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

2015/06/08 by Anthony Perret, Grégory Schehr, Gregory Schehr · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Combinatorics #Conjecture #Distribution (mathematics) