA) = P(A). The following version of Wiener’s ergodic theorem suffices for our purposes: Wiener’s ergodic theorem: Suppose ( Ω, S,P) is a probability space on which Rm acts by measure-preserving transformations τv, v ∈ Rm. Suppose that Φ ∈ L1(P), and that the function (v, ω) 7→ Φ ◦ τv is measurable with respect to the product σ-algebra B(Rm)×S, where B(Rm) is the Borel σ-algebra generated by open sets in Rm. Suppose that S ⊂ Rm is a bounded open convex set containing the origin. Then the limit lim R→∞ 1 volS(R) ∫ S(R) Φ(τvω) d vol(v) = Φ̄(ω) exists with probability 1 and in L1(P). The limiting random variable Φ̄ is τ - invariant (i.e., for each v ∈ Rm, Φ̄ ◦ τv = Φ̄), and does not depend on the choice of the convex set S. This is a special case of a theorem proven in Becker [1, Theorems 2 and 3]. Note that Becker’s formulation of this theorem deals with rather general increasing families (UR) of open sets in Rm satisfying two conditions: (A) the Hardy-Littlewood maximal operator associated with the family (UR) is of weak type (1, 1), and (B) for each t ∈ Rm, lim R→∞ vol((t + UR)4UR)/ vol(UR) = 0 , where 4 denotes the symmetric difference. In the case when S is the unit ball, condition (A) reduces to the classical Hardy–Littlewood maximal theorem, after which it remains to note that the maximal function associated with the family S(R) is dominated (up to a constant factor) by the one corresponding to the unit ball. The verification of condition (B) is straightforward. Note that Becker’s presentation does not formally contain the claim that the limiting random variable Φ does not depend on the family (UR) but in our 234 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components situation it can be easily established by applying Becker’s theorem to a family UR containing arbitrarily large homothetic images of two bounded convex sets S′ and S′′. Next, recall that the action of Rm is called ergodic? if for every set A ∈ S satisfying P((τvA)4A) = 0, either P(A) = 0, or P(A) = 1. In the ergodic case, the limiting random variable Φ̄ is a constant function. Due to the L1(P)- convergence, the value of this constant equals the expectation of Φ: Φ̄ = E{ Φ } . Let X ⊂ C(Rm) be an invariant set of continuous functions (i.e., G ∈ X implies G ◦ τv ∈ X for all v ∈ Rm). Let S be the minimal σ-algebra on X containing all “intervals” I(u; a, b) = { G ∈ X : G(u) ∈ [a, b) } . Let γ be a Gaus- sian probability measure on (X, S) meaning that for every finitely many points u1, . . . , uk ∈ Rm, the push-forward of γ by the mapping G 7→ [G(u1), . . . , G(uk)] is a (centered) Gaussian, possibly degenerate, measure on Rk. If γ is invariant under the introduced action of Rm on X, then Rm ×X 3 (u, G) 7→ G(u) ∈ R is a translation-invariant Gaussian function on the probability space (X,S, γ) with continuous trajectories and continuous covariance kernel and we can talk about its spectral measure ρ. Fomin–Grenander–Maruyama theorem: Suppose that ρ has no atoms. Then the action of Rm on (X, S, γ) by translations is ergodic. For the reader’s convenience, we remind the proof of this theorem?? in AppendixB. Now, let F be a Gaussian function on Rm satisfying the assumptions of Theorem 1. By the moment assumption (ρ1), with probability 1 it is C2−- smooth. Hence, it generates a Gaussian measure γF on (C1(Rm), B(C1(Rm))) where B(C1(Rm)) is the Borel σ-algebra generated by open sets in C1(Rm). In what follows, it will be convenient to pass from C1(Rm) to its subset C1 ∗ (Rm) = { G ∈ C1(Rm) : \|G\|+ \|∇G\| 6= 0 } , which consists of functions for which 0 is not a critical value. Note that C1∗ (Rm) is a Borel subset of C1(Rm) and, by the first statement in Lemma 6, γF ( C1(Rm) \ C1 ∗ (Rm) ) = 0. ?a.k.a. metric-transitive ??The full version of the Fomin–Grenander–Maruyama theorem states that the continuity of the spectral measure ρ is necessary and sufficient for the ergodicity of the action of Rm on (X, S, γ) by translations. We will use (and prove) only the sufficiency part. The proof we present follows the argument for the univariate case given in [13, Section 5.10]. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 235 F. Nazarov and M. Sodin Furthermore, Rm acts on ( C1∗ (Rm),B(C1∗ (Rm)), γF ) by translations and, since the distribution of F is translation invariant, the action is measure-preserving. Thus, Wiener’s theorem applies in this setting. To apply the Fomin–Grenander– Maruyama theorem, we only need to note that the Borel σ-algebra B(C1∗ (Rm)) coincides with the σ-algebra S generated by the intervals I(u; a, b) (see Ap- pendix A.1.). We conclude that • under the assumption (ρ1) of Theorem 1, for any random variable Φ ∈ L1(γF ) such that the function (v, G) 7→ Φ(τvG) is measurable, the ergodic averages (AS RΦ)(G) def= 1 volS(R) ∫ S(R) Φ(τvG) d vol(v) converge to a τ -invariant limit Φ̄ with probability 1, as well as in L1(γF ), as R →∞. Moreover, under assumption (ρ2), we have Φ̄ = E{ Φ } . We split the proof of Theorem 1 into two parts: first, we prove the convergence of (volS(R))−1NS(R; F ) to a limit ν. Then, assuming condition (ρ4), we show that this limit is positive. 6.2. Existence of the limit 6.2.1. The sandwich estimate for NS(R; G)/ volS(R). Without loss of generality, we assume that S ⊃ B(1). Then, the integral-geometric Lemma 1 provides us with the “sandwich estimate”: 1 volS(R) ∫ S(R−r) N(v, r; G) volB(r) d vol(v) 6 NS(R; G) volS(R) 6 1 volS(R) ∫ S(R+r) N∗(v, r; G) volB(r) d vol(v) . The difference N∗(v, r; G)−N(v, r; G) = N∗(r; τvG)−N(r; τvG) is bounded by N#(r; τvG), where N#(r;G) def= { N(∂B(r);G) if G is non-degenerate on ∂B(r), +∞ otherwise, and N(∂B(r);G) is the number of connected components of ∂B(r)\Z(G). Recall that we say that G is non-degenerate on the sphere ∂B(r) if G and ∇∂B(r)G do not vanish simultaneously anywhere on ∂B(r). 236 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components We introduce the functionals Φr(G) def= N(r; G) volB(r) , Ψr(G) def= N#(r; G) volB(r) . Then the sandwich estimate takes the form ( 1− r R )m (AS R−rΦr)(G) 6 NS(R; G) volS(R) 6 ( 1 + r R )m[ (AS R+rΦr)(G) + (AS R+rΨr)(G) ] . (6.2.1) 6.2.2. Checking measurability. We need to check that, given r > 0, the functions (v, G) 7→ Φr(τvG), (v, G) 7→ Ψr(τvG) are measurable with respect to the product σ-algebra B(Rm)×B(C1∗ (Rm)). The function (v, G) 7→ τvG is a measurable (even continuous) map ( Rm × C1 ∗ (Rm), B(Rm)×B(C1 ∗ (Rm) ) → ( C1 ∗ (Rm), B(C1 ∗ (Rm)) ) . Since the composition of measurable functions is measurable, it remains to show that, given r > 0, the functions G 7→ N(r,G) and G 7→ N#(r,G) are measurable as maps from ( C1∗ (Rm), B(C1∗ (Rm)) ) to ( [0,+∞], B([0, +∞]) ) . The measurability of the map G 7→ N(r,G) follows from its lower semiconti- nuity on C1∗ (Rm). To see that G 7→ N#(r,G) is measurable, first, consider the set Degen(r) of functions G ∈ C1∗ (Rm) for which there exists a point x ∈ ∂B(r) such that ∇G(x) is orthogonal to the tangent space to ∂B(r) at x. This set is closed in C1∗ (Rm) with respect to the C1-topology and, therefore, is B(C1∗ (Rm))- measurable. On the other hand, our map G 7→ N#(r,G) is lower semi-continuous on C1∗ (Rm) \Degen(r). 6.2.3. Integrability. Next, we note that, for every fixed r > 0, the functions Φr and Ψr on C1∗ (Rm) are γF -integrable. This readily follows from Lemma 10 and Lemma 9, correspondingly. 6.2.4. Proof of convergence. By the sandwich estimate ((6.2.1)), for every function G ∈ C1∗ (Rm), we have ∣∣∣NS(R; G) volS(R) − (AS RΦr)(G) ∣∣∣ 6 ∣∣∣ ( 1− r R )m (AS R−rΦr)(G)− (AS RΦr)(G) ∣∣∣ + ∣∣∣ ( 1 + r R )m (AS R+rΦr)(G)− (AS RΦr)(G) ∣∣∣ + ( 1 + r R )m (AS R+rΨr)(G) . (6.2.2) Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 237 F. Nazarov and M. Sodin By the Wiener ergodic theorem, there exist τ -invariant functions Φ̄r and Ψ̄r such that lim R→∞ AS RΦr = Φ̄r and lim R→∞ AS RΨr = Ψ̄r both γF -almost everywhere and in L1(γF ). Letting R → ∞ on both sides of ((6.2.2)), we get lim R→∞ ∣∣∣NS(R;G) volS(R) − (AS RΦr)(G) ∣∣∣ 6 Ψ̄r(G) for γF -almost every G, (6.2.3) and lim R→∞ ∫ ∣∣∣NS(R;G) volS(R) − (AS RΦr)(G) ∣∣∣dγF (G) 6 ∫ Ψ̄r dγF = E{N#(r; F )} volB(r) . (6.2.4) By Lemma 9, the RHS of (6.2.4) is . r−1 for r > 1. So taking a sequence rk ↑ ∞, we observe that lim k→∞ ∫ Ψ̄rk dγF = 0 . and, consequently, inf k Ψ̄rk = 0 γF -almost everywhere . Since AS RΦr(G) converge to Φ̄r for γF -almost every G, the second observation together with (6.2.3) imply that (volS(R))−1 NS(R;G) is Cauchy for γF -almost every G. Similarly, the convergence of AS RΦr(G) to Φ̄r in L1(γF ) together with the first observation and (6.2.4) imply that (volS(R))−1 NS(R;G) is Cauchy in L1(γF ). Thus, the limit ν def= lim R→∞ NS(R; G) volS(R) exists γF -almost everywhere and in L1(γF ). It follows from ((6.2.1)) that, for every r > 0, Φ̄r 6 ν 6 Φ̄r + Ψ̄r γf − almost everywhere . If, in addition, the action of Rm on ( C1∗ (Rm),B(C1∗ (Rm)), γF ) is ergodic, then Φ̄r = E{Φr}, Ψ̄r = E{Ψr}. Therefore, E{ Φ̄r } 6 ν 6 E{ Φ̄r } + E{ Ψ̄r } γf − almost everywhere , (6.2.5) whence, for every r > 0, γF -essential oscillation of ν does not exceed E{Ψr}. Recalling that E{Ψr} . r−1 and letting r →∞, we see that ν is a (non-random) constant. This completes the proof of convergence in Theorem 1. 