Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matr...

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spelling irk-123456789-1478152019-02-17T01:24:13Z Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices Etingof, P. Ma, X. Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is βz2, this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type Âm-1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model. 2007 Article Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices / P. Etingof, X. Ma // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 12 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 15A52 http://dspace.nbuv.gov.ua/handle/123456789/147815 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is βz2, this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type Âm-1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.
format Article
author Etingof, P.
Ma, X.
spellingShingle Etingof, P.
Ma, X.
Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Etingof, P.
Ma, X.
author_sort Etingof, P.
title Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
title_short Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
title_full Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
title_fullStr Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
title_full_unstemmed Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
title_sort density of eigenvalues of random normal matrices with an arbitrary potential, and of generalized normal matrices
publisher Інститут математики НАН України
publishDate 2007
url http://dspace.nbuv.gov.ua/handle/123456789/147815
citation_txt Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices / P. Etingof, X. Ma // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 12 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
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AT max densityofeigenvaluesofrandomnormalmatriceswithanarbitrarypotentialandofgeneralizednormalmatrices
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last_indexed 2025-07-11T02:53:38Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 048, 13 pages Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices? Pavel ETINGOF ∗ and Xiaoguang MA Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 USA E-mail: etingof@math.mit.edu, xma@math.mit.edu ∗ URL: http://www-math.mit.edu/∼etingof/ Received December 05, 2006, in final form March 03, 2007; Published online March 14, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/048/ Abstract. Following the works by Wiegmann–Zabrodin, Elbau–Felder, Hedenmalm–Maka- rov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is β|z|2, this is done by Wiegmann–Zabrodin and Elbau–Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to ∗-representations of the deformed preprojective algebra of the affine quiver of type Âm−1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model. Key words: Hele-Shaw flow; equilibrium measure; random normal matrices 2000 Mathematics Subject Classification: 15A52 1 Introduction The normal matrix model became a focus of attention for many mathematical physicists after the recent discovery (see e.g. [11, 6, 7, 8]) of its unexpected connections to the 2-dimensional dispersionless Toda hierarchy and the Laplacian growth model (which is an exactly solvable model describing free boundary fluid flows in a Hele-Shaw cell or porous medium). The original normal matrix model contained a potential function whose Laplacian is a positive constant, but later in [12], Wiegmann and Zabrodin considered a more general model, where the potential function was arbitrary. This is the model we will consider in this paper. In the normal matrix model with an arbitrary potential function, one considers the ran- dom normal matrices of some size N with spectrum restricted to a compact domain D1 