Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

We study the probability distribution function (PDF) of the smallest eigenvalue of Laguerre-Wishart matrices $W = X^\dagger X$ where $X$ is a random $M \times N$ ($M \geq N$) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large $N$, large $M$ with $M/N \to 1$ -- i.e. for quasi-square large matrices $X$ -- we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlev\'e III equation, as found by Tracy and Widom, using Fredholm operators techniques. Furthermore, our method allows us to compute explicitly the first $1/N$ corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and P\'ech\'e. We also study the soft edge limit, when $M-N \sim {\cal O}(N)$, for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

[1]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[2]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III , 1981 .

[3]  Kazuo Okamoto.  On the τ-function of the Painlevé equations , 1981 .

[4]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[5]  Verbaarschot,et al.  Spectral density of the QCD Dirac operator near zero virtuality. , 1993, Physical Review Letters.

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

[7]  Craig A. Tracy,et al.  Mathematical Physics © Springer-Verlag 1994 Fredholm Determinants, Differential Equations and Matrix Models , 2022 .

[8]  A. Ronveaux,et al.  Laguerre-Freud's equations for the recurrence coefficients of semi-classical orthogonal polynomials , 1994 .

[9]  C. Tracy,et al.  Level-spacing distributions and the Airy kernel , 1992, hep-th/9211141.

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

[11]  Peter J. Forrester,et al.  Complex Wishart matrices and conductance in mesoscopic systems: Exact results , 1994 .

[12]  Alphonse P. Magnus,et al.  Painleve´-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials , 1995 .

[13]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[14]  T. H. Baker,et al.  Random matrix ensembles with an effective extensive external charge , 1998 .

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

[16]  K. Johansson.  Shape Fluctuations and Random Matrices , 1999, math/9903134.

[17]  Spohn,et al.  Universal distributions for growth processes in 1+1 dimensions and random matrices , 2000, Physical review letters.

[18]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[19]  Noureddine El Karoui.  A rate of convergence result for the largest eigenvalue of complex white Wishart matrices , 2004, math/0409610.

[20]  Yang Chen,et al.  Orthogonal polynomials with discontinuous weights , 2005, math-ph/0501057.

[21]  S. Majumdar,et al.  Exact asymptotic results for the Bernoulli matching model of sequence alignment. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[23]  Craig A. Tracy,et al.  Nonintersecting Brownian Excursions , 2006, math/0607321.

[25]  G. Biroli,et al.  On the top eigenvalue of heavy-tailed random matrices , 2006, cond-mat/0609070.

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

[27]  I. Johnstone.  MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY-WIDOM LIMITS AND RATES OF CONVERGENCE. , 2008, Annals of statistics.

[28]  Yang Chen,et al.  Painlevé V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles , 2008, 0807.4758.

[29]  S. Majumdar,et al.  Exact distribution of the maximal height of p vicious walkers. , 2008, Physical review letters.

[30]  Satya N Majumdar,et al.  Nonintersecting Brownian interfaces and Wishart random matrices. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  H. Spohn,et al.  One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. , 2010, Physical review letters.

[32]  P. Forrester.  Log-Gases and Random Matrices (LMS-34) , 2010 .

[33]  Renormalization-group theory for finite-size scaling in extreme statistics. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Isaac Pérez Castillo,et al.  Large deviations of the smallest eigenvalue of the Wishart-Laguerre ensemble. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  P. Forrester.  Log-Gases and Random Matrices , 2010 .

[36]  Alberto Rosso,et al.  Free-energy distribution of the directed polymer at high temperature , 2010, 1002.4560.

[37]  V. Dotsenko.  Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers , 2010, 1003.4899.

[38]  P. Forrester,et al.  Non-intersecting Brownian walkers and Yang–Mills theory on the sphere , 2010, 1009.2362.

[39]  Henry P. McKean,et al.  Fredholm determinants , 2011 .

[40]  S. Majumdar,et al.  A simple derivation of the Tracy–Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix , 2011, 1102.0738.

[41]  J. Quastel,et al.  Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions , 2010, 1003.0443.

[42]  Zongming Ma,et al.  FAST APPROACH TO THE TRACY-WIDOM LAW AT THE EDGE OF GOE AND GUE. , 2011, The annals of applied probability : an official journal of the Institute of Mathematical Statistics.

[43]  Karl Liechty.  Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials , 2012, Journal of statistical physics.

[44]  Zongming Ma,et al.  Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices , 2012, 1203.0839.

[45]  Peter J. Forrester,et al.  Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N , 2012, 1210.4438.

[46]  J. Baik,et al.  Limiting distribution of maximal crossing and nesting of Poissonized random matchings , 2011, 1111.0269.

[47]  S. Majumdar,et al.  Top eigenvalue of a random matrix: large deviations and third order phase transition , 2013, 1311.0580.

[48]  Completing the picture for the smallest eigenvalue of real Wishart matrices. , 2014, Physical review letters.

[49]  Todd Kemp.  Beyond universality in random matrix theory , 2015 .

[50]  W. Hachem,et al.  A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices , 2015, 1509.04910.

[51]  A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge , 2015, 1504.00235.