238 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components 6.3. Positivity of ν It remains to show that condition (ρ4) yields the positivity of the limiting constant ν. We prove that if assumption (ρ4) holds, then P{N(r; F ) > 0} > 0 when r is sufficiently big. Since the LHS of estimate (6.2.5) can be rewritten as ν > E{N(r; F )}/ volB(r) for each r > 0, this will yield the positivity of ν. 6.3.1. A Gaussian lemma Lemma 11. Let µ be a compactly supported Hermitian measure with spt(µ) ⊂ spt(ρ). Then for each closed ball B̄ ⊂ Rm and for each ε > 0, P{‖F − µ̂‖C(B̄) < ε } > 0 . P r o o f of Lemma 11. The part of the theory of continuous Gaussian functions developed in Appendix (A.7. and A.12.) yields the statement of the lemma for all measures µ absolutely continuous with respect to ρ with density h ∈ L2 H(ρ) def= { g ∈ L2(ρ) : g(−x) = g(x) for all x ∈ Rm } . In the general case, we can approximate the measure µ in the weak topology by measures dµ = h dρ with spt(h) contained in a fixed compact neighbourhood of spt(µ). Then it remains to recall that for measures supported on a fixed compact set, the weak convergence yields locally uniform convergence of their Fourier integrals. 6.3.2. Proof of the positivity of ν(ρ). We take a Hermitian compactly supported measure µ with spt(µ) ⊂ spt(ρ) and a bounded domain D ⊂ Rm so that µ̂ ∣∣ ∂D < 0 and µ̂(u0) > 0 for some u0 ∈ D. Choose r so big that D̄ ⊂ B(r). If ε > 0 is sufficiently small, then G(u0) > 0 and G ∣∣ ∂D < 0 for every function G satisfying ‖G− µ̂‖C(B̄(r)) < ε. Thus, for every such function G, the zero set Z(G) has at least one connected component in D. Applying Lemma 11, we see that P{ N(r; F ) > 0 } > P{‖F − µ̂‖C(B̄(r)) < ε } > 0 completing the proof of Theorem 1. 7. Recovering the Function ν̄ by a Double Scaling Limit The proof of Theorem 2 will rely upon the following lemma, which is of independent interest. Let (fL) be a tame parametric Gaussian ensemble, that is, an ensemble satisfying the assumptions of Theorem 2. As above, we put fx,L(u) = f(x + L−1u) and Kx,L(u, v) = E{ fx,L(u)fx,L(v) } = KL(x + L−1u, x + L−1v) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 239 F. Nazarov and M. Sodin Till the end of this section, we fix a point x ∈ U so that lim L→∞ Kx,L(u, v) = kx(u− v) pointwise in Rm × Rm , where the Hermitian positive-definite function kx is the Fourier integral of a mea- sure ρx satisfying assumptions (ρ1)–(ρ3). By Fx we denote the limiting Gaussian function on Rm, and put ν = ν̄(x) = ν(Fx). Lemma 12. For every ε > 0, lim R→∞ lim L→∞ P {∣∣∣N(R; fx,L) volB(R) − ν ∣∣∣ > ε } = 0 . P r o o f . Fix R > 2 and ε > 0. Our goal will be to show that, for every t, lim L→∞ P { N(R; fx,L) > t } 6 P { N(R + 1; Fx) > t } , lim L→∞ P { N(R; fx,L) < t } 6 P { N(R− 1;Fx) < t } . Applying these inequalities with t = (ν + ε) volB(R) and t = (ν − ε) volB(R) respectively, and then combining the results with Theorem 1, we get the conclu- sion of Lemma 12. The proofs of these two relations are very similar, so we will present only the proof of the first one. We choose a big constant M and a small constant κ so that the kernels kx and Kx,L (with L > L0) satisfy the “(M, κ)-conditions” introduced in the beginning of Section 5.1. For the kernel kx this is possible due to conditions (ρ1) and (ρ3). For the scaled kernels Kx,L this is possible due to the controllability of (f L ). Given positive constants A and a, we put E(A, a) = { g ∈ C1(B(R+1.1)) : ‖g‖C1(B̄(R+1)) 6 A, min B̄(R+1) max{\|g\|, \|∇g\|} > a } . Introduce the events Ω′L = { fx,L /∈ E(A, a) } and Ω′′ = { Fx /∈ E(A, a) } . By Lemma 7, the aforementioned “(M, κ)-conditions” imply that, for a given δ > 0, we can make the probabilities of both events less than δ if we choose sufficiently big A and sufficiently small a. We fix a finite a/(2A)-net X in B̄(R + 1) and denote by E′ ⊂ R\|X\| the set of traces on X of functions g ∈ E(A, a) satisfying N(R; g) > t. This is a bounded subset of R\|X\|. Note that if g, h ∈ E(A, a) and \|g − h\| < a/2 on X, then \|g − h\| < a everywhere on B̄(R + 1), and by Lemma 3 (applied with α = β = a), N(R + 1;h) > N(R; g). We fix a function ϕ ∈ C∞ 0 (R\|X\|) satisfying 0 6 ϕ 6 1 everywhere, ϕ ≡ 1 on E′ and ϕ ≡ 0 on R\|X\| \ E′ +a/2 (as usual, by E′ +s we denote the s-neighbourhood 240 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components of E′), and consider the finite dimensional Gaussian vectors fx,L\|X and Fx\|X . First, we note that { ω : N(R; fx,L) > t } ⊂ { ω : fx,L\|X ∈ E′, fx,L ∈ E(A, a) } ∪ Ω′L ⊂ { ω : ϕ(fx,L\|X) = 1} ∪ Ω′L , whence, P{ N(R; fx,L) > t } < E{ϕ(fx,L\|X)}+ δ . The pointwise convergence of the scaled kernels Kx,L(u, v) to the limiting kernel kx(u− v) yields? E{ϕ(fx,L\|X)} L→∞→ E{ϕ(Fx \|X)} 6 P{ ϕ(Fx\|X) > 0 } (in the inequality we used that ϕ 6 1 everywhere). Now, {ω : ϕ(Fx \|X) > 0} ⊂ { ω : Fx \|X ∈ E′ +a/2 } ⊂ { ω : Fx \|X ∈ E′ +a/2, Fx ∈ E(A, a) }∪Ω′′ ⊂ { ω : N(R + 1;Fx) > t } ∪ Ω′′ . In the last step we used that, by our construction, if Fx ∈ E(A, a) and Fx \|X ∈ E′ +a/2, then there is a function g ∈ E(A, a) such that N(R, g) > t, and \|Fx−g\| < 1 2a on X, whence, N(R + 1;Fx) > N(R; g) > t. Hence, P{ϕ(Fx \|X) > 0} < P{ N(R + 1;Fx) > t } + δ . Thus, for sufficiently large L, we have P{ N(R; fx,L) > t } < E{ϕ(fx,L\|X)}+ δ < P{ ϕ(Fx\|X) > 0 } + 2δ < P{ N(R + 1; Fx) > t } + 3δ , completing the argument. ?If ξ L are Gaussian n-dimensional vectors and the entries of the covariance matrices K L of ξL converge to the entries of the covariance matrix K of ξ, then E{ ϕ(ξL) } = E {∫ Rn ϕ̂(λ)e2πiλ·ξL dλ } = ∫ Rn ϕ̂(λ)E { e2πiλ·ξL } dλ = ∫ Rn ϕ̂(λ)e−πKLλ·λ dλ → ∫ Rn ϕ̂(λ)e−πKλ·λ dλ = E{ ϕ(ξ) } , where the convergence holds by the dominated convergence theorem. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 241 F. Nazarov and M. Sodin 8. Proof of Theorem 2 It remains to tie the ends together. Let (fL) be a tame parametric Gaussian ensemble on an open set U ⊂ Rm. This implies that, • for every compact set Q ⊂ U , there exist constants M < ∞ and κ > 0 such that the covariance kernels of the functions fx,L on B̄(R + 1) satisfy the (M,κ)-conditions from Sec. 5.1 whenever x ∈ Q, R > 0, and L > L0(Q,R). Fix a Borel set U ′ ⊂ U of full volume on which the scaled functions fx,L have translation invariant limits Fx. Then, by Appendix A.12, • the covariance kernels kx(u − v) of the limiting functions Fx satisfy the (M,κ)-conditions whenever x ∈ Q ∩ U ′. 8.1. ν̄ ∈ L∞loc(U) First, we show that ν̄ is locally uniformly bounded on U ′ and then that it is measurable. 8.1.1. Boundedness of ν̄. Recall that ν̄(x) = lim R→∞ E{N(R;Fx)} volB(R) , x ∈ U ′ . Given any compact set Q ⊂ U , Lemma 10 implies that, for every x ∈ U ′ ∩ Q, we have E{N(R; Fx)} 6 C(Q) volB(R). Thus, the function ν̄ is locally bounded on U ′. 8.1.2. Measurability of ν̄. Put νR,L(x, ω) = N(R; fx,L) volB(R) . The function νR,L is defined on the set U−(R+1)/L × Ω′, where U−r = { x ∈ U : dist(x, ∂U) > r } and Ω′ = { ω ∈ Ω: fL ∈ C1∗ (U) } , P(Ω \ Ω′) = 0. It is measurable as a composition of a lower semicontinuous mapping C1 ∗ (B(R + 1)) 3 g 7→ N(R; g) volB(R) ∈ R , a continuous mapping U−(R+1)/L × C1 ∗ (U) 3 (x, g) 7→ gx,L ∣∣ B(R+1) ∈ C1 ∗ (B(R + 1)) , and a measurable mapping U−(R+1)/L × Ω′ 3 (x, ω) 7→ (x, fL) ∈ U−(R+1)/L × C1 ∗ (U) . 242 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Fix x ∈ U ′. By Lemma 10, there exist q > 1 and C < ∞ such that ∫ Ω′ νq R,L dP < C for all sufficiently large L. Given ε > 0, put Ωε(R, L, x) = { ω ∈ Ω′ : \|νR,L(x, ω)− ν̄(x)\| > ε } . Then ∫ Ωε \|νR,L(x, ω)− ν̄(x)\| dP(ω) 6 ∫ Ωε νR,L dP + ν̄(x)P{Ωε} 6 (P{Ωε})1− 1 q (∫ Ωε νq R,L dP ) 1 q + ν̄(x)P{Ωε} 6 C(P{Ωε})1− 1 q . Therefore, ∣∣∣ ∫ Ω′ νR,L(x, ω) dP(ω)− ν̄(x) ∣∣∣ 6 ∫ Ω′ \|νR,L(x, ω)− ν̄(x)\| dP(ω) 6 ε + C(P{Ωε})1− 1 q and lim R→∞ lim L→∞ ∣∣∣ ∫ Ω′ νR,L(x, ω) dP(ω)− ν̄(x) ∣∣∣ 6 ε + C lim R→∞ lim L→∞ (P{Ωε})1− 1 q . By Lemma 12, the double limit on the RHS vanishes, so lim R→∞ lim L→∞ ∣∣∣ ∫ Ω′ νR,L(x, ω) dP(ω)− ν̄(x) ∣∣∣ = 0 . It follows from here that the function ν̄(x) can be represented as, say, ν̄(x) = lim R→∞ lim L→∞ ∫ Ω′ νR,L(x, ω) dP(ω) . Since the functions νR,L(x, ω) are non-negative and measurable in (x, ω), their integrals with respect to ω over a fixed set Ω′ are also measurable as functions of x ∈ U ′. Thus, the function ν̄ is also measurable. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 243 F. Nazarov and M. Sodin 8.2. Towards the proof of Theorem 2: another sandwich estimate Without loss of generality we assume that the continuous compactly supported function ϕ in the assumptions of Theorem 2 is non-negative. We denote Q = spt(ϕ). Fix δ > 0 such that Q+4δ ⊂ U and put Q1 = Q+δ, Q2 = Q+2δ. For x ∈ Q1, let ϕ−(x) = min B̄(x,δ) ϕ, ϕ+(x) = max B̄(x,δ) ϕ . Note that ϕ−(x) 6 ϕ(y) 6 ϕ+(x) whenever x ∈ Q1, y ∈ B(x, δ). Fix the parameters D,R, L so that 1 < D < R < δL. We have L−m ∫ U ϕdnL = ∫ Q1 ( ∫ B(x,R/L) ϕ(y) dnL(y) volB(R) ) d vol(x) , whence, ∫ Q1 ϕ−(x) nL(B(x,R/L)) volB(R) d vol(x) 6 L−m ∫ U ϕdnL 6 ∫ Q1 ϕ+(x) nL(B(x,R/L)) volB(R) d vol(x) . (8.2.1) Since the total nL-mass of each connected component of Z(fL) equals 1, the LHS of (8.2.1) cannot be less than ∫ Q1 ϕ−(x)νR,L(x, ω) d vol(x), where, as above, νR,L(x, ω) = (volB(R))−1N(R; fx,L). In order to estimate the RHS of (8.2.1), we cover Q2 by ' vol(Q2) ( L D )m open balls of diameter D/L. Denote by { Sj } the collection of boundary spheres of these balls. Due to the second statement in Lemma 6, with probability 1 there is no point x such that, for some j, x ∈ Sj ∩ Z(fL) and ∇SjfL(x) = 0. Under this non-degeneracy condition, the number of connected components of Z(fL) that intersect the sphere Sj is bounded by the number N(Sj ; fL ) of connected components of Sj \ Z(fL). Denote by n∗L the part of the component counting measure nL supported on the connected components of Z(fL) intersecting at least one of the spheres Sj . Since every other component of Z(fL) intersecting a 244 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components ball B(x,R/L) centered at x ∈ Q1 is contained in B(x, (R + D)/L), we see that the RHS of (8.2.1) does not exceed (R + D R )m ∫ Q1 ϕ+(x)νR+D,L(x, ω) d vol(x) + ∫ Q1 ϕ+(x) n∗L(B(x,R/L)) volB(R) d vol(x) . By Fubini, the second integral on the RHS is bounded by (maxU ϕ)L−mn∗L(Q2). In turn, n∗L(Q2) 6 ∑ j N(Sj ; fL ) with probability 1. Thus, for almost every ω, we have ∫ Q1 ϕ−(x)νR,L(x, ω) d vol(x) 6 L−m ∫ U ϕdnL 6 ( 1 + D R )m ∫ Q1 ϕ+(x)νR+D,L(x, ω) d vol(x) + (max U ϕ) L−m ∑ j N(Sj ; fL ) . 