and probability measure PN (M)dM = Z−1 N exp(−NtrW (M))dM, ?This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html 1The compactness of D is needed to guarantee convergence of the arising integrals. mailto:etingof@math.mit.edu mailto:xma@math.mit.edu http://www-math.mit.edu/~etingof/ http://www.emis.de/journals/SIGMA/2007/048/ http://www.emis.de/journals/SIGMA/kuznetsov.html 2 P. Etingof and X. Ma where dM is the measure on the space of normal matrices induced by the Euclidean metric on all complex matrices, W is a potential function (a real function on C with some regularity properties, e.g. continuous), and ZN is a normalizing factor. In the original works on the normal matrix model, the potential was W (z) = β|z|2 − P (z)− P (z), where P is a complex polynomial of some degree d, and β a positive real number. For this type of potential, it was shown in the works [11, 6, 7, 8] (and then proved rigorously in [3]) that under some conditions on the potential, the asymptotic density of eigenvalues is uniform with support in the interior domain of a closed smooth curve. This curve is a solution of an inverse moment problem, appearing in the theory of Hele-Shaw flows with a free boundary. Thus, applying the conformal mapping method (see [10] and references therein), one discovers that the conformal map of the unit disk onto the outside of this curve which maps 0 to ∞ is a Laurent polynomial of degree d. This allows one to find the curve explicitly up to finitely many parameters, which can be found from a finite system of algebraic equations. In [12], Wiegmann and Zabrodin generalized this analysis to an arbitrary potential function. They showed that the density of eigenvalues is the Laplacian of the potential function, and the eigenvalues are concentrated in the domain which can be determined from an appropriate inverse moment problem. This was proved rigorously in the paper [5], which extends the Elbau–Felder work to the case of an arbitrary potential. One of the goals of the present paper is to use the generalized conformal mapping method, developed in [4] by Entov and the first author for studying Hele-Shaw flows with moving bound- ary for curved surfaces, to calculate the boundary of the region of eigenvalues explicitly in the case when W (z) = Φ(|z|2)− P (z)− P (z), (1) where Φ is a function of one variable. In this case, the conformal map of the disk onto the outside of the curve is no longer algebraic, but one can still give an explicit answer in terms of a contour integral. Another goal is to extend the above results to the case of generalized normal matrix model. In this model, we consider block complex matrices of a certain kind with commutatation relations similar to the definition of a normal matrix; they give rise to ∗-representations of the deformed preprojective algebra of the affine quiver of type Âm−1. We prove that the problem of computing the asymptotic eigenvalue distribution for this model, as the size of the matrices goes to infinity, is equivalent to the same problem for the usual normal matrix model. This allows one to find the boundary of the eigenvalue region explicitly if the potential is given by (1). The structure of this paper is as follows. In Section 2, we state some basic facts about the normal matrix model. In Section 3, we define the generalized normal matrix model, and write down the probability measure in this model. In Section 4, we recall some facts about the equilibrium measure and explain that the asymptotic eigenvalue distribution tends to the equilibrium measure in the normal matrix model and the generalized normal matrix model. In Section 5, we use the singular point method from [4, 10] to reconstruct the boundary of the support domain of the equilibrium measure. 2 The normal matrix model with an arbitrary potential function Let D be a domain in the complex plane C. We consider the set N (D) = {M ∈ MatN (C)|[M,M †] = 0, spectrum(M) ⊂ D} Density of Eigenvalues of Random Normal Matrices 3 of normal matrices with spectrum in the domain D. Let dM be the measure on N (D) induced by the Euclidean metric on MatN (C). It is well known (see e.g. [9, 1]), that in terms of the eigenvalues this measure on N (C) is given by the formula dM = ∏ 1≤i<j≤N |zi − zj |2 N∏ i=1 d2zidU, where M = Udiag(z1, . . . , zN )U †, U ∈ U(N), and dU denotes the normalized U(N)-invariant measure on the flag manifold U(N)/U(1)N . Now let W : C → R be a continuous function. If M is a normal matrix, then we can define W (M) to be diag(W (z1), . . . ,W (zN )) in an orthonormal basis in which M = diag(z1, . . . , zN ). It follows from the above that the probability measure on N (D) with potential function W is given by PN (M)dM = Z−1 N e−N ∑ i W (zi) ∏ 1≤i<j≤N |zi − zj |2 N∏ i=1 d2zidU, (2) where ZN = ∫ DN e−N ∑ i W (zi) ∏ 1≤i<j≤N |zi − zj |2 N∏ i=1 d2zi. Here we assume that the integral is convergent (this is the case, for instance, if D is compact). 3 The generalized normal matrix model 3.1 Generalized normal matrices Let us consider the following generalization of normal matrices. Let m ≥ 1 be an integer. For a fixed collection λ = (λ1, . . . , λm) of real numbers such that ∑ i λi = 0, and a domain D, we define Nm(λ, D) to be the subset of A ∈ MatmN (C) satisfying the following conditions: for any A ∈ Nm(λ, D), • If Aij , i, j = 1, . . . ,m are N ×N blocks of A, then Aij = 0 unless j − i = 1 mod m; • The spectrum of A12A23 · · ·Am1 is contained in D; • [A,A†] =  λ1IN . . . λmIN , where IN is the identity matrix of size N . Note that N1(0, D) = N (D), thus elements of Nm(λ, D) are a generalization of normal matrices. We will thus call them generalized normal matrices. Remark 1. Generalized normal matrices are related in the following way to quiver representa- tions. Let Q be the cyclic quiver of type Âm−1, and Q̄ its double. Let ΠQ(λ) be the deformed preprojective algebra of Q with parameters λ (see [2]). By definition, this algebra is the quotient of the path algebra of Q̄ by the relation ∑ a∈Q [a, a∗] = ∑ λiei, where ei are the vertex idempotents. The algebra ΠQ has a ∗-structure, preserving ei and sending a to a∗ and a∗ to a. It is easy to see that Nm(λ, D) is the set of all matrix ∗-representations of ΠQ of dimension Nδ (where δ = (1, 1, . . . , 1) is the basic imaginary root) such that the spectrum of the monodromy operator a1 · · · am is in D. 4 P. Etingof and X. Ma Denote Ai,i+1 by Ai. The group U(N)m = U(N)× U(N)× · · · × U(N)︸ ︷︷ ︸ m times acts naturally on Nm(λ, C) by the formula (S1, . . . , Sm)A =  S1A1S † 2 S2A2S † 3 . . . Sm−1Am−1S † m SmAmS†1  . We have the following lemma, which is a generalization of the fact that a normal matrix diagonalizes in an orthonormal basis: Lemma 1. For any element A ∈ Nm(λ, D), we can find an element (S1, . . . , Sm) ∈ U(N)m such that (S1, . . . , Sm)A =  D1 D2 . . . Dm−1 Dm  , where Di are diagonal matrices. Proof. From the definition, we have AiA † i − A† i−1Ai−1 = λiIN , where the index is considered modulo m, and m∑ i=1 λi = 0. Now consider a collection of N -dimensional unitary spaces {Vi}m i=1, and let us regard Ai as a linear map Ai : Vi+1 → Vi. So AiA † i is a hermitian endomorphism of Vi. Now suppose that v is an eigenvector of AiA † i with eigenvalue ν. We claim that then Ai−1v (if it is nonzero) is an eigenvector of Ai−1A † i−1 with eigenvalue ν − λi. Indeed, Ai−1A † i−1Ai−1v = Ai−1(AiA † i − λi)v = (ν − λi)Ai−1v. Thus, denoting by Vi(ν) the eigenspace of AiA † i with eigenvalue ν, we find that Ai−1 : Vi(ν) → Vi−1(ν−λi). Since Vi = ⊕ν∈RVi(ν) (as AiA † i is hermitian), it suffices to prove the lemma in the case when AiA † i is a scalar in Vi, in which case