The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k-th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom formulas involving solutions of the Painlev\'e II equation. Limit distributions for quantities involving two or more near-extreme eigenvalues, such as the gap between the k-th and the \ell-th largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy-Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlev\'e II equations, and we investigate the asymptotic behavior of these solutions.

[1]  V. Pérez-García,et al.  Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities , 2010, 1001.2425.

[2]  Dan Dai,et al.  Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System , 2017, Communications in Mathematical Physics.

[3]  W. Van Assche,et al.  The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] , 2001 .

[4]  A. Hone.  Coupled Painlevé systems and quartic potentials , 2001 .

[5]  K. Johansson.  The Arctic circle boundary and the airy process , 2003, math/0306216.

[6]  M. P. Pato,et al.  Deformations of the Tracy-Widom distribution. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  C. Charlier,et al.  Thinning and conditioning of the Circular Unitary Ensemble , 2016, 1604.08399.

[8]  Vladimir E. Korepin,et al.  Differential Equations for Quantum Correlation Functions , 1990 .

[9]  A. Okounkov.  Random matrices and ramdom permutations , 1999, math/9903176.

[10]  J. Baik,et al.  Combinatorics and Random Matrix Theory , 2016 .

[11]  Hamiltonian Structure of Equations Appearing in Random Matrices , 1993, hep-th/9301051.

[12]  S. V. Manakov.  On the theory of two-dimensional stationary self-focusing of electromagnetic waves , 1973 .

[13]  A. Kuijlaars,et al.  Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights , 2007, 0712.1333.

[14]  P. Moerbeke,et al.  Completely Integrable Systems, Euclidean Lie-algebras, and Curves , 1980 .

[15]  Александр Борисович Сошников,et al.  Детерминантные случайные точечные поля@@@Determinantal random point fields , 2000 .

[16]  Craig A. Tracy,et al.  Mathematical Physics © Springer-Verlag 1994 Level Spacing Distributions and the Bessel Kernel , 1993 .

[17]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[18]  Peter J. Forrester,et al.  The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble , 2007, 0704.1926.

[19]  D. Romik.  The Surprising Mathematics of Longest Increasing Subsequences , 2015 .

[20]  M. Vanlessen,et al.  Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory , 2005 .

[21]  Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases , 2011, 1104.4940.

[22]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[23]  S. Zohren,et al.  On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble , 2016, 1611.00631.

[24]  J. Møller,et al.  Determinantal point process models and statistical inference , 2012, 1205.4818.

[25]  S. P. Hastings,et al.  A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation , 1980 .

[26]  P. Forrester,et al.  Asymptotic correlations at the spectrum edge of random matrices , 1995 .

[27]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[28]  P. Forrester.  The spectrum edge of random matrix ensembles , 1993 .

[29]  Shuai-Xia Xu,et al.  Painlevé XXXIV Asymptotics of Orthogonal Polynomials for the Gaussian Weight with a Jump at the Edge , 2011 .

[30]  G. Schehr,et al.  Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices , 2013, 1312.2966.

[31]  H. Spohn,et al.  Scale Invariance of the PNG Droplet and the Airy Process , 2001, math/0105240.

[32]  K. Johansson.  Random matrices and determinantal processes , 2005, math-ph/0510038.

[33]  V. Karpman.  An asymptotic solution of the Korteweg-De Vries equation , 1967 .

[34]  T. Claeys,et al.  Hankel Determinant and Orthogonal Polynomials for a Gaussian Weight with a Discontinuity at the Edge , 2015, Communications in Mathematical Physics.

[35]  Benjamin Young,et al.  Asymptotic domino statistics in the Aztec diamond , 2012, 1212.5414.

[36]  P. J. Forrester,et al.  Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles , 2012, 1209.2190.

[37]  Alexei Borodin,et al.  Determinantal point processes , 2009, 0911.1153.

[38]  Alexander Its,et al.  A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics , 1997 .

[39]  P. Deift,et al.  Universality for the Toda Algorithm to Compute the Largest Eigenvalue of a Random Matrix , 2016, 1604.07384.

[40]  Jinho Baik,et al.  A Fredholm Determinant Identity and the Convergence of Moments for Random Young Tableaux , 2001 .

[41]  A. Its,et al.  Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy , 2009, 0901.2473.

[42]  B. Eynard,et al.  Random matrices. , 2015, 1510.04430.

[43]  Thomas Bothner,et al.  Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics , 2017, 1702.04462.

[44]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[45]  Missing levels in correlated spectra , 2004, nucl-th/0403006.

[46]  Stephanos Venakides,et al.  Strong asymptotics of orthogonal polynomials with respect to exponential weights , 1999 .

[47]  S. Majumdar,et al.  Truncated Linear Statistics Associated with the Top Eigenvalues of Random Matrices , 2016, 1609.08296.

[48]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[49]  G. Olshanski,et al.  Asymptotics of Plancherel measures for symmetric groups , 1999, math/9905032.