8.3. Completing the proof of Theorem 2 To juxtapose the integrals L−m ∫ U ϕdnL and ∫ U ϕν̄ d vol , we note that, since pointwise ϕ+ 6 ϕ + ωϕ(δ), where ωϕ is the modulus of continuity of ϕ, we have ∫ U ϕν̄ d vol > ∫ U ϕ+ν̄ d vol−ωϕ(δ)‖ν̄‖L∞(Q1) vol(Q1) > ( 1 + D R )m ∫ U ϕ+ν̄ d vol − [( 1 + D R )m − 1 ] (max U ϕ)‖ν̄‖L∞(Q1) vol(Q1)− ωϕ(δ)‖ν̄‖L∞(Q1) vol(Q1) , whence, for almost every ω, L−m ∫ U ϕ dnL − ∫ U ϕν̄ d vol 6 2m (max U ϕ) ∫ Q1 \|νR+D,L(x, ω)− ν̄(x)\| d vol(x) + [( 1 + D R )m − 1 ] (max U ϕ)‖ν̄‖L∞(Q1) vol(Q1) + ωϕ(δ)‖ν̄‖L∞(Q1) vol(Q1) + (max U ϕ) L−m ∑ j N(Sj ; fL ) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 245 F. Nazarov and M. Sodin The matching lower bound is similar but somewhat simpler: for almost every ω, we have L−m ∫ U ϕ dnL − ∫ U ϕν̄ d vol > −(max U ϕ) ∫ Q1 \|νR,L(x, ω)− ν̄(x)\|d vol(x)− ωϕ(δ)‖ν̄‖L∞(Q1) vol(Q1) . Gathering the upper and lower bounds and taking the upper expectation, we obtain E∗ ∣∣∣L−m ∫ U ϕdnL − ∫ U ϕν̄ d vol ∣∣∣ 6 2m(max U ϕ) ∫ Q1 E{\|νR+D,L(x)− ν̄(x)\|+ \|νR,L(x)− ν̄(x)\|}d vol(x) + (max U ϕ) L−m ∑ j E∗{N(Sj ; fL ) } + (max U ϕ) ‖ν̄‖L∞(Q1) vol(Q1) [( 1 + D R )m − 1 ] + 2ωϕ(δ)‖ν̄‖L∞(Q1) vol(Q1) . It remains to estimate the terms on the RHS. Fix ε > 0 and choose δ so small that ωϕ(δ) < ε. This takes care of the last term on the RHS. To treat the second term we use Lemma 9, which yields E∗{N(Sj ; fL ) } . Dm−1 uniformly in j. Therefore, L−m ∑ j E∗{N(Sj ; fL ) } . L−m vol(Q2)(L/D)m Dm−1 . D−1 vol(Q2) . Let U ′ ⊂ U be a Borel subset of full volume on which the scaled functions fx,L have translation invariant limits. The functions ν̄ and E{ νR,L } are locally uniformly bounded on U ′ by a constant independent of R and L. Let ηR(x) def= lim L→∞ E∣∣νR,L(x)− ν̄(x) ∣∣ . The function ηR is uniformly bounded on Q1 ∩ U ′ by a constant independent of R. Then, applying Fatou lemma, we obtain 246 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components lim L→∞ E∗ ∣∣∣L−m ∫ U ϕdnL − ∫ U ϕν̄ d vol ∣∣∣ 6 C(ϕ,Q) (∫ Q1 [ ηR+D(x)+ηR(x) ] d vol(x)+‖ν̄‖L∞(Q1) [( 1+D R )m−1+ε ] +D−1 ) . For any x ∈ U ′, we have ηR(x) → 0 as R →∞. Using the dominated convergence theorem, we get lim L→∞ E∗ ∣∣∣L−m ∫ U ϕdnL − ∫ U ϕν̄ d vol ∣∣∣ 6 C(ϕ,Q) ( ε‖ν̄‖L∞(Q1) + D−1 ) . Letting ε → 0 and D →∞, we finish off the proof of Theorem 2. 9. The Manifold Case. Proof of Theorem 3 9.1. Smooth Gaussian functions and their covariance kernels under C2-changes of variable Suppose that U , V are open subsets of Rm and ψ : V → U is a C2-diffeomor- phism. Suppose that f : U → R is a continuous Gaussian function on U with a C2,2 covariance kernel K(x, y). Then f ◦ ψ is a continuous Gaussian function on V with the reproducing kernel K̃(x, y) = K(ψ(x), ψ(y)). Note that for every pair of the multi-indices α, β, the mixed partial derivative ∂α x ∂β y K̃(x, y) is a linear combination of finitely many expressions of the kind [ ∂α′ x ∂β′ y K ] (ψ(x), ψ(y))Qα′,β′ , where α′, β′ are multi-indices with 1 6 \|α′\| 6 \|α\|, 1 6 \|β′\| 6 \|β\|, and Qα′,β′ is a certain polynomial expression of partial derivatives of order at most max(\|α\|, \|β\|) of coordinate functions of ψ. In particular, if K is Ck,k-smooth, then so is K̃. Since the maxima of higher order derivatives in the definition of the norm ‖K‖L,Q,k are multiplied by higher negative powers of L, we conclude that for every compact Q ⊂ V and L > 1, ‖K̃‖L,Q,k 6 C(ψ,Q, k) ‖K‖L,ψ(Q),k , where C(ψ,Q, k) is some factor depending on max \|γ\|6k max Q \|∂γψ\|. Next, let CK x be the matrix with the entries CK x (i, j) = ∂xi ∂yj K(x, y). Then detCK̃ x = (det(Dψ)(x))2 det CK ψ(x) . One immediate consequence of these observations is that Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 247 F. Nazarov and M. Sodin • the local controllability of K can be verified after any C2-change of vari- ables ψ, and moreover, the corresponding constants at x will change only by bounded factors depending on the first and second derivatives of ψ and ψ−1 at x and ψ(x) respectively. Now, let us see what the C2-change of variable ψ does to translation invariant scaling limits. Let z = ψ(z′) ∈ U . Assume that we have a sequence of kernels KL such that the corresponding scaled kernels Kz,L(u, v) = KL(z + L−1u, z + L−1v) converge to k(u − v), where k is a continuous function on Rm. Assume that for some r > 0, there is a closed ball B̄ = B̄(z, r) ⊂ U such that sup L>1 L−1 max B̄×B̄ (\|∇xKL\|+ \|∇yKL\|) = M < ∞ . Let u′, v′ ∈ Rm. Then, for sufficiently large L, we have ∣∣ψ(z′ + L−1u′)− ψ(z′)− 1 L(Dψ)(z′)u′ ∣∣ 6 C(ψ) L−2\|u′\|2 , and similarly for v′, where the constant C(ψ) depends only on the bounds for the second partial derivatives of ψ in an arbitrarily small (but fixed) neighbourhood of z′. Moreover, if u′ and v′ are fixed and L is large, then the points ψ(z′ + 1 Lu′), z + 1 L(Dψ)(z′)u′ , together with similar two points taken with v′ instead of u′, belong to the ball B. So we obtain ∣∣K̃(z′ + 1 Lu′, z′ + 1 Lv′)−K(z + 1 L(Dψ)(z′)u′, z + 1 L(Dψ)(z′)v′) ∣∣ 6 LMC(ψ)L−2(\|u′\|2 + \|v′\|2) → 0, as L →∞ . Since K(z + 1 L(Dψ)(z′)u′, z + 1 L(Dψ)(z′)v′) converge to k((Dψ)(z′)(u′ − v′)), we conclude that K̃(z′ + L−1u′, z′ + L−1v′) converge to k̃(u′ − v′), where k̃(u′) = k((Dψ)(z′)(u′)). Since a non-degenerate linear change of variable on the space side corresponds to a non-degenerate linear change of variable and renormalization on the Fourier side, the spectral measures ρ and ρ̃, corresponding to k and k̃, respectively, do or do not have atoms simultaneously. This shows that the Gaussian parametric ensembles fL on U and f̃L = fL ◦ ψ on V = ψ−1U are or aren’t tame simulta- neously. Finally, the corresponding limiting Gaussian functions Fz and F̃z′ are related by F̃z′ = Fz ◦ (Dψ)(z′), whence, ν̄ F̃z′ (z′) = \| det(Dψ)(z′)\| ν̄Fz(z). 248 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components 9.2. Possibility to verify tameness in charts From the previous discussion it becomes clear why it suffices to check that fL ◦ πα is tame on Uα for some atlas A = (Uα, πα) to be sure that fL ◦ π is tame on U for any chart π : U → X. Indeed, take any compact Q ⊂ π(U) and cover it by a finite union of open charts ⋃ j παj (Uαj ). Then we can choose compact sets Qj ⊂ Q ∩ παj (Uαj ) so that ⋃ j Qj = Q. However, on each Qj the computations of all relevant quantities in the charts (U, π) and (Uαj , παj ) give essentially the same results (up to bounded factors) because all partial derivatives of order 1 and 2 of the transition mappings π−1 αj ◦ π and π−1 ◦ παj are bounded on π−1(Qj) and π−1 αj (Qj), respectively. If the atlas A has uniformly bounded distortions, our observations show that for every point x ∈ X, all computations in all charts (U, π) of A such that x ∈ π(U) yield essentially the same results. Thus, for every point x ∈ X, we can compute the relevant quantities in its own chart from A (the most convenient one) without affecting the existence of uniform bounds for them, but, of course, affecting the best possible values of those bounds. 9.3. Completing the proof of Theorem 3 Take two charts π1 : U1 → X and π2 : U2 → X and consider the corresponding Gaussian parametric ensembles f1,L = fL ◦ π1 and f2,L = fL ◦ π2 on U1 and U2 respectively. For every x ∈ π(U1) ∩ π(U2) ⊂ X, we have ν̄1(π−1 1 (x)) = ∣∣det ( [D(π−1 2 π1)](π−1 1 (x)) )∣∣ ν̄2(π−1 2 (x)) in the sense that if one side is defined, then so is the other and the equality holds. Therefore, the push-forwards (π1)∗(ν̄1 d vol) and (π2)∗(ν̄2 d vol) coincide on π(U1)∩π(U2), which allows us to define a Borel measure n∞ on X unambiguously and to justify the formula for its density with respect to any volume volX on X compatible with the smooth structure. The only thing that remains to do to establish Theorem 3 as stated, is to show that lim L→∞ E∗ {∣∣∣L−m ∫ X ϕ dnL − ∫ X ϕdn∞ ∣∣∣ } = 0 . The standard partition of unity argument allows us to reduce the problem to the case when the support of the test function ϕ is contained in one chart π(U). Hence, the desired result would be an immediate consequence of Theorem 2 applied to the pull-back measures π∗nL and the test-function ϕ◦π, if not for one minor nuisance: the pull-back to U of a component counting measure of (fL) on X by the chart mapping π may fail to be a component counting measure of fL ◦π Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 249 F. Nazarov and M. Sodin on U because the connected components of Z(fL) on X that stretch outside π(U) may get truncated or split into several pieces when mapped to U by π−1. So the pull-back π∗nL may have mass less than 1 on some connected components of fL◦π that stretch to the boundary of U . We circumvent this difficulty by noticing that the closed support spt(ϕ◦π) is contained in U . Thus, if we “beef up” the measure of each “defective” component γ by adding an appropriate positive multiple of a point mass at any point u ∈ γ \ spt(ϕ ◦ π), the pull-back π∗nL will turn into a component counting measure n′L but