the statement is easy. � 3.2 The Euclidean measure on generalized normal matrices First, let us consider the N = 1 case. Pick real numbers αi such that λi = αi − αi−1, and let Q(x) = m∏ i=1 (x + αi). A matrix A =  r1e iθ1 r2e iθ2 . . . rm−1e iθm−1 rmeiθm  , (where rj ≥ 0, θj ∈ [0, 2π)) is in Nm(λ, C) if and only if r2 1 − r2 m = λ1 = α1 − αm, Density of Eigenvalues of Random Normal Matrices 5 r2 2 − r2 1 = λ2 = α2 − α1, ... r2 m − r2 m−1 = λm = αm − αm−1. Thus to each A ∈ Nm(λ, C) we can attach a real number x = r2 i − αi, which is independent of i, and a complex number z = m∏ j=1 rje iθj . It is easy to see that the point (z, x) belongs to the surface Σ = { (z, x) ∈ C× R |x + αi ≥ 0 ∀ i, zz̄ = m∏ i=1 (x + αi) } . Moreover, it is clear that any point of Σ corresponds to some A, and two matrices A, A′ giving rise to the same point (z, x) are conjugate. This implies that we have a bijection between the equivalence classes in Nm(λ, C) under the action of U(1)m and points of Σ. Writing z = reiθ, we see that x, θ are coordinates on Σ, so we may write the Euclidean measure on Nm(λ, C) using the coordinates x, θ. Theorem 1. The Euclidean measure on Nm(λ, C) for N = 1 is: µ = 1 2 √ Q′(x)dxdθdU, where dU is the Haar measure on U(1)m/U(1). Proof. We have ri = √ x + αi. Thus the parametrized real curve {(r1(x), . . . , rm(x))|x ∈ R} has length element ds = √∑ i r ′ i(x)2dx = 1 2 √∑ i 1 x + αi dx. This implies that the Euclidean measure on Nm(λ, C) is dµ = 1 2 √∑ i 1 x + αi dx ∏ j rjdθj = 1 2 √∑ i ∏ j 6=i (x + αj)dxdθ1 · · ·dθm = 1 2 √ Q′(x)dxdθ1 · · ·dθm = 1 2 √ Q′(x)dxdθdU, as desired. � Let us now consider the case of general N . From Lemma 1, we know that under the action of U(N)m, the equivalence class of A ∈ Nm(λ, C) can be represented by m diagonal matrices M =  D1 D2 . . . Dm−1 Dm  , (3) where Di = diag(zi 1, . . . , z i N ). From the definition D1D † 1 −D† mDm = λ1IN , 6 P. Etingof and X. Ma D2D † 2 −D† 1D1 = λ2IN , ... DmD† m −D† m−1Dm−1 = λmIN . So we have zi jz i j − zi+1 j zi+1 j = αi − αi+1. Let xj = zi jz i j − αi and zj = ∏ i z i j , then we have |zj |2 = ∏ i (xj + αi), j = 1, . . . , N. Thus ((z1, x1), . . . , (zN , xN )) is a point on ΣN/SN . Similarly to N = 1 case, it is easy to show that this gives rise to a bijection between conjugacy classes of elements of Nm(λ, C) and points of ΣN/SN . Using this fact and combining the method of computation for usual normal matrices with the N = 1 case, one gets the following result. Theorem 2. The Euclidean measure on Nm(λ, C) has the form dM = 1 2N ∏ i √ Q′(xi) ∏ i<j |zi − zj |2dx1 · · ·dxNdθ1 · · ·dθNdU, where dU is the normalized invariant measure on U(N)m/U(1)N . Proof. At first, consider the subset N diag m (λ, C) of Nm(λ, C) consisting of the elements M of the form (3). Then by Theorem 1, the measure on N diag m (λ, C) induced by the Euclidean metric is the product measure: µdiag = 1 2N ∏ i √ Q′(xi)dx1 · · ·dxNdθ1 · · ·dθNdUdiag, (4) where dUdiag is the Haar measure on U(1)Nm/U(1)N . Now consider the contribution of the off-diagonal part. Consider the elements {vi,j = Ei,j − Ej,i, wi,j = √ −1(Ei,j + Ej,i) | 0 6 i < j 6 N} of the Lie algebra of U(N). Let Vi,j,k, Wi,j,k be the derivatives of (exp(tvi,j))kM and (exp(twi,j))kM at t = 0, where ak := (1, . . . , 1, a, 1, . . . , 1) ∈ U(N)m, with a ∈ U(N) in the k-th place. Then by formula (4), we have dM = φ · 1 2N ∏ i √ Q′(xi)dx1 · · ·dxNdθ1 · · ·dθNdU, where φ = | ∧i<j,k (Vi,j,k ∧Wi,j,k)|. (5) To calculate φ, let us denote by Bi,j,k, i 6= j, the derivative of (exp(tEi,j))kM (note that since Ei,j lies only in the complexified Lie algebra of U(N)m, we have (exp(tEi,j))kM /∈ Nm(λ, C), but this is not important for our considerations). Then equation (5) takes the form φ = | ∧i6=j,k Bi,j,k|. Now φ can be easily calculated. To do so, we note that for a given i, j, the transformation (exp(tEi,j))k changes only the entries ap i,j of M . On these entries, it acts by ap i,j → ap i,j + t(zp j δp,k − zp i δp,k−1). Density of Eigenvalues of Random Normal Matrices 7 This means that for each i, j, | ∧k Bi,j,k| = |Ji,j |, where Ji,j := det  z1 j −z1 i 0 · · · 0 0 z2 j −z2 i . . . ... ... 