Moments Match between the KPZ Equation and the Airy Point Process

The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point pr...

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Datum:	2016
Hauptverfasser:	Borodin, A., Gorin, V.
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Sprache:	English
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spelling	irk-123456789-1480032019-02-17T01:25:55Z Moments Match between the KPZ Equation and the Airy Point Process Borodin, A. Gorin, V. The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations. 2016 Article Moments Match between the KPZ Equation and the Airy Point Process / A. Borodin, V. Gorin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60B20; 60H15; 33C10 DOI:10.3842/SIGMA.2016.102 http://dspace.nbuv.gov.ua/handle/123456789/148003 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	The results of Amir-Corwin-Quastel, Calabrese-Le Doussal-Rosso, Dotsenko, and Sasamoto-Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations.
format	Article
author	Borodin, A. Gorin, V.
spellingShingle	Borodin, A. Gorin, V. Moments Match between the KPZ Equation and the Airy Point Process Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Borodin, A. Gorin, V.
author_sort	Borodin, A.
title	Moments Match between the KPZ Equation and the Airy Point Process
title_short	Moments Match between the KPZ Equation and the Airy Point Process
title_full	Moments Match between the KPZ Equation and the Airy Point Process
title_fullStr	Moments Match between the KPZ Equation and the Airy Point Process
title_full_unstemmed	Moments Match between the KPZ Equation and the Airy Point Process
title_sort	moments match between the kpz equation and the airy point process
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/148003
citation_txt	Moments Match between the KPZ Equation and the Airy Point Process / A. Borodin, V. Gorin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT borodina momentsmatchbetweenthekpzequationandtheairypointprocess AT gorinv momentsmatchbetweenthekpzequationandtheairypointprocess
first_indexed	2025-07-11T03:29:26Z
last_indexed	2025-07-11T03:29:26Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 102, 7 pages Moments Match between the KPZ Equation and the Airy Point Process? Alexei BORODIN †‡ and Vadim GORIN †‡ † Department of Mathematics, Massachusetts Institute of Technology, USA E-mail: borodin@math.mit.edu, vadicgor@gmail.com ‡ Institute for Information Transmission Problems of Russian Academy of Sciences, Russia Received August 09, 2016, in final form October 21, 2016; Published online October 26, 2016 http://dx.doi.org/10.3842/SIGMA.2016.102 Abstract. The results of Amir–Corwin–Quastel, Calabrese–Le Doussal–Rosso, Dotsenko, and Sasamoto–Spohn imply that the one-point distribution of the solution of the KPZ equa- tion with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple direct proof of those via a combinatorial match of their multivariate integral representations. Key words: KPZ equation; Airy point process 2010 Mathematics Subject Classification: 60B20; 60H15; 33C10 1 Introduction Since Tracy and Widom’s discovery of the ASEP solvability eight years ago [18, 19, 20], the rela- tionship between the “determinantal” and “non-determinantal” solvable models in the (1 + 1)- dimensional KPZ (Kardar–Parisi–Zhang) universality class has largely remained a