the total integral of ϕ◦π will not be affected in any way. Now we can just apply Theorem 2 to n′L instead of π∗nL and reach the desired conclusion. Appendices A. Smooth Gaussian Functions In this appendix, we collect well-known facts about smooth Gaussian func- tions, which have been used throughout this paper. Our smooth Gaussian func- tions will be defined on open subsets of Rm. For a topological space X, by B(X) we denote the Borel σ-algebra generated by all open subsets of X. As everywhere else in the paper, all Hilbert spaces are real and separable and all Gaussian random variables have zero mean. A.1. The space Ck(V ) Let V ⊂ Rm be an open set. For k ∈ Z+, we denote by Ck(V ) the space of Ck- smooth functions on V . The topology in Ck(V ) is generated by the seminorms? ‖g‖Q,k = max Q max \|α\|6k ∣∣∂αg ∣∣ where Q runs over all compact subsets of V . If Qn is an increasing sequence of compact subsets of V such that every compact set K ⊂ V is contained in each Qn with n > n0(K), then the countable family of the seminorms ‖g‖Qn,k, n = 1, 2, . . . , gives the same topology, so Ck(V ) is metrizable. Since it is separable as well, every open set in Ck(V ) can be written as a countable union of “standard neighbourhoods” B(Q, g0, ε) = { g ∈ Ck(V ) : ‖g − g0‖Q,k < ε } . We will need two simple lemmas. ?The reader should be aware that the same notation was used in the main text for the seminorm in Ck,k(V × V ). This should not lead to a confusion because the functions measured in these seminorms have different numbers of variables. 250 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Lemma A.1. The Borel σ-algebra B = B(Ck(V )) coincides with the least σ- algebra on Ck(V ) containing all “intervals” I(x; a, b) = { g ∈ Ck(V ) : a 6 g(x) < b } , i.e., B is generated by point evaluations g 7→ g(x). P r o o f of Lemma A.1. Denote by B′ the least σ-algebra on Ck(V ) containing all intervals I(x; a, b). We need to show that the σ-algebras B and B′ coincide. Since the mapping Ck(V ) 3 g 7→ g(x) ∈ R is continuous and, thereby, measurable, every interval I(x; a, b) is Borel, that is B′ ⊂ B. To show that B ⊂ B′, it suffices to check that every standard neighourhood B(Q, g0, ε) belongs to B′, or, which is the same, that the mapping Ck(V ) 3 g 7→ ‖g − g‖Q,k is B-measurable. Since for every fixed x ∈ V and every multiindex α with \|α\| 6 k, the mapping g 7→ ∂αg(x) can be represented as a pointwise (on Ck(V )) limit of finite linear combinations of point evaluations, it is measurable as well. It remains to note that ‖g − g0‖Q,k = sup x∈Q′ max \|α\|6k ∣∣∂αg(x)− ∂αg0(x) ∣∣, where Q′ is any countable dense (in Q) subset of Q. Lemma A.2. Ck(V ) is a Borel subset of C(V ). P r o o f of Lemma A.2. Take any function ϕ1 ∈ C∞ 0 (B), where B is the unit ball in Rm, put ϕj = jmϕ(jx) and consider the sequence of continuous mappings C(V ) 3 g 7→ g ∗ ϕj ∈ Ck(V−1/j). Note that g ∈ Ck(V ) if and only if g ∗ ϕj converge in Ck uniformly on every compact set Q ⊂ V . Taking a countable exhaustion Qn of V and choosing j(n) so that Qn ⊂ V−1/j(n), we get the representation Ck(V ) = ⋂ q>1 ⋂ n>1 ⋃ j>j(n) ⋂ s′,s′′>j { g ∈ C(V ) : ‖g ∗ ϕs′ − g ∗ ϕs′′‖j, Qn < 1 q } . Clearly, the RHS is Borel in C(V ) (since each “basic set” on the RHS is open in C(V )). A.2. The definition and basic properties of Ck-smooth Gaussian functions Definition A.1. Let (Ω, S,P) be a probability space. The function f : V × Ω → R is a Gaussian function on V if (i) for each x ∈ V , the mapping ω 7→ f(x, ω) is measurable as a mapping from( Ω,S ) to ( R, B(R) ) ; Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 251 F. Nazarov and M. Sodin (ii) for each finite set of points x1, . . . , xn ∈ V and for each c1, . . . , cn ∈ R, the sum ∑ j cjf(xj , ω) is a Gaussian random variable (maybe, degenerate). Let k ∈ Z+. The Gaussian function f is called Ck-smooth (or just Ck) if (iii) for almost every ω ∈ Ω, the function x 7→ f(x, ω) belongs to the space Ck(V ). Removing a subset of zero probability from Ω, we may (and will) just demand that the function x 7→ f(x, ω) is in Ck(V ) for all ω ∈ Ω. Every Ck-Gaussian function f generates two mappings Ω 3 ω 7→ f( · , ω) ∈ Ck(V ) and V 3 x → f(x, · ) ∈ L2(Ω,P) . With some abuse of notation, we denote these mappings by the same letter f . Lemma A.3. Suppose that f is a Ck-smooth Gaussian function on V . Then (a) the mapping f : ( V × Ω, B(V )×S) → ( R, B(R) ) is measurable; (b) the mapping f : ( Ω, S ) → ( Ck(V ),B(Ck(V )) ) is measurable; (c) the mapping f : V → L2(Ω,P) is Ck-smooth. P r o o f of Lemma A.3. (a) We partition V into countably many Borel sets Vj of diameter 6 1/n each, fix an arbitrary collection of points xj ∈ Vj , and define a function fn : (V ×Ω) → R by fn(x, ω) = f(xj , ω) for x ∈ Vj . The mappings fn : ( V × Ω,B(V ) × S ) → ( R, B(R) ) are measurable and f = lim n→∞ fn pointwise on V × Ω. (b) It is an immediate consequence of Lemma A.1 combined with fact that, for every x ∈ V , the mapping ω 7→ f(x, ω) is measurable. (c) Recall that if a sequence ξn : Ω → R of Gaussian random variables converges pointwise to ξ, then ξ is also a Gaussian random variable. It follows that for every multiindex α with \|α\| 6 k, the mapping (x, ω) 7→ ∂αf(x, ω) is a continuous Gaussian function. Since the pointwise convergence of Gaussian random variables yields convergence in in L2(Ω,P), we see that the mapping V 3 x 7→ ∂αf(x, · ) ∈ L2(Ω,P) is continuous and gives the corresponding partial derivative of the mapping V 3 x 7→ f(x, · ) considered as a function on V with values in L2(Ω,P). 252 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Definition A.2. Let f be a Ck Gaussian function on V . Let γf def= f∗P be the push-forward of the probability measure P to Ck(V ) by f . We say that two Ck Gaussian functions f1 and f2 are equivalent if γf1 = γf2. We do not distinguish between equivalent Gaussian functions. In principle, we can forget about the original probability space (Ω, S,P) and consider the probability space ( Ck(V ), B(Ck(V )), γf ) and the mapping V × Ck(V ) 3 (x, g) 7→ g(x) ∈ R instead. We can go one step further and remove any Borel subset of γf -measure 0 from Ck(V ) in this representation. A.3. Positive-definite kernels Let f be a Ck Gaussian function on V . Let Kf (t, s) def= E{ f(t)f(s) } be the corresponding covariance kernel. It is a positive-definite symmetric function? on V × V . The function f is uniquely determined by Kf up to equivalence. Indeed, since a Gaussian distribution in Rn is determined by its covariance matrix, this fact is evident for the sets of the form S = { g ∈ Ck(V ) : ( g(x1), . . . , g(xn) ) ∈ B } where x1, . . . , xn ∈ V and B ∈ B(Rn). The general case follows immediately because the fact that B(Ck(V )) is generated by point evaluations implies that every set S ∈ B(Ck(V )) can be approximated by sets of such kind up to an arbitrary small γf -measure. Next, we observe that if g is a continuous Gaussian function on V with Kg = Kf , then g is equivalent, as a continuous Gaussian function, to the Gaussian function f̃ : Ω f→ Ck(V ) ↪→ C(V ). The function f̃ generates a measure γ f̃ on C(V ): γ f̃ (S) = γf (S ∩ Ck(V )) , S ∈ B(C(V )) . Furthermore, K f̃ (x, y) = E{ f̃(x)f̃(y) } = E{ f(x)f(y) } = Kf (x, y) = Kg(x, y) . Therefore, by the previous remark, γ f̃ = γg. In this situation, almost surely, g is a Ck Gaussian function. Indeed, by Lemma A.2, C(V ) \ Ck(V ) ∈ B(C(V )), whence, γg(C(V ) \ Ck(V )) = γ f̃ (C(V ) \ Ck(V )) = γf (∅) = 0 . ?That is, the symmetric matrix ( Kf (xi, xj) )n i,j=1 is positive definite for every choice of x1, . . . , xn ∈ V . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 253 F. Nazarov and M. Sodin That is, any continuous Gaussian function whose covariance kernel coincides with the one of a Ck Gaussian function, almost surely, is a Ck function itself. Also observe that since the mapping V 3 x 7→ f(x, · ) ∈ L2(Ω,P) is Ck, the partial derivative ∂α x ∂β y Kf (x, y) exists and is continuous on V × V for any multiindices α, β with \|α\|, \|β\| 6 k. Moreover, ∂α x ∂β y Kf (x, y) = E{ ∂α x f(x) ∂β y f(y) } . A.4. From positive-definite kernels to reproducing kernel Hilbert spaces In this section, we shall only assume that we are given a continuous positive- definite symmetric kernel K on V ×V . We shall describe a canonical construction of the Hilbert space H = H(K) of continuous functions on V such that K is the reproducing kernel in that space, that is, for every g ∈ H and every x ∈ V , we have g(x) = 〈g,Kx〉H where Kx(y) = K(x, y). Consider the linear space L of all mappings h : V → R such that h(x) 6= 0 only for finitely many x ∈ V . Define the semi-definite scalar product on L by 〈h1, h2〉 = ∑ x,y∈V K(x, y) h1(x)h2(y) (this sum is actually finite). Since K is positive-definite, we have 〈h, h〉 > 0 for every h ∈ L. Define the Hilbert seminorm on L by ‖h‖ = √ 〈h, h〉. Then 〈 · , · 〉 and ‖ · ‖ become a nondegenerate scalar product and the associated Hilbert norm on L/L0 where L0 is the linear subspace of L consisting of all h ∈ L with ‖h‖ = 0. Let H be the Hilbert space completion of the pre-Hilbert space (L/L0, 〈 · , · 〉 ) . For x ∈ V , denote by hx the vector in H corresponding to the function hx(y) = { 0, y 6= x, 1, y = x. Note that ‖hx − hy‖2 = K(x, x) + K(y, y) − 2K(x, y) → 0 as y → x, so the mapping V 3 x 7→ hx ∈ H is continuous. Since span{hx : x ∈ V } is dense in H by construction and since for every countable dense subset V ′ ⊂ V , the set {hx : x∈V ′} is dense in {hx : x∈V }, H is separable. Now, define a linear map Φ: H → C(V ) by