0 z3 j . . . 0 0 . . . 0 . . . −zm i −zm i 0 · · · 0 zm j  = m∏ s=1 zs j − m∏ s=1 zs i = zj − zi. This implies that φ = ∏ i6=j |Ji,j | = ∏ i<j |zi − zj |2, as desired. � 3.3 The probability measure with potential function on generalized normal matrices Let W : C → R be a potential function. The probability measure on Nm(λ, D) corresponding to this function is defined similarly to the case of usual normal matrices: PN (M)dM = Z−1 N exp(−NtrW (M1 · · ·Mm))dM, M ∈ Nm(λ, D), where Mi are the blocks of M . Thus in terms of eigenvalues PN (M)dM = 1 2NZN exp −N ∑ j W (zj)  × ∏ i √ Q′(xi) ∏ i<j |zi − zj |2dx1 · · ·dxNdθ1 · · ·dθNdU. Example 1. Let us calculate the potential function corresponding to the quadratic potential Tr(MM †). We have Tr(MM †) = ∑ i,j |zi j |2 = ∑ i,j (xj + αi) = N ∑ i αi + m ∑ j xj . Thus if we choose αi so that ∑ i αi = 0 (this can be done in a unique way), then Tr(MM †) = m ∑ j xj , so the corresponding potential function is W (z) = mQ−1(|z|2) (the function Q is invertible on the interval [−α,∞), where α = minαi). 4 Equilibrium measure 4.1 Some basic facts about equilibrium measure Let D be a compact subset of the complex plane C, and W (z) a potential function (a continuous function on D). Denote by M(D) the set of the Borel probability measures σ on D without point masses, and define the energy of σ to be Iσ = ∫ D W (z)dσ(z) + ∫ D ∫ D log |z − w|−1dσ(z)dσ(w). 8 P. Etingof and X. Ma An equilibrium measure for W on D is a measure σ ∈M(D) such that Iσ = inf µ∈M(D) Iµ. Theorem 3. The equilibrium measure σ exists and is unique. It satisfies the equation W (z)− 2 ∫ D log |z − w|dσ(w) = C, (6) where C is a constant, almost everywhere with respect to σ. The proof of this theorem can be found in [3]. Note that equation (6) does not have to hold outside the support of σ. Note also that if σ is absolutely continuous with respect to the Lebesgue measure near a point z0 in the interior of D, and dσ = g(z)d2z, where g is continuous near z0 and g(z0) > 0, then ∆W = 4πg near z0. This clearly cannot happen at points where ∆W ≤ 0. In particular, if ∆W ≤ 0 everywhere, then dσ tends to be concentrated on the boundary of D. 4.2 Asymptotic eigenvalue distribution in the normal matrix model In Section 2, we defined a measure PN (M)dM = JN (z1, . . . , zN )d2z1 · · ·d2zNdU. by formula (2). We are interested in the behavior of this measure when N → ∞. Let δz = 1 N N∑ j=1 δzj be the measure on D corresponding to the points zj . Then − log(ZNJN (z1, . . . , zN )) = N2 (∫ W (ξ)dδz(ξ) + ∫∫ ξ 6=ζ log |ξ − ζ|−1dδz(ζ)dδz(ξ) ) . This shows that the leading contribution to the integral with respect to the measure PN (M)dM comes from configurations of eigenvalues z1, . . . , zN for which the expression in parentheses in the last equation is minimized. This means that in the limit N → ∞, we should expect the measures δz for optimal configurations to converge to the equilibrium measure with potential function W . This indeed turns out to be the case, as shown by the following theorem, proved in [3]. Theorem 4. Let the k-point correlation function be R (k) N ((zi)k i=1) = ∫ DN−k JN (z1, . . . , zN ) N∏ i=k+1 d2zi. Then the measure R (k) N ((zi)k i=1) k∏ i=1 d2zi on Dk converges weakly to dσ⊗k, where dσ is the equilibrium measure on D, corresponding to the potential function W . In particular, if k = 1, it means that the eigenvalue distribution tends to the equilibrium measure in D as N →∞. Density of Eigenvalues of Random Normal Matrices 9 4.3 Asymptotic eigenvalue