mystery. One step towards solving this mystery is the celebrated result of Amir–Corwin–Quastel [1], Calabrese–Le Doussal–Rosso [8], Dotsenko [10], and Sasamoto–Spohn [17], that provides an ex- plicit expression for the distribution (or its Laplace transform) of one-point value of the solution of the KPZ equation with the so-called narrow wedge initial condition. It can be re-interpreted as saying that this Laplace transform coincides with the average of a multiplicative statistics of the Airy determinantal random point process. Although this restatement seems to be known to experts, we couldn’t find it in this form in the literature, so we give an exact formulation as Theorem 2.1 below. Such a result is very useful as it immediately implies that this solution of the KPZ equation asymptotically at large times has the GUE Tracy–Widom distribution, which is a display of the KPZ universality, cf. Corwin’s survey [9]. Finding other facts of similar nature has been a challenge so far. Imamura and Sasamoto [12] proved a similar statement for the O’Connell–Yor semi-discrete Brownian directed polymer. Unfortunately, the associated determinantal point process was not governed by a positive measure. Still, taking the edge limit of this process, they were able to recover Theorem 2.1. Another representation of the Laplace transform of the O’Connell–Yor partition function as the average of a multiplicative functional over a signed determinantal point process can be found in [14]. Very recently, one of the authors found in [4] an identity that relates a single point height distribution of the (higher spin inhomogeneous) stochastic six vertex model in a quadrant on ?This paper is a contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy. The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html mailto:borodin@math.mit.edu mailto:vadicgor@gmail.com http://dx.doi.org/10.3842/SIGMA.2016.102 http://www.emis.de/journals/SIGMA/Deift-Tracy.html 2 A. Borodin and V. Gorin one side, and multiplicative statistics of the Macdonald measures on the other. The ASEP limit of this identity was worked out in [7]. Taking the KPZ limit of both leads to Theorem 2.1 again. The goal of this note is to look at Theorem 2.1 from the point of view of moments, rather than the corresponding distributions. One can study both the KPZ equation and the Airy point process via their exponential moments. Those are computationally tractable but they are of limited mathematical use because the corresponding moment problems are indeterminate. Still, on the KPZ side physicists were able to consistently use the moments to access the distributions via the (non-rigorous) replica trick, see [10] and [8] for early examples. Our Theorem 2.2 proves the moments identity that corresponds to Theorem 2.1. The ar- gument is a combinatorial match between known multivariate integral representations of the moments on both sides. Interestingly, these integral representations were known long be- fore [1, 8, 10, 17], but their similarity had not been exploited. We are hoping that the moments point of view will be beneficial for finding other similar correspondence. We did attempt to extend the moments correspondence to a two-point identity, as integral representations on both side are again known. Unfortunately, we have not been successful in that so far. 2 The one-point equality Let a1 ≥ a2 ≥ a3 ≥ · · · be points of the Airy point process1 at β = 2 (see, e.g., [2, 11]) which is a determinantal point process on R with correlation kernel KAiry(x, y) = Ai(x)Ai′(y)−Ai′(x)Ai(y) x− y = ∫ ∞ 0 Ai(x+ a)Ai(y + a)da. Here Ai(x) is the Airy function. From the opposite direction, let Z(T,X) denote the solution of the stochastic heat equation (see, e.g., [9, 16]) ∂ ∂T Z = 1 2 ∂2 ∂X2 Z − ZẆ, Z(0, X) = δ(X = 0), where Ẇ is the space–time white noise. H := − log(Z) is the Hopf–Cole solution of the Kardar– Parisi–Zhang stochastic partial differential equation with the narrow-edge initial data. The following statement is a reformulation of results of [1, 8, 10, 17]. Theorem 2.1. Set T 2 = C3. Then for each real C > 0, u ≥ 0 we have EAiry [ ∞∏ k=1 1 1 + u exp (Cak) ] = EKPZ [ exp ( −uZ(T, 0) exp ( T 24 ))] . (1) On the other hand, we could not find the following statement in the existing literature. Theorem 2.2. Set T 2 = C3, and let hk(x1, x2, . . . ) = ∑ i1≤i2≤···≤ik xi1xi2 · · ·xik be the complete symmetric homogeneous function in variables x1, x2, . . . . Then for each C > 0, k = 1, 2, . . . we have EAiry [hk (exp (Ca1) , exp (Ca2) , . . . )] = EKPZ [ Z(T, 0)k k! ] exp ( k T 24 ) . (2) 1It should not be confused with Airy2 process; the latter is a random continuous curve rather than a random point process. Moments Match between the KPZ Equation and the Airy Point Process 3 Our proof of Theorem 2.2 is based on direct comparison of contour integral formulas: for the right-hand side of (2) such formula is known as a solution for the attractive delta Bose gas equation, cf. the discussion in [6, Section 6.2], while for the left-hand side it can be computed through the Laplace transform of the correlation kernel KAiry. Remark 2.3. Expanding formally the result of Theorem 2.1 into power series in u and evaluating the coefficients, one gets the result of Theorem 2.2 and vice versa. However, these theorems are not equivalent: The u power series expansion of the left-hand side of (2.1) fails to converge for any u 6= 0. Below we provide two different proofs for Theorems 2.1 and 2.2, respectively. The following corollary is present in [1, 8, 10, 17], but it seems reasonable for us to give a proof using Theorem 2.1 above only, without appealing to the explicit evaluation of either side. Corollary 2.4. The following convergence in distribution holds: lim T→+∞ [( 2 T )1/3( ln(Z(T, 0)) + T 24 )] = a1. Proof. Take a ∈ R and set u = exp ( −(T/2)1/3a ) . Then the left-hand side of (1) is E [ ∞∏ k=1 1 1 + exp ( (T/2)1/3(ak − a) )] . (3) We claim that the random variable under expectation in (3) almost surely converges as T →∞ to the indicator function of the event a1 < a. Indeed, if a1 > a, then the expression in (3) converges to 0 as T → ∞. On the other hand, if a1 < a, then (taking into the account that ∞∑ k=1 exp(ak) is almost surely finite, as follows from the finiteness of its expectation, which is explicitly known, see, e.g., [15, Section 2.6.1]) the same expression converges to 1. Since the random variable under expectation is almost surely between 0 and 1, the almost sure convergence implies convergence of expectations, and therefore lim T→+∞ E [ ∞∏ k=1 1 1 + exp ( (T/2)1/3(ak − a) )] = Prob(a1 < a). (4) On the other hand, the right side of (1) is E [ exp ( − exp ( (T/2)1/3 ( ln(Z(T, 0)) + T/24 (T/2)1/3 − a )))] . (5) Observe that for any random variable