Φ[h](x) = 〈h, hx〉, h ∈ H. If Φ[h] = 0, then 〈h, hx〉 = 0 for all x ∈ V , so h = 0. Thus, we can identify H with a linear subspace H = Φ(H) of C(V ). Note also that Φ[hx](y) = 〈hx, hy〉 = K(x, y) = Kx(y), 254 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components so hx is identified with Kx. Transferring the scalar product from H to H, we turn H into a Hilbert space of continuous functions on V with the reproducing kernel K. Observe, finally, that such a Hilbert space is unique. Indeed, if H1 ⊂ C(V ) is another Hilbert space of continuous functions with the same reproducing kernel K, then the linear span H0 of the functions Kx, x ∈ V , is contained and dense in H1 with respect to the Hilbert norm in H1 (because if g ∈ H1 is orthogonal to all Kx in H1, then g(x) = 〈g, Kx〉H1 = 0 for all x ∈ V , whence, g = 0) and for every pair of functions g1 = ∑ finite axKx, g2 = ∑ finite byKy in H0, we have 〈g1, g2〉H1 = ∑ x,y K(x, y) axby = 〈g1, g2〉H . Thus the identity mapping H0 → H0 can be extended to a bijective isometry H → H1. Let now g′ ∈ H1 be the image of g ∈ H under this isometry. Then g′(x) = 〈g′,Kx〉H1 = 〈g, Kx〉H = g(x) , x ∈ V , so H1 and H consist of exactly the same functions on V and are endowed with the same scalar product. We end this section with a useful observation. Let {ek} be an arbitrary orthonormal basis in H. For every g ∈ H, we put ĝ(k) = 〈g, ek〉H. Then the Fourier series ∑ k ĝ(k)ek converges to g in H. For every y ∈ V , we have ∣∣∣g(y)− ∑ 16k6N ĝ(k)ek(y) ∣∣∣ = ∣∣∣ 〈 g − ∑ 16k6N ĝ(k)ek,Ky 〉∣∣∣ 6 ∥∥∥g − ∑ 16k6N ĝ(k)ek ∥∥∥ H ‖Ky‖H → 0 as N →∞ . Since ‖Ky‖H = √ K(y, y) is a continuous function of y on V , this yields the locally uniform convergence of the Fourier series ∑ k ĝ(k)ek to g. Taking g = Kx and observing that 〈Kx, ek〉 = ek(x), we conclude that for every x, y ∈ V , we have ∑ k ek(x)ek(y) = K(x, y) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 255 F. Nazarov and M. Sodin A.5. Canonical series representation of continuous Gaussian functions Let H0 be any Gaussian subspace of L2(Ω,P) and let V 3 x 7→ fx ∈ H0 be a continuous mapping such that for every x ∈ V , the random variable fx is Gaus- sian. The corresponding covariance kernel K(x, y) = E{fxfy} = 〈fx, fy〉L2(Ω,P) is also continuous. Let H be the closed linear span of {fx}x∈V in L2(Ω,P). It is a Gaussian subspace of L2(Ω,P). For h ∈ H, define Φ[h](x) = 〈h, fx〉L2(Ω,P). Note that Φ[h] ∈ C(V ) and Φ[h] = 0 if and only if h = 0. Also, Φ[fx] = K(x, · ) = Kx. Thus, H = {Φ[h] : h ∈ H} is a linear subspace of C(V ) and if we endow it with the scalar product 〈Φ[h],Φ[h′]〉H = 〈h, h′〉L2(Ω,P), it will become a Hilbert space H(K) of continuous functions with the reproducing kernel K. Now, take any orthonormal basis {ej} in H and choose ξj ∈ H such that ej = Φ[ξj ]. Note that 〈ξi, ξj〉L2(Ω,P) = 〈ei, ej〉H = { 0, i 6= j , 1, i = j , so ξj are orthogonal and, thereby, independent standard Gaussian. For every x ∈ V , we have fx = ∑ j 〈fx, ξj〉 ξj = ∑ j Φ[ξj ](x)ξj = ∑ j ej(x)ξj . The upshot is that, • given any Gaussian subspace H0 ⊂ L2(Ω,P), any continuous mapping x 7→ fx from V to H0, and any orthonormal basis ej in the reproduc- ing kernel Hilbert space H(K), where K(x, y) = E{f(x)f(y)}, we can de- fine independent standard Gaussian real variables on (Ω,S,P) such that fx = ∑ j ξj ej(x) for all x ∈ V . Assume now that we start with a continuous Gaussian function f with some underlying probability space (Ω, S,P). Applying the above construction to the induced mapping x 7→ fx = f(x, · ), we get f(x, ω) = ∑ j ξj(ω) ej(x) in L2(Ω,P) for all x ∈ V . (A.1) Implementing ξj as some everywhere defined functions on Ω and taking into account that L2(Ω,P)-convergence yields convergence in probability, we have, in particular, that for every x ∈ V, n∑ j=1 ξj(ω) ej(x) → f(x) in probability as n →∞ . (A.2) 256 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Now, put Xj = ξjej(x), Sn = ∑n j=1 Xj and note that for every compact Q ⊂ V , the random variables Xj , Sn and S = f can be viewed as random vectors in the Banach space C(Q). The random variables Xj are symmetric and independent, and (A.2) means that for every point evaluation functional zx ∈ C(Q)∗ given by 〈zx, g〉 = g(x) for x ∈ Q, we have 〈z, Sn〉 → 〈zx, S〉 in probability. By the classical Ito–Nisio theorem, which we will recall in the next section, the series ∑ j Xj converges to S in C(Q). Thus, • the canonical series representation (A.1) actually converges in C(V ). A.6. The Ito–Nisio theorem Let X be a separable Banach space. An X-valued random variable on a proba- bility space (Ω, S,P) is just a measurable mapping from (Ω, S,P) to (X, B(X) ). Everywhere below, Xj is a sequence of independent X-valued random variables, S is an X-valued random variable on the same probability space, and Sn =∑ j6n Xj . We denote by ‖ · ‖ the norm in X, and by 〈·, ·〉 the natural coupling of the dual space X∗ and X. First, we recall a classical P. Levý’s lemma. If Sn converges to S in probability, then Sn converges to S almost surely. P r o o f of Levý’s lemma. We will check that, for almost every ω ∈ Ω, Sn(ω) is a Cauchy sequence. Take ε ∈ (0, 1 2) and take m so large that P{‖Sk − S‖ > 1 2 ε } < 1 2 ε , k > m . Then take any positive integer n > m. For k = m, . . . , n, let Ak = { ω ∈ Ω: ‖S` − Sm‖ 6 2ε for all ` = m, . . . , k − 1, but ‖Sk − Sm‖ > 2ε } . Note that the events Ak are disjoint, and each Ak is independent of Sn − Sk. Also, if ω ∈ Ak and ‖Sn − Sk‖ 6 ε, then ‖Sn − Sm‖ > ‖Sk − Sm‖ − ‖Sn − Sk‖ > 2ε− ε = ε . Furthermore, since {‖Sk − Sn‖ > ε } ⊂ {‖Sk − S‖ > 1 2 ε }⋃{‖Sn − S‖ > 1 2 ε } , we have P{‖Sk − Sn‖ 6 ε } = 1− P{‖Sk − Sn‖ > ε } > 1− ε . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 257 F. Nazarov and M. Sodin Therefore, (1− ε)P{ max m6k6n ‖Sk − Sm‖ > 2ε} = (1− ε) n∑ k=m P (Ak) < n∑ k=m P{‖Sn − Sk‖ 6 ε}P(Ak) = n∑ k=m P{Ak and ‖Sn − Sk‖ 6 ε} 6 P{‖Sn − Sm‖ > ε} 6 ε . Since n and ε are arbitrary, we see that, with probability 1, Sn is Cauchy, so it converges almost surely. Clearly, the almost sure limit and the limit in probability must be the same. We call a subset Z ⊂ X∗ normalizing if it is countable and ‖x‖ = sup {〈z, x〉 : z ∈ Z} (then automatically Z is contained in the unit ball of X∗). Now, we can state the part of the Ito–Nisio theorem that we need: Ito–Nisio theorem. Suppose that the random variables Xj are symmetric and that there exists a normalizing set Z ⊂ X∗ such that 〈z, Sn〉 → 〈z, S〉 in probability for every z ∈ Z. Then Sn → S in X almost surely. P r o o f. By P. Levý’s lemma, it is enough to show that Sn converges to S in probability. First of all, note that the Borel σ-algebra B(X) coincides with the σ-algebra B′(X) generated by the events {x : 〈z, x〉 ∈ [a, b)}, z ∈ Z, a, b ∈ R. Indeed, for every x0 ∈ X, the mapping x 7→ ‖x − x0‖ = supZ ‖〈z, x〉 − 〈z, x0〉‖ is B′(X)-measurable. Thus, every open ball in X is B′(X)-measurable. Since X is separable, every open set in X is B′(X)-measurable, so B(X) ⊂ B′(X). The inverse inclusion is obvious. Next, we show that Sn and S − Sn are independent for every n. We need to check that P{Sn ∈ C1, S − Sn ∈ C2} = P{Sn ∈ C1}P{S − Sn ∈ C2} for every C1, C2 ∈ B(X). Since B(X) = B′(X), it suffices to check this for the events of the form C = {(〈z1, x〉, . . . , 〈zq, x〉 ) ∈ B } , B ∈ B(Rq), z1, . . . , zq ∈ Z , in which case it follows from the independence of Sm−Sn and Sn for m > n and the fact that, for m →∞, (〈z1, Sm−Sn〉, . . . , 〈zq, Sm−Sn〉 ) → (〈z1, S−Sn〉, . . . , 〈zq, S−Sn〉 ) in probability . 258 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components For a set X′ ⊂ X, denote X′+ε = ⋃ x∈X′ B(x, ε) . We claim that for every finite set X′ ⊂ X, there exists a finite “separating set” of functionals Z ′ ⊂ Z such that max z∈Z′ \|〈z, y′ − y′′〉\| > ε , whenever y′, y′′ ∈ X′+ε and ‖y′ − y′′‖ > 8ε . Indeed, consider all differences x′ − x′′ with x′, x′′ ∈ X′ and for each of them choose z = z(x′, x′′) ∈ Z such that \|〈z, x′−x′′〉\| > 1 2 ‖x′−x′′‖. Since y′, y′′ ∈ X′+ε, we can find x′, x′′ ∈ X′ so that ‖x′ − y′‖, ‖x′′ − y′′‖ < ε. Then ‖x′ − x′′‖ > 6ε. Taking z = z(x′, x′′), we get \|〈z, y′ − y′′〉\| > \|〈z, x′ − x′′〉\| − 2ε > 3ε− 2ε = ε , proving the claim. Now, comes the crux of the proof. Suppose that A and B are X-valued independent random variables and A is symmetric. Then, for every finite X′ ⊂ X and every ε > 0, we can write P{ A /∈( 1 2(X′−X′) ) +ε } 6P{ A+B /∈X′+ε } +P{−A+B /∈X′+ε } =2P{ A+B /∈X′+ε } . The inequality here is due to the observation that if a, b ∈ X and a + b,−a + b ∈ X′+ε, then a ∈ ( 1 2(X′ − X′) ) +ε . The equality follows at once from the symmetry of A and the independence of A and B. To finish the proof, we take ε > 0 and let x1, x2, . . . be a countable dense set in X. Put X′N = {x1, . . . , xN}. Since (X′N )+ε ↑ X as N → ∞, we have P{S /∈ (X′N )+ε} < ε for large enough N . We fix such N and, to simplify notation, let X′ = X′N . Since Sn and S−Sn are independent, and Sn is symmetric, we can use them as A and B in the argument above and get P{ Sn /∈ ( 1 2(X′−X′) ) +ε } < 2ε for all n. Let Z ′ ⊂ Z be a finite separating set for X′∪ 1 2(X′−X′). Then for every n, P{‖Sn − S‖ > 8ε } 6 P{ Sn /∈ ( 1 2(X′ − X′) ) +ε } + P{ S /∈ X′+ε } + P{ max z∈Z′ \|〈z, Sn〉 − 〈z, S〉\| > ε } 6 3ε + ∑ z∈Z′ P{\|〈z, Sn〉 − 〈z, S〉\| > ε } . Since each term in the finite sum on the RHS tends to 0 as n →∞ and ε can be taken as small as we want, the desired convergence in probability follows. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 259 F. Nazarov and M. Sodin A.7. The local behavior of continuous Gaussian functions Suppose that f is a continuous Gaussian function on V with the covariance kernel K. As before, we denote by γf the corresponding Gaussian measure on C(V ). The (closed) set S(f) of functions g ∈ C(V ) for which P{f ∈ U} = γf (U) > 0 for every open neighbourhood U of g in C(V ) is called the topological support of the measure γf . The following lemma gives a simple and useful description of the topological support of γf : Lemma A.4. S(f) = ClosC(V )H(K). P r o o f of Lemma A.4. First, we show that for every g ∈ H(K), every compact Q ⊂ V , and every ε > 0, we have P{‖f − g‖C(Q) < ε} > 0. We choose an orthonormal basis {ej} in H(K) so that g = te1 for some t ∈ R, and represent f as ∑ j ξjej where ξj are independent Gaussian random variables. Since, by the Ito–Nisio theorem, the series converges in C(Q) with probability 1, there exists N = N(ε) such that ‖ ∑ j>N ξjej‖C(Q) = ‖f − ∑ j6N ξjej‖C(Q) < 1 2 ε with positive probability. Next, we choose η so small that η ∑ j6N ‖ej‖C(Q) < 1 2 ε . Now, suppose that ‖ ∑ j>N ξjej‖C(Q) < 1 2 ε , ξ1 ∈ (t− η, t + η) , and ξ2, . . . , ξn ∈ (−η, η) . Then ‖f − g‖C(Q) 6 \|ξ1 − t\| ‖e1‖C(Q) + ∑ 26j6N \|ξj \| ‖ej‖C(Q) + ∥∥∑ j>N ξjej ∥∥ C(Q) 6 η ∑ j6N ‖ej‖C(Q) + ∥∥∑ j>N ξjej ∥∥ C(Q) < ε . Hence, P{‖f − g‖C(Q) < ε} > P{‖ ∑ j>N ξjej‖C(Q) < 1 2 ε } × P{ ξ1 ∈ (t− η, t + η), ξ2, . . . , ξN ∈ (−η, η) } > 0 . 260 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Thus, H(K) ⊂ S(f). Since S(f) is closed in C(V ), we get S(f) ⊃ ClosC(V )H(K). To show the converse, assume that g ∈ C(V ) and P{‖f − g‖C(Q) < 1 2 ε} = p > 0 . We fix an orthonormal basis {ej} in H(K) and choose N so large that P{‖f − ∑ j6N ξjej‖C(Q) > 1 2 ε} < 1 2 p . Then P{‖g − ∑ j6N ξjej‖C(Q) < ε} > p− 1 2 p > 0 . Since ∑ j6N ξjej ∈ H(K) , we conclude that the ε-neighbourhood of g in C(Q) intersects H(K). Since ε and Q are arbitrary, we see that S(f) ⊂ ClosC(V )H(K). A.8. Fernique’s theorem The next result we state was proven by Fernique and independently by Landau and Shepp. It allows one to pass from some very weak estimates for various norms and semi-norms of Gaussian functions to almost as strong bounds for tails as possible in principle. Fernique’s theorem. Let X be a random variable with values in a Banach space X, and let {ϕj} ⊂ X∗ be an at most countable set of linear functionals on X such that, for every choice of finitely many ϕj’s, the joint distribution of ϕj(X) is Gaussian. Suppose that, for some λ > 0 and µ < 1 2 , P{ sup j \|ϕj(X)\| > λ } 6 µ . (A.1) Then, for all t > 1, P{ sup j \|ϕj(X)\| > λt } 6 e−at2 (A.2) with a positive constants a depending only on µ. Here, we present Fernique’s original proof, which is short and elegant. Landau and Shepp [21] gave a different proof based on the Gaussian isoperimetry. The Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 261 F. Nazarov and M. Sodin advantage of the latter proof is that it does not need a priori assumption µ < 1 2 and gives the optimal RHS of (A.2), which is Φ ( tΦ−1(µ) ) , where Φ(s) = √ 2 π ∞∫ s e−x2/2 dx . P r o o f of Fernique’s theorem. Without loss of generality, we assume that λ = 1. Let Ωn(t) be the event { sup16j6n \|ϕj(X)\| > t } , n ∈ N ∪ {∞}. We need to estimate P{Ω∞(t)} assuming that P{Ω∞(1)} 6 µ. Since Ωn(t) ⊂ Ωm(t) for n 6 m, and Ω∞(t) = ⋃ n>1 Ωn(t), it will suffice to prove estimate (A.2) for every finite n with constants A and a independent of n. In what follows, we fix n ∈ N, and put ϕ = ( ϕ1(X), . . . , ϕn(X) ) . This is a finite-dimensional Gaussian random vector. We let ‖ϕ‖ def= max 16j6n \|ϕj(X)\|. Fernique’s proof is based on the following classical observation: if ψ is an n- dimensional Gaussian vector, which has the same distribution as ϕ and which is independent of ϕ, then 1√ 2 (ϕ + ψ) and 1√ 2 (ϕ − ψ) are two Gaussian vectors, which have the same distribution as ϕ and which are independent of each other. The proof of this statement reduces to a routine verification of coincidence of all relevant covariance matrices. Now, take t > 0 and τ > 0 and write P{‖ψ‖ 6 τ }P{‖ϕ‖ > t } = P{‖ 1√ 2 (ϕ− ψ)‖ 6 τ }P{‖ 1√ 2 (ϕ + ψ)‖ > t } = P{‖ 1√ 2 (ϕ− ψ)‖ 6 τ, ‖ 1√ 2 (ϕ + ψ)‖ > t } 6 P{‖ϕ‖ > 1√ 2 (t− τ), ‖ψ‖ > 1√ 2 (t− τ) } = (P{‖ϕ‖ > 1√ 2 (t− τ) })2 . Letting τ = 1 and recalling that P{‖ψ‖ 6 1 } > 1− µ, we get P{‖ϕ‖ > t } 6 1 1− µ ( P{‖ϕ‖ > 1√ 2 (t− τ) })2 . Put p(t) = P{‖ϕ‖ > t } . This is a non-increasing function of t, which satisfies p(t) 6 1 1− µ p2 ( 1√ 2 (t− 1) ) , t > 1 , p(1) 6 µ . 262 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Let tk = ( √ 2)k+1 − 1√ 2− 1 , k > 0 , that is, t0 = 1, and tk = 1√ 2 (tk+1 − 1). Then, by induction on k, we have p(tk) 6 (1− µ) ( µ 1− µ )2k . Since t2k+1 = [( √ 2)k+2 − 1√ 2− 1 ]2 < 2k+2 ( √ 2− 1)2 = 4 ( √ 2− 1)2 2k , we see that for tk 6 t 6 tk+1, p(t) 6 p(tk) < e−at2 with a = 1 4 ( √ 2− 1)2 log µ 1− µ > 0 , completing the proof. A.9. Kolmogorov’s theorem Here, we formulate a version of the classical Kolmogorov’s theorem for Ck,k kernels. Let k ∈ N and let, as before, V ⊂ Rm be an open set. Definition A.3. We say that a symmetric function K : V ×V → R belongs to Ck,k(V ×V ) if all partial derivatives of K including at most k differentiations in x variables and at most k differentiations in y variables exist and are continuous on V × V (in which case, the order of differentiations does not matter and we can denote these derivatives ∂α x ∂β y K(x, y) as usual). Kolmogorov’s theorem. Let k ∈ N. Suppose that K : V × V → R is a positive definite symmetric function of class Ck,k(V × V ) and, in addition, that NV,k(K) def= max \|α\|,\|β\|6k sup x,y∈V ∣∣∂α x ∂β y K(x, y) ∣∣ < ∞ . Then there exists a (unique up to an equivalence) Ck−1 Gaussian function f on V with the covariance kernel K. Moreover, for every γ ∈ (0, 1) and every closed ball B̄ ⊂ V , we have E{‖f‖B̄, k−1+γ } 6 C(B̄, V, k, γ) √ NV,k(K) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 263 F. Nazarov and M. Sodin Note that since every compact set Q ⊂ V can be covered by a finite union of closed balls contained in V , the latter estimate immediately implies that, for any compact set Q ⊂ V , E{‖f‖Q, k−1 } 6 C(Q,V, k) √ NV,k(K) . The same is true for the Hölder norm, but the cover should be chosen carefully so that any two sufficiently close points x, y ∈ Q are covered by a single ball. Then the resulting bound on Q depends on both the bounds on the balls and the geometry of the cover. We will never need to estimate the Hölder norms on any compact set other than a ball, so we will not go into the details here. It is also worth noting that in the assumptions of Kolmogorov’s theorem we use that NV,k(K) < ∞ instead of the more natural for a function defined on an open set assumption that NQ,k(K) < ∞ for every compact set Q ⊂ V . This allows us to reduce the number of nested compact sets we need to choose before doing any estimate. Of course, this replacement it is harmless. A.10. Proof of Kolmogorov’s theorem To prove Kolmogorov’s theorem we will use a “convolution approach”. As far as high order derivatives are concerned, this approach allows one to pass to the limits in a family of covariance kernels easier than the more usual approach based on nets (see, for instance, [15, Sec. 3.1]). We split the proof into several steps. A.10.1. As we have seen in A.4., there exists a separable Hilbert spaceH and a continuous mapping V 3 x 7→ fx ∈ H such that K(x, y) = 〈fx, fy〉. Without loss of generality, we assume that H is a Gaussian subspace of L2(Ω,P), where (Ω,S,P) is a probability space. Our first task is to implement the mapping x 7→ fx as a B(V )×S-measurable function of (x, ω). We start with implementing each fx as an everywhere defined function on Ω. Then we pick a compact exhaustion Qn of V , a sequence εn > 0 with ∑ n ε2 n < ∞, and choose ρn > 0 so small that ‖fx− fy‖2 L2(Ω,P) < ε2 n for all x ∈ Qn, y ∈ V with \|x−y\| < ρn. We fix a countable partition of V into Borel sets Vj,n of diameter less than ρn each, choose some point xj,n in every Vj,n and put fn(x, ω) = fxj,n(ω) if x ∈ Vj . Then, for every x ∈ Qn, ‖fn−fx‖2 L2(Ω,P) < ε2 n, so for each t > 0, we have P{\|fn(x, ω)− fx(ω)\| > t } < t−2 ε2 n . Since ∑ n ε2 n < +∞ and Qn exhaust V , the functions fn(x, · ) converge to fx both P-almost surely and in L2(Ω,P). 