distribution in the generalized normal matrix model As we have seen above, the eigenvalue distribution in the generalized normal matrix model is PN (M)dM = JN,m(z1, . . . , zN )d2z1 · · ·d2zNdU, where − log(2NZNJN,m) = N2 (∫ W (ξ)dδz(ξ) + ∫∫ ξ 6=ζ log |ξ − ζ|−1dδz(ζ)dδz(ξ) ) − N 2 ∫ log Q′(Q−1(|ξ|2))dδz(ξ). In the limit N → ∞ the second term becomes unimportant compared to the first one, which implies that Theorem 4 is valid for the generalized normal matrix model. Thus in the limit N → ∞, the usual and the generalized normal matrix models (with the same potential) are equivalent. 5 Reconstruction of the boundary of the domain In previous sections, we showed that in the normal matrix model and the generalized normal matrix model, when N → ∞, the eigenvalue distribution converges to an equilibrium measure on D corresponding to some potential function W . In this section, we will try to find this measure explicitly in some special cases. More specifically, we will consider the case when ∆W > 0. In this case, if the region D is sufficiently large, it turns out that the equilibrium measure is often absolutely continuous with respect to Lebesgue measure, and equals dσ = (4π)−1χE∆Wd2z, where E is a region contained in D (the region of eigenvalues), and χE is the characteristic function of E. More precisely, it follows from Proposition 3.4 in [3] that if there exists a region E ⊂ D such that dσ satisfies equation (6) in E, and the left hand side of this equation is ≥ C on D \ E, then dσ is the equilibrium measure on D for the potential function W . Moreover, note that if E works for some D then it works for any smaller D′ such that E ⊂ D′ ⊂ D. So, in a sense, E is independent of D. (Here we refer the reader to [5], section 4, where there is a much more detailed and precise treatment of equilibrium measures, without the assumption ∆Φ > 0). Thus let us assume that E exists, and consider the problem of finding it explicitly given the potential W . 5.1 The reconstruction problem We will consider the case when D = D(R) is the disk of radius R centered at the origin, and W (z) = Φ(zz̄)− P (z)− P (z), where Φ is a function of one variable continuous on [0,∞) and twice continuously differentiable on (0,∞), and P a complex polynomial. We assume that (sΦ′(s))′ is positive, integrable near zero, and satisfies the boundary condition lims→0 sΦ′(s) = 0. Computing the Laplacian of W , we get (taking into account that ∆ = 4∂∂̄): g(s) := (4π)−1∆W = π−1(Φ′(s) + sΦ′′(s)), where s = zz̄. Define the measure dσ = gd2z. 10 P. Etingof and X. Ma Suppose that the region E exists, and contains the origin. In this case, differentiating equa- tion (6) with respect to z, we have inside E: z̄Φ′(zz̄)− P ′(z) = ∫ E g(ww̄) z − w d2w. (7) On the other hand, inside the disk D, the function W0(z) := 2 ∫ D g(ww̄) log |z − w|d2w satisfies the equation ∆W0 = 4πg, and is rotationally invariant, so W0(z) = Φ(zz̄) + C ′, where C ′ is a constant. Hence, differentiating, we get, inside D: z̄Φ′(zz̄) = ∫ D g(ww̄) z − w d2w. (8) Thus, subtracting (7) from (8), we obtain inside E: P ′(z) = ∫ D\E g(ww̄) z − w d2w. (9) Let I(s) = π ∫ s 0 g(t)dt = sΦ′(s). Then ∂̄I(zz̄) = πzg(zz̄)dz̄. Thus, using Green’s formula, we get from (9): P ′(z) = 1 2πi ∫ ∂D−∂E I(ww̄) w(z − w) dw, where the boundaries are oriented counterclockwise. The integral over the boundary of D is zero by Cauchy’s formula, so we are left with the equation P ′(z) = 1 2πi ∫ ∂E I(ww̄) w(w − z) dw. This equation appeared first in the theory of Hele-Shaw flows on curved surfaces in [4], and it can be solved explicitly by the method of singular points developed in the same paper. Let us recall this method. 