ξ, the expression exp ( − exp((T/2)1/3(ξ − a)) ) is almost surely between 0 and 1, converges to 0 as T → +∞ if ξ > a, and converges to 1 if ξ < a. Therefore, using the fact that a1 has a continuous distribution, (5) as T → ∞ behaves as (see, e.g., [6, Lemma 4.1.39] for more details) Prob ( ln(Z(T, 0)) + T/24 (T/2)1/3 < a ) + o(1). (6) Equating (4) to (6) we are done. � 4 A. Borodin and V. Gorin 3 Proofs of Theorems 2.1 and 2.2 Proof of Theorem 2.1. The expectation E exp(−uZ(T, 0) admits a formula as a Fredholhm determinant, which was discovered in [1, 8, 10, 17]. Following [5, Section 2.2.1], this formula reads E [ exp ( −uZ(T, 0) exp(T/24) )] = 1 + ∞∑ L=1 (−1)L L! ∞∫ 0 dx1 · · · ∞∫ 0 dxL det [Ku(xi, xj)] L i,j=1 , (7) where Ku(x, x ′) = ∫ ∞ −∞ dr 1 + 1 u exp ( (T/2)1/3r ) Ai(x− r)Ai(x′ − r). On the other hand, the left-hand side of (1) is a multiplicative function of a determinantal point process and, therefore, also admits a Fredholm determinant formula, see, e.g., [3, equation (2.4)]: E [ ∞∏ k=1 1 1 + u exp (Cak) ] = det [ 1− ( 1− 1 1 + u exp(Cr) ) KAiry(r, r ′) ] L2(R) = 1 + ∞∑ L=1 (−1)L L! ∫ ∞ −∞ dy1 · · · ∫ ∞ −∞ dyL ( L∏ k=1 1 1 + 1 u exp(−Cyk) ) det [KAiry(yi, yj)] L i,j=1 . (8) One immediately sees that upon the change of variables ri = −yi and identification T 2 = C3, the formulas (7) and (8) are the same. � Proof of Theorem 2.2. The moments EZ(T, 0)k are known through solving the attractive delta Bose gas equation, cf. the discussion in [6, Section 6.2]. Following [5, Lemma 4.1], we have E [ Z(T, 0)k ] = ∫ a1+i∞ a1−i∞ dz1 2πi · · · ∫ ak+i∞ ak−i∞ dzk 2πi ∏ 1≤A<B≤k zA − zB zA − zB − 1 · k∏ j=1 exp ( T 2 z2j ) , (9) where the real numbers a1, . . . , ak satisfy a1 � a2 � · · · � ak. It is convenient for us to modify the contours of integration in (9) to the imaginary axis iR. One collects certain residues in such a deformation, and the final result is read from [5, equation (13)] to be E [ Z(T, 0)k k! ] = ∑ λ`k λ=1m12m2 ··· 1 m1!m2! · · · ∫ i∞ −i∞ dw1 2πi · · · ∫ i∞ −i∞ dw`(λ) 2πi det [ 1 wj + λj − wi ]`(λ) i,j=1 × `(λ)∏ j=1 exp ( T 2 ( w2 j + (wj + 1)2 + · · ·+ (wj + λj − 1)2 )) , (10) where λ = (λ1 ≥ λ2 ≥ . . . ) is a partition of k and `(λ) is the number of non-zero parts λj . Let us now produce a similar expression for the left-hand side of (2). Define the Laplace transform of the correlation functions of the Airy point process through R(c1, . . . , cn) = ∫ Rn e(c·x) det[KAiry(xi, xj)] n i,j=1dx1 · · · dxn, c1, c2, . . . , cn > 0. The definition of the Airy point process implies that for a partition λ = (λ1 ≥ λ2 · · · ≥ λ`) = 1m12m2 · · · one has E [ mλ(exp(Ca1), exp(Ca2), . . . ) ] = 1 m1!m2! · · · R(Cλ1, . . . , Cλ`), Moments Match between the KPZ Equation and the Airy Point Process 5 wheremλ(y1, y2, . . . ) is the monomial symmetric function in variables y1, y2, . . . , as in [13, Chap- ter I]. Expanding hk into linear combination of mλ’s, we can then write E [ hk(exp(Ca1), exp(Ca2), . . . ) ] = ∑ λ`k λ=1m12m2 ··· 1 m1!m2! · · · R ( Cλ1, . . . , Cλ`(λ) ) , (11) where the summation goes over all partitions of k. Comparing (11) with (10), we see that it remains to identify the contour integrals over imaginary axis in (10) with R ( Cλ1, . . . , Cλ`(λ) ) . The rest is based on the following identity that can be found in [15, Lemma 2.6]:∫ +∞ −∞ exz Ai(z + a)Ai(z + b)dz = 1 2 √ πx exp ( x3 12 − a+ b 2 x− (a− b)2 4x ) , x > 0. Its immediate corollary is (we use an agreement zn+1 = z1 and sn+1 = s1 here, and also assume c1, . . . , cn > 0) E(c1, . . . , cn) := ∫ Rn e(c·z) n∏ i=1 KAiry(zi, zi+1)dz = 1 2nπn/2 e ∑ c3i /12 n∏ i=1 √ ci × ∫ s1≥0 · · · ∫ sn≥0 exp ( − n∑ i=1 (si − si+1) 2 4ci − n∑ i=1 si + si+1 2 ci ) n∏ i=1 dsi. Using the Gaussian integrals in variables z1, . . . , zn, the last formula is converted into E(c1, . . . , cn) = 1 (2π)n e ∑ c3i /12 ∫ s1≥0 ds1 · · · ∫ sn≥0 dsn ∫ z1∈R dz1 · · · ∫ zn∈R dzn × exp ( n∑ i=1 ( −ciz2i + i(zi − zi+1)si − (ci + ci+1)si/2 )) . (12) Since i(zi − zi+1) has zero real part, we can integrate over si in (12), arriving at the formula: E(c1, . . . , cn) = e ∑ x3i /12 (2π)n ∫ z1∈R · · · ∫ zn∈R exp ( − n∑ i=1 ciz 2 i ) n∏ i=1 dzi −i(zi − zi+1) + ci+ci+1 2 . (13) We can now write the formula for R(c1, . . . , cn) (we subdivide a permutation into cycles, use (13) and then combine back): R(c1, . . . , cn) = ∫ Rn dx1 · · · dxn ∑ σ∈S(n) (−1)σ n∏ j=1 exjcjKAiry(xj , xσ(j)) = e ∑ c3i /12 (2π)n ∫ z1∈R · · · ∫ zn∈R exp ( − n∑ i=1 ciz 2 i ) ∑ σ∈S(n) (−1)σ n∏ i=1 dzi −i(zi − zσ(i)) + ci+cσ(i) 2 = e ∑ c3i /12 (2π)n ∫ z1∈R dz1 · · · ∫ zn∈R dzn exp ( − n∑ i=1 ciz 2 i ) det [ 1( −izi + ci 2 ) + ( izj + cj 2 ) ]n i,j=1 . Remark 3.1. We can use the Cauchy determinant formula det [ 1 ai + bj ]n i,j=1 = n∏ i=1 1 ai + bi ∏ 1≤i<j≤n (ai − aj)(bi − bj) (ai + bj)(aj + bi) with ai = −izi + ci/2, bi = izi + ci/2 to simplify the last determinant. 6 A. Borodin and V. Gorin We now take a partition λ ` k with `(λ) = n, set ci = Cλi and make a change of variables izj = Cwj + Cλj 2 − C 2 to get (note that we deformed the contours to the imaginary axis; we do not pick up any residues in such a deformation) R(Cλ1, . . . , Cλn) = exp ( C3 n∑ i=1 λ3i /12 ) (2πi)n ∫ i∞ −i∞ dw1 · · · ∫ i∞ −i∞ dwn × exp ( C3 n∑ i=1 λi(wi + λi/2− 1/2)2 ) det [ 1 wj + λj − wi ]n i,j=1 . (14) It remains to simplify the exponents: exp [ C3 n∑ i=1 ( λ3i 12 + λi ( wi + λi 2 − 1 2 )2 )] = exp [ C3 n∑ i=1 ( λiw 2 i + λ3i 3 + λi 4 + wiλ 2 i − wiλi − λ2i 2 )] = exp [ C3 n∑ i=1 ( λiw 2 i + λi(λi − 1)wi + λi(λi − 1)(2λi − 1) 6 − λi 12 )] = exp [ C3 n∑ i=1 ( w2 i + (wi + 1)2 + · · ·+ (wi + λi − 1)2 ) − C3 k 12 ] . (15) Combining (11) with (14), (15) and identifying C3 = T 2 . we arrive at (10) multiplied by exp(−kT/24). � Acknowledgements A.B. was partially supported by the NSF grants DMS-1056390 and DMS-1607901. V.G. was partially supported by the NSF grant DMS-1407562 and by the Sloan Research Fellowship. References [1] Amir G., Corwin I., Quastel J., Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions, Comm. Pure Appl. Math. 64 (2011), 466–537, arXiv:1003.0443. [2] Anderson G.W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Vol. 118, Cambridge University Press, Cambridge, 2010. [3] Borodin A., Determinantal point processes, in The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011, 231–249, arXiv:0911.1153. [4] Borodin A., Stochastic higher spin six vertex model and Macdonald measures, arXiv:1608.01553. [5] Borodin A., Bufetov A., Corwin I., Directed random polymers via nested contour integrals, Ann. Physics 368 (2016), 191–247, arXiv:1511.07324. [6] Borodin A., Corwin I., Macdonald processes, Probab. 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Moments Match between the KPZ Equation and the Airy Point Process

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