264 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components Let now E = {(x, ω) : lim n→∞ fn(x, ω) exists }. Since fn is B(V )×S-measurable, so is E. Also, for every x ∈ V , we have P{ω : (x, ω) /∈ E} = 0. Thus, f(x, ω) def= { limn→∞ fn(x, ω), (x, ω) ∈ E, 0, otherwise is a measurable representation of the mapping x 7→ fx. A.10.2. Denote Fω(x) = f(x, ω). By Fubini, for every compact Q ⊂ V , E {∫ Q \|Fω\|2 d vol } = ∫ Q ‖fx‖2 L2(Ω,P) d vol(x) 6 max x∈Q K(x, x) volQ 6 NV,k(K)2 volQ . Thus Fω ∈ L2 loc(V ) for every ω ∈ Ω1 ⊂ Ω with P(Ω1) = 1. Replacing f(x, ω) by f(x, ω)1lΩ1(ω), we will assume that f(x, ω) is such that Fω ∈ L2 loc(V ) for all ω ∈ Ω. A.10.3. Next, we note that for every ϕ ∈ C∞ 0 (B(r)), the convolution in the x variable (f ∗x ϕ)(x, ω) def= (Fω ∗ ϕ)(x) is a Ck (actually, C∞) Gaussian function on V−r for every ω ∈ Ω. The only non-trivial part of this claim is the Gaussian distribution property. To see it, observe that, as an element of H ⊂ L2(Ω,P), (f ∗x ϕ)(x, · ) = ∫ B(r) fx+y ϕ(y) d vol(y) . The integral on the RHS can be understood as the usual Riemann integral of a continuous L2(Ω,P)-valued function, and hence, it can be approximated in L2(Ω,P) by finite Riemann sums ∑ j cjfx+yj ∈ H and, therefore, lies in H itself. In what follows, we write f ∗ ϕ instead of f ∗x ϕ and view f ∗ ϕ as a random Gaussian function. A.10.4. We shall need a few estimates for f ∗ϕ and its derivatives ∂α(f ∗ϕ) = f ∗ ∂αϕ for \|α\| 6 k − 1. First of all, by Fubini, E{[ ∂α(f ∗ ϕ)(z) ]2} = ∫∫ B(r)×B(r) K(z + x, z + y)∂αϕ(x)∂αϕ(y) d vol(x) d vol(y) = ∫∫ B(r)×B(r) ∂α x ∂α y K(z + x, z + y)ϕ(x)ϕ(y) d vol(x) d vol(y) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 265 F. Nazarov and M. Sodin The expression on the right is trivially bounded by ‖ϕ‖2 L1 NV,k(K). If, in addition, the function ϕ has zero integral mean, we can improve our trivial bound to Cr2 ‖ϕ‖2 L1 NV,k(K). To see this, we put Eα(z;x, y) = ∂α x ∂α y K(z+x, z+y)−∂α x ∂α y K(z, z+y)−∂α x ∂α y K(z+x, z)+∂α x ∂α y K(z, z) and note that by “bilinear” Lagrange’s Mean-Value Theorem, \|Eα(z; x, y)\| 6 Cr2 NV,k(K). Then, writing ∂α x ∂α y K(z+x, z+y)=−∂α x ∂α y K(z, z)+∂α x ∂α y K(z, z+y)+∂α x ∂α y K(z+x, z)+Eα(z; x, y) and integrating in x and y against ϕ(x)ϕ(y) d vol(x) d vol(y), we obtain E{[ ∂α(f ∗ ϕ)(z) ]2} = ∫∫ B(r)×B(r) Eα(z; x, y) d vol(x) d vol(y) 6 Cr2 ‖ϕ‖2 L1 NV,k(K) . A.10.5. We shall also need the following “entropy bound”: Lemma A.5 (entropy bound). Let r > 0. Let g be a continuous Gaussian function on V and ψ be any C∞ 0 (B(r))-function. Then g ∗ ψ is a continuous Gaussian function on V−r and for every two compact sets Q,Q′ ⊂ V such that Q+r ⊂ Q′, we have E{‖g ∗ ψ‖C(Q) } 6 5‖ψ‖L1 √ 1 + log ‖ψ‖L∞ volQ′ ‖ψ‖L1 √ sup Q′ E{\|g\|2} . P r o o f of the entropy bound. Without loss of generality, we assume that supQ′ E{\|g\|2} = 1. Then for every x ∈ Q′, g(x) is a Gaussian random variable with E{g(x)2} 6 1. Hence, Ee 1 4 g(x)2 6 1√ 2π ∫ R e 1 4 x2 e− 1 2 x2 dx = √ 2 . Take ρ > √ 2. Noting that the function ρe− 1 4 ρ2 decreases on [ √ 2,+∞), we esti- mate the convolution by (g ∗ ψ)(x) = ∫ B(r) g(x + y)ψ(y) d vol(y) = ∫ B(r)∩{\|g\|6ρ} g(x + y)ψ(y) d vol(y) + ∫ B(r)∩{\|g\|>ρ} g(x + y)ψ(y) d vol(y) 6 ρ‖ψ‖L1 + ρe− 1 4 ρ2‖ψ‖L∞ ∫ Q′ e 1 4 g2 d vol , 266 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components so E‖g ∗ ψ‖C(Q) 6 ρ [‖ψ‖L1 + e− 1 4 ρ2‖ψ‖L∞ √ 2 volQ′]. Taking ρ = 2 √ 1 + log ‖ψ‖L∞ volQ′ ‖ψ‖L1 (which is > 2 because ‖ψ‖L1 6 ‖ψ‖L∞ volB(r) 6 ‖ψ‖L∞ volQ′) and using that 2(1 + √ 2) < 5, we get the desired bound. A.10.6. Now, we fix ϕ > 0 in C∞ 0 (B(1)) with ∫ ϕd vol = 1. For r > 0, let ϕr(x) = r−mϕ(r−1x) and note that ‖ϕr‖L1 = 1 and ‖ϕr‖L∞ 6 Cr−m for all r > 0. Take a sequence rj = 2−j−1 and put fj = f ∗ ϕrj ∗ ϕrj . Then fj are Ck Gaussian functions on V−2rj , and fj(x, · ) → fx, as j → ∞, in L2(Ω,P) for all x ∈ V . Next, we fix a closed ball B̄ = B̄(x, r) ⊂ V and choose j0 so large that B̄(x, r + 2rj0) ⊂ V . Consider the series fj0 + ∑ j>j0 (fj+1 − fj) . (A.1) If we show that for every α with \|α\| 6 k − 1, the expression E‖∂αfj0‖C(B̄) + ∑ j>j0 E‖∂αfj+1 − ∂αfj‖C(B̄) is bounded by C(B̄, j0) √ NV,k(K), and that for every α with \|α\| = k − 1 and every γ ∈ (0, 1), the expression E‖∂αfj0‖B̄,γ + ∑ j>j0 E‖∂αfj+1 − ∂αfj‖B̄,γ is bounded by C(B̄, j0, γ) √ NV,k(K), then we will be done because then the series (A.1) will converge in Ck−1(V ) almost surely, its sum will be a Gaussian function f with the covariance kernel K, and the desired bounds for E‖f‖B̄,k−1+γ will hold as well. A.10.7. For a multi-index α with \|α\| 6 k, we write ∂αfj0 = ∂α(f ∗ϕrj0 )∗ϕrj0 and note that the function g = ∂α(f ∗ ϕrj0 ) satisfies E{g(x)2} 6 NV,k(K). So Lemma A.5 yields the bound E‖∂αfj0‖C(B̄) 6 C(B̄, j0) √ NV,k(K). The interest- ing part is E‖∂αfj+1 − ∂αfj‖C(B̄). If \|α\| 6 k − 1, writing ∂αfj+1 − ∂αfj = ∂α(f ∗ (ϕrj+1 − ϕrj )) ∗ (ϕrj+1 + ϕrj+1) applying the entropy bound with g = ∂α(f ∗ (ϕrj+1 − ϕrj )) and ψ = ϕrj+1 + ϕrj , and recalling that, by A.10., E{\|g\|2} 6 Cr2 j NV,k(K), we see that E{‖∂α(fj+1 − fj)‖C(B̄) } 6 C rj √ 1 + log(Cr−m j+1) √ NV,k(K) Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 267 F. Nazarov and M. Sodin with some C = C(B̄). Since ∑ j rj √ 1 + log(Cr−m j+1) < ∞, this takes care of the first of the series in A.10. including the uniform norms of the derivatives of f of order up to k − 1. A.10.8. To get convergence of the series ∑ j>j0 E{‖∂αfj+1 − ∂αfj‖B̄,γ } (A.2) for a multi-index α with \|α\| = k − 1, we need the bound for E‖∇∂αfj+1 − ∇∂αfj‖C(B̄). Note that despite we still have convolutions with mean zero func- tions in the representation of ∇∂αfj , we cannot use our trick from A.10. because the kernel smoothness is totally exhausted. Thus, we can use only the trivial estimate from A.10. without the factor rj , and the entropy bounds yields E{‖∇∂αfj+1 −∇∂αfj‖C(B̄) } 6 C √ 1 + log(Cr−m j+1) √ NV,k(K) . There is no hope to choose rj so that these terms will form a convergent series, so there is no chance to show on this way that the k − 1-st order derivatives are Lipschitz. Fortunately, we do not need that much. All we really need is Hölder continuity. Using a classical trick, we observe that for any function h that is C1 in some neighbourhood of B̄, and for any two points x, y ∈ B̄, we have? \|h(x)−h(y)\| 6 min [ 2‖h‖C(B̄), ‖∇h‖C(B̄)\|x−y\|] 6 21−γ‖h‖1−γ C(B̄) ‖∇h‖γ C(B̄) \|x−y\|γ . By Hölder’s inequality, E { ‖∂αfj+1 − ∂αfj‖1−γ C(B̄) ‖ ‖∇∂αfj+1 −∇∂αfj‖γ C(B̄) } 6 ( E‖∂αfj+1 − ∂αfj‖C(B̄) )1−γ ( E‖∇∂αfj+1 −∇∂αfj‖C(B̄) )γ , and, by the entropy bound, the RHS is . r1−γ j ( 1 + log(Cr−m j+1) ) √ NV,k(K) . Hence, the series (A.2) converges and the proof of Kolmogorov’s theorem is com- plete. ?We use the inequality min(a, b) 6 a1−γbγ valid for positive a and b and for γ ∈ (0, 1). 268 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components A.11. Remarks to Kolmogorov’s theorem A.11.1. Kolmogorov’s theorem, as stated and proved, allows us to estimate E‖f‖B̄,k−1+γ . However, applying then Fernique’s theorem, we immediately see that, in assumptions of Kolmogorov’s theorem, we can estimate any moment E‖f‖p B̄,k−1+γ we want (that would be exactly as much as we use in this paper), and even prove that the distribution tail P{‖f‖B̄,k−1+γ > t } at t → +∞ is Gaussian with controllable bounds. Indeed, we take X = Ck−1(B̄), X = f , fix a countable dense set B′ ⊂ B̄, and put ϕα,x(f) = ∂αf(x), \|α\| 6 k − 1, x ∈ B′ , ϕα,γ,x(f) = ∂αf(x)− ∂αf(y) \|x− y\|γ , \|α\| = k − 1, x, y ∈ B′ , x 6= y . Note that this is a countable system of linear functionals {ϕj} ⊂ X∗ satisfying the assumptions of Fernique’s theorem, and that ‖f‖B̄,k−1+γ = supj \|ϕj(f)\|. By Kolmogorov’s theorem, there exists a positive constant λ = λ(B̄, V, k, γ) such that P{‖f‖B̄,k−1+γ > λ √ NV,k(K) } < 1 4 . Then Fernique’s theorem tells us that P{‖f‖B̄,k−1+γ > tλ √ NV,k(K) } < e−at2 , t > 1 , whence, P{‖f‖B̄,k−1+γ > t } < C(B, V, k, γ)e−c(B,V,k,γ)t2/NV,k(K) , t > 0 . (A.1) In particular, E{‖f‖p B̄,k−1+γ } 6 C(B, V, k, γ) N p/2 V,k (K) . It is worth mentioning that one can also arrive at estimate (A.1) directly after a certain modification of the proof of Kolmogorov’s theorem we gave. A.11.2. We have to distinguish between Ck Gaussian functions on U and Gaussian functions with Ck,k(U × U) covariance kernels: the former are always the latter but, in general, not vice versa. However, by Kolmogorov’s theorem, the continuous Gaussian functions with Ck,k(U ×U) covariance kernels fail to be in Ck themselves just barely: they all are in Ck−(U) = ⋂ 0<γ<1 Ck−1+γ(U). A.11.3. The “convolution approach” to Kolmogorov’s theorem allows one to approximate Gaussian functions of finite smoothness by C∞ ones. This approxi- mation can be used to establish some properties of the kernel. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 269 F. Nazarov and M. Sodin Using this idea, we will show now that every semi-norm ‖K‖Q,k of a positive definite Ck,k(U × U) kernel can be read from the “diagonal”: max \|α\|,\|β\|6k max x,y∈Q ∣∣∂α x ∂β y K(x, y) ∣∣ = max \|α\|6k max x∈Q ∣∣∂α x ∂α y K(x, y) ∣∣ y=x ∣∣ . Indeed, if \|α\|, \|β\| 6 k − 1, then we can write ∣∣∂α x ∂β y K(x, y) ∣∣2 = ∣∣E{ ∂αf(x) ∂βf(y) }∣∣2 6 E{ [∂αf(x)]2 } E{ [∂βf(y)]2 } = ( ∂α x ∂α y K(x, y) ∣∣ x=y ) ( ∂β x ∂β y K(x, y) ∣∣ y=x ) for the Ck− Gaussian function f with the covariance kernel K, thus estimating the off-diagonal values by the square root of the product of the two corresponding diagonal ones. We cannot do the same estimate directly for the highest order derivatives, but we can consider the convolutions f ∗ϕ that are infinitely smooth and get the inequality ∣∣∂α x ∂β y Kϕ(x, y) ∣∣2 6 ( ∂α x ∂α y Kϕ(x, y) ∣∣ x=y ) ( ∂β x ∂β y Kϕ(x, y) ∣∣ y=x ) (A.2) for the corresponding covariance kernels Kϕ(x, y) = ∫∫ K(x + x′, y + y′)ϕ(x′)ϕ(y′) d vol(x′)d vol(y′) . Taking ϕ1 ∈ C0(B(1)) and ϕ(x) = ϕr(x) = r−mϕ1(r−1x), we can pass to the limit ∂α x ∂β y Kϕr(x, y) → ∂α x ∂β y K(x, y) as r → 0 , for any \|α\|, \|β\| 6 k, x, y ∈ U , we conclude that (A.2) holds for K as well. Of course, here one can also work with the kernel directly, approximating the derivatives by finite difference ratios and passing to the limit in some inequalities for long sums. A.11.4. The convolutions also facilitate convergence: if the kernels K` ∈ Ck,k(U × U) are uniformly bounded on compact subsets of U × U and converge pointwise to some kernel K on U×U , then (K`)ϕ → Kϕ in C∞(U−r×U−r) for any ϕ ∈ C∞ 0 (B(r)). If we know, in addition, that for \|α\|, \|β\| 6 k, the partial deriva- tives ∂α x ∂β y K`(x, y) are uniformly locally bounded as well, we can use elementary analysis to show that K ∈ Ck−1,k−1(U × U) and ∂α x ∂β y K`(x, y) → ∂α x ∂β y K(x, y) for \|α\|, \|β\| 6 k − 1 uniformly on compact subsets of U ×U . However, in general, it is impossible to conclude that K ∈ Ck,k(U × U). Surprisingly, this conclusion holds if the limiting kernel K is translation invariant, i.e., K(x, y) = κ(x− y) for some κ : Rm → R. This will be shown in the next section. 