5.2 The singular point method Define the Cauchy transform hE of E with respect to the measure dσ by hE(z) = ∫ D\E dσ(w) z − w , z ∈ E. This is a holomorphic function of z which (as we have just seen) is independent of the radius R of D. As we have seen, it is also given by the contour integral hE(z) = 1 2πi ∫ ∂E I(ww̄) w(w − z) dw, (10) and in our case we have hE(z) = P ′(z). Let f : D(1) → C \ E be a conformal map, such that f(0) = ∞, and (1/f)′(0) = a ∈ R+ (such a map is unique). Density of Eigenvalues of Random Normal Matrices 11 Lemma 2. The function φ(ζ) = I(f(ζ)f(ζ))− f(ζ)hE(f(ζ)) continues analytically from the unit circle to a holomorphic function outside the unit disk. Proof. By the Cauchy formula, we have hE(z) = 1 2πi ∫ ∂E hE(w) w − z dw, for any z ∈ E. So by formula (10), we have 1 2πi ∫ ∂E I(ww̄)/w − hE(w) w − z dw = 0, for any z ∈ E. It follows that the function I(zz̄)/z−hE(z), defined along ∂E, can be analytically continued to a holomorphic function outside E, which vanishes at infinity. This implies the lemma. � Similarly to [4], this lemma implies the following theorem. Theorem 5. The function hE is rational if and only if the function θ(ζ) = I(f(ζ)f(1/ζ̄)) is. Moreover, the number of poles of θ is twice of the number of poles of zhE(z). More specifically, if ζ0 and 1/ζ̄0 are poles of θ of order m, then z0 = f(ζ0) is a pole of order m for hE(z), and vice versa. Thus, if h is a rational function, then θ can be determined from h up to finitely many parameters. After this, f can be reconstructed from θ using the Cauchy formula. For this, note that the function I is invertible, since I ′ = g > 0. Also, θ takes nonnegative real values on the unit circle. Thus, we have f(ζ)f(1/ζ̄) = I−1(θ(ζ)). Taking the logarithm of both sides, we obtain log(ζf(ζ)) + log ( ζ−1f(1/ζ̄) ) = log I−1(θ(ζ)). Thus we have f(ζ) = aζ−1 exp ( 1 2πi ∫ |u|=1 log I−1(θ(u)) u− ζ du ) , a = exp ( − 1 4πi ∫ |u|=1 log I−1(θ(u)) u du ) . The unknown parameters of θ can now be determined from the cancellation of poles in Theorem 5, similarly to the procedure described in [10]. We note that the knowledge of the function hE is not sufficient to determine E (for example if E is a disk of any radius centered at the origin then hE = 0). To determine the parameters completely, we must also use the information on the area of E:∫ E dσ = − 1 2πi ∫ |u|=1 θ(u) f(u) f ′(u)du = 1. 12 P. Etingof and X. Ma 5.3 The polynomial case In particular, in our case, hE(z) = P ′(z) = a1 + a2z + · · ·+ adz d−1, which implies that θ(ζ) = d∑ j=−d bjζ j , and b̄j = b−j . So we get f(ζ) = aζ−1 exp  1 2πi ∫ |u|=1 log I−1 ( d∑ j=−d bju j ) u− ζ du  , a = exp − 1 4πi ∫ |u|=1 log I−1 ( d∑ j=−d bju j ) u du  . (11) Finally, note that if the coefficients of the polynomial P are small enough, then all our assumptions are satisfied: the region E exists (in fact, it is close to a disk), and contains the origin. Also, in this case the left hand side of equation (6) is ≥ C, which implies that the equilibrium measure in this case (and hence, the asymptotic eigenvalue distribution) is the measure dσ in the region E. Example 2. Consider Example 1: the generalized normal matrix model with the density exp(−βtr(MM † − P (M)− P (M)†). As we showed, in this case Φ(s) = mβQ−1(s). So a short computation shows that πm−1β−1g(Q(x)) = Q(x)2 Q′(x)3 ∑ i 1 (x + αi)2 . This implies that g > 0, i.e. our analysis applies in this case. 5.4 Some explicit solutions Consider the case Φ(s) = Csb, C, b > 0. For example, in the generalized normal matrix model with αi = 0 and potential term as in Example 1, one has Φ(s) = ms1/m, which is a special case of the above. We have g(s) = π−1Cb2sb−1, so our analysis applies (note that if b < 1 then g is singular at zero, but the singularity is integrable and thus nothing really changes in our considerations), and I(s) = Cbsb. Thus the integral in (11) can be computed explicitly (by factoring θ), and the formula for the conformal map f simplifies as follows: f(ζ) = (aζ)−1 