270 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components A.12. Translation-invariant Gaussian functions A continuous Gaussian function on Rm is translation-invariant if its covariance kernel K(x, y) depends on x−y only, i.e., K(x, y) = κ(x−y) for some continuous positive definite κ : Rm → R. In this case, κ can be written as a Fourier integral of some finite symmetric positive Borel measure ρ on Rm, i.e., κ(x) = ∫ Rm e2πi(λ x) dρ(λ) . Consider the Hilbert space L2 H(ρ) of all Hermitean (h(−x) = h(x)) functions h : Rm → C with ∫ \|h\|2 dρ < ∞. The standard L2(ρ) scalar product 〈h1, h2〉 =∫ h1h̄2 dρ is real on L2 H(ρ). Also, for every x ∈ Rm, the function fx(λ) = e2πi(λ x) belongs to L2 H(ρ) and 〈fx, fy〉 = κ(x−y). Finally, the linear span of the functions fx is dense in L2 H(ρ). Indeed, if h ∈ L2 H(ρ), then Φ[h](x) = 〈h, fx〉 is the Fourier transform of the finite Borel measure hdρ. Hence, it vanishes identically only if h = 0 ρ-a.e. . Bringing all these observations together, we conclude that • the Hilbert space H(K) coincides with the Fourier image F L2 H(ρ). Now, we discuss the smoothness properties of translation invariant Gaussian functions and covariance kernels. First of all, note that if K(x, y) = κ(x − y), then ∂α x ∂β y K(x, y) = (−1)\|β\| ( ∂α+βκ ) (x− y) . Thus, K is in Ck,k(Rm ×Rm) if and only if κ ∈ C2k(Rm), that is, if and only if, ∫ Rm \|λ\|2k dρ(λ) < ∞ . (A.1) We end this section with a curious and quite useful observation: • if a sequence of positive definite kernels K` ∈ Ck,k(U`, U`) with U` exhausting Rm has a pointwise translation invariant limit κ(x−y) and ∂α x ∂α y K`(x, y) ∣∣ x=y=0 and stays bounded for \|α\| 6 k, then κ ∈ C2k(Rm). P r o o f. For ϕ ∈ C∞ 0 (Rm), ϕ(−x) = ϕ(x) and put Kϕ(x, y) = (κ∗ϕ∗ϕ)(x− y). Since κ = Fρ implies that κ ∗ ϕ ∗ ϕ = Fρϕ, where dρϕ = ϕ̂ 2 dρ, we see that (−1)k ∑ \|α\|=k ∂2α(κ ∗ ϕ ∗ ϕ)(0) = (2π)2k ∑ \|α\|=k ∫ Rm λ2α1 1 . . . λ2αm m dρϕ(λ) = (2π)2k ∫ Rm \|λ\|2k dρϕ(λ) . Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 271 F. Nazarov and M. Sodin If we know in advance that κ ∈ C2k(Rm), then the quantities ∂2α(κ ∗ ϕ ∗ ϕ)(0), \|α\| = k, are uniformly bounded when ϕ runs over even non-negative C∞ 0 functions supported on a small ball centered at the origin and normalized by ∫ Rm ϕd vol = 1. Then, taking as before, ϕr(x) = r−mϕ(r−1x), letting r → 0, and applying Fatou’s lemma, we get ∫ Rm \|λ\|2k dρ(λ) 6 lim r→0 ∫ Rm \|λ\|2k dρϕr(λ) < ∞ . Now, observe that the quantities ∂2α(κ∗ϕ∗ϕ)(0) stay uniformly bounded even if κ(x− y) is a pointwise limit of Ck,k positive definite symmetric kernels K`(x, y) that are defined only in a neighbourhood of the origin in Rm×Rm and that have uniformly bounded derivatives ∂α x ∂α y K`(x, y) ∣∣ x=y=0 . So, in this case, we still get ∫ Rm \|λ\|2k dρ(λ) < ∞ , completing the proof of our observation. ˘ B. Proof of the Fomin–Grenander–Maruyama Theorem Assuming that ρ has no atoms, we need to show that if A ∈ S is a set satisfying γ ( (τvA)4A ) = 0 for every v ∈ Rm, then γ(A) is either 0 or 1. As before, we use the notation (τvG)(u) = G(u + v), where v ∈ Rm and G ∈ X. Since S is generated by the intervals I(u; a, b), given ε > 0, we can take finitely many points u1, . . . , un ∈ Rm and a Borel set B ⊂ Rn so that γ{A4P} < ε, where P = P (u1, . . . , un; B) def= { G ∈ X : (G(u1), . . . , G(un)) ∈ B } . Without loss of generality, we may assume that the distribution of the Gaussian vector ( G(u1), . . . , G(un) ) is non-degenerate?. In this case, we can write γ ( P (u1, . . . , un; B) ) = (2π)−n/2 ( detΛ )− 1 2 ∫ B e− 1 2 (Λ−1t t) d vol(t) , ?Otherwise one of the values, say, G(un), is a linear combination of other values with prob- ability 1. If G(un) = ∑n−1 j=1 cjG(uj) is such a representation, then γ ({ G ∈ X : (G(u1), . . . , G(un)) ∈ B }4{ G ∈ X : (G(u1), . . . , G(un−1)) ∈ B′}) = 0 , where B′ = { (t1, . . . , tn−1) ∈ Rn−1 : ( t1, . . . , tn−1, ∑n−1 j=1 cjtj ) ∈ B } is a Borel set in Rn−1, so we can remove the point un from the consideration at no cost. 272 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components where Λ = ( k(ui−uj) )n i,j=1 is the covariance matrix of the vector ( G(u1), . . . , G(un) ) . As before, we denote by k the Fourier integral of the spectral measure ρ. Since τvP = P (u1 + v, . . . , un + v; B), we have P ∩ τvP = P (u1, . . . , un, u1 + v, . . . , un + v; B ×B) . Then γ ( P ∩ τvP ) = (2π)−n ( det Λ̃ )− 1 2 ∫ B×B e− 1 2 (Λ̃−1(v)t̃ t̃) d vol( t̃ ) where Λ̃(v) = ( Λ Θ(v) Θ∗(v) Λ ) with Θi,j(v) = k(ui − v − uj) . Note that the matrix Λ̃(v) is invertible and ( Λ̃(v) )−1 is close to ( Λ−1 0 0 Λ−1 ) if ‖Θ(v)‖ is small enough. Next, we observe that we can choose a sequence v` ∈ Rm so that ‖Θ(v`)‖ → 0 as ` →∞. Indeed, letting ∆ = maxi,j \|ui − uj \|, we have 1 volB(R) ∫ B(R) ∑ i,j k(ui − v − uj)2 d vol(v) 6 n2 volB(R) ∫ B(R+∆) k2 d vol , while by Wiener’s theorem [16, VI.2.9], the absence of atoms in ρ is equivalent to lim R→∞ 1 volB(R) ∫ B(R) k2 d vol = 0 . Then, using the dominated convergence theorem, we conclude that lim `→∞ γ ( P ∩ τv` P ) = γ ( P )2 . Recalling that A ∩ τv` A = A up to γ-measure 0, we obtain γ(A) = γ(A ∩ τv` A) 6 γ(P ∩ τv` P ) + 2ε `→∞→ γ(P )2 + 2ε 6 [ γ(A) ]2 + 2ε . Since ε > 0 is arbitrary, we conclude that γ(A) 6 γ(A)2, whence γ(A) = 0 or γ(A) = 1. C. Condition (ρ4) Here, we collect several observations that, in many instances, help to verify condition (ρ4). Recall that this condition asserts that • there exist a finite compactly supported Hermitian measure µ with spt(µ) ⊂ spt(ρ) and a bounded domain D ⊂ Rm such that Fµ ∣∣ ∂D < 0 and (Fµ)(u0) > 0 for some u0 ∈ D. Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 273 F. Nazarov and M. Sodin Throughout this section, we assume that condition (ρ3) is satisfied, that is, that the measure ρ is not supported on a hyperplane in Rm. C.1. Quadratic hypersurface criterion The support of any measure ρ not satisfying condition (ρ4) must be contained in a quadratic variety Aλλ = b, where A is an m × m symmetric matrix and b ∈ Rm. P r o o f. Suppose that spt(ρ) is not contained in any quadratic variety of the above form. Then 1 2m(m + 1) + 1-dimensional vectors v(λ) = { 1, λ(i)λ(j) : 1 6 i 6 j 6 m } , λ ∈ spt(ρ), span R 1 2 m(m+1)+1 (here λ(i) denotes the i-th coordinate of λ). Then we can create two finite linear combinations of cosines: f(x) = ∑ λ∈spt(ρ) aλ cos ( 2πλ x ) , g(x) = ∑ λ∈spt(ρ) bλ cos ( 2πλx ) , such that f(0) = 1, (D2f)(0) = 0 , and g(0) = 0, (D2g)(0) = I , where D2f is the matrix with the entries ∂2 xi xj f and I is the unit matrix. Note that we also automatically have Df(0) = Dg(0) = 0, Then the function h = ε2f − g will satisfy h(0) = ε2 and h(x) < 0 on {\|x\| = 2ε } , provided that ε is small enough. C.2. Pjetro Majer’s interior point criterion The next observation is due to Pietro Majer. Let the interior of the convex hull of spt(ρ) contain a point from spt(ρ). Then condition (ρ4) is satisfied. In particular, condition (ρ4) is satisfied when 0 ∈ spt(ρ). P r o o f. Let υ be such a point. Since υ lies in the interior of the convex hull of spt(ρ), there are λ1, . . . , λn ∈ spt(ρ) that span the whole space Rm, such that υ = ∑ i tiλi , ti > 0, ∑ i ti = α < 1 . Consider the function f(x) = ∑ i bi cos ( 2πλi x )− cos(2πυ x) 274 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 Asymptotic Laws for the Number of Connected Components with bi = αti + n−1(1− α2 + ε), where ε > 0. Then, for x → 0, f(x) = [ ∑ i bi − 1 ] −2π2 [ ∑ i bi(λi x)2 − (υ x)2 ] +o(\|x\|2) . In particular, f(0) = ∑ i bi − 1 = ε > 0 . Next, we note that (υ x)2 = (∑ i tiλi x )2 = (∑ i t 1/2 i t 1/2 i λi x )2 6 (∑ i ti )(∑ i ti(λi x)2 ) = α ∑ i ti(λi x)2 . Now, suppose that x belongs to the non-degenerate ellipsoid E = {∑ i (λi x)2 = εn π2(1− α2) } . Since λ1, . . . , λn span Rn, we have \|x\|2 = O(ε) , ε → 0, x ∈ E . Therefore, for x ∈ E and ε → 0, we have f(x) 6 ε− 2π2 ∑ i (bi − αti)(λi x)2 + o(ε) = ε− 2π2(1− α2 + ε) n ∑ i (λi x)2 + o(ε) < ε− 2ε + o(ε) < 0 , completing the proof. C.3. Analytic closure criterion Our last observation is that • the requirement spt(µ) ⊂ spt(ρ) in condition (ρ4) can be relaxed to the requirement spt(µ) ⊂ spt r.a.(ρ) where spt r.a.(ρ) is the intersection of all real-analytic varieties containing spt(ρ). Note that every quadratic variety is an analytic variety as well, so if spt(ρ) ⊂ V then spt r.a.(ρ) ⊂ V too. Sometimes, spt r.a.(ρ) is much larger that spt(ρ) and satisfy the assumption of C.2. (or some other condition sufficient for establishing (ρ4) without spt(ρ) doing so). For instance, suppose that m = 2, S ⊂ R2 is the unit circumference, and spt(ρ) ⊂ S is an infinite set. Since infinite subsets of S Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 3 275 F. Nazarov and M. Sodin are uniqueness sets for real-analytic functions on S, we see that spt r.a.(ρ) = S. Then, taking µ = m1 (the Lebesgue measure on S), we conclude that condition (ρ4) is satisfied. P r o o f. Let Q ⊂ Rm be a compact set. Consider two linear subspaces of the space C(Q) of real-valued continuous functions on Q: X = {Fµ : µ is Hermitian, compactly supported, spt(µ) ⊂ spt(ρ) } and X r.a. = {Fµ : µ is Hermitian, compactly supported, spt(µ) ⊂ spt r.a.(ρ) } . We need to show that the C(Q)-closure of X contains X r.a.. We will be using a simple duality argument. Suppose that a signed measure ν supported by Q annihilates X, that is, ∫ Q (Fµ ) dν = 0 for all admissible µ . 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