d∏ j=1 (1− ζζ−1 j )1/b. The parameters a > 0 and ζj are determined from the singularity conditions and the area condition. Consider for simplicity the example d = 1. In this case we have hE(z) = K, Density of Eigenvalues of Random Normal Matrices 13 and we can assume without loss of generality that K ∈ R. Then f(ζ) = (aζ)−1(1 + βζ)1/b, β ∈ R, and θ(ζ) = Cba−2b(1 + βζ)(1 + βζ−1). The residue of θ at zero is thus Cbβa−2b. Thus the singularity condition says Cbβa1−2b = K. The area condition is 1 = Cba−2b(1 + β2(1− b−1)). Thus we find β = KC−1b−1a2b−1, and the equation for a has the form Cba−2b + C−1b−1K2a2b−2(1− b−1) = 1. Remark 2. This example shows that to explicitly solve the generalized (as opposed to the usual) normal matrix model in the N → ∞ limit with the quadratic (Gaussian) potential, one really needs the technique explained in Section 5 of this paper, and the techniques of [3] are not sufficient. Acknowledgements P.E. is grateful to G. Felder and P. Wiegmann for useful discussions. The work of P.E. was partially supported by the NSF grant DMS-0504847. References [1] Chau L.-L., Zaboronsky O., On the structure of correlation functions in the normal matrix model, Comm. Math. Phys. 196 (1998), 203–247, hep-th/9711091. [2] Crawley-Boevey W., Holland M.P., Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), 605–635. [3] Elbau P., Felder G., Density of eigenvalues of random normal matrices, Comm. Math. Phys. 259 (2005), 433–450, math.QA/0406604. [4] Entov V.M., Etingof P.I., Viscous flows with time-dependent free boundaries in a non-planar Hele-Shaw cell, Euro. J. Appl. Math. 8 (1997), 23–35. [5] Hedenmalm H., Makarov N., Quantum Hele-Shaw flow, math.PR/0411437. [6] Kostov I.K., Krichever I., Mineev-Weinstein M., Wiegmann P.B., Zabrodin A., The τ -function for analytic curves, in Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge Univ. Press, Cambridge, 2001, 285–299. [7] Krichever I., Marshakov A., Zabrodin A., Integrable structure of the Dirichlet boundary problem in multiply- connected domains, Comm. Math. Phys. 259 (2005), 1–44, hep-th/0309010. [8] Marshakov A., Wiegmann P.B., Zabrodin A., Integrable structure of the Dirichlet boundary problem in two dimensions, Comm. Math. Phys. 227 (2002), 131–153, hep-th/0109048. [9] Oas G., Universal cubic eigenvalue repulsion for random normal matrices, Phys. Rev. E 55 (1997), 205–211, cond-mat/9610073. [10] Varchenko A.N., Etingof P.I., Why the boundary of a round drop becomes a curve of order four, AMS, Providence, 1992. [11] Wiegmann P.B., Zabrodin A., Conformal maps and integrable hierarchies, Comm. Math. Phys. 213 (2000), 523–538, hep-th/9909147. [12] Wiegmann P.B., Zabrodin A., Large scale correlations in normal non-Hermitian matrix ensembles, J. Phys. A: Math. Gen. 36 (2003), 3411–3424, hep-th/0210159. http://arxiv.org/abs/hep-th/9711091 http://arxiv.org/abs/math.QA/0406604 http://arxiv.org/abs/math.PR/0411437 http://arxiv.org/abs/hep-th/0309010 http://arxiv.org/abs/hep-th/0109048 http://arxiv.org/abs/cond-mat/9610073 http://arxiv.org/abs/hep-th/9909147 http://arxiv.org/abs/hep-th/0210159 1 Introduction 2 The normal matrix model with an arbitrary potential function 3 The generalized normal matrix model 3.1 Generalized normal matrices 3.2 The Euclidean measure on generalized normal matrices 3.3 The probability measure with potential function on generalized normal matrices 4 Equilibrium measure 4.1 Some basic facts about equilibrium measure 4.2 Asymptotic eigenvalue distribution in the normal matrix model 4.3 Asymptotic eigenvalue distribution in the generalized normal matrix model 5 Reconstruction of the boundary of the domain 5.1 The reconstruction problem 5.2 The singular point method 5.3 The polynomial case 5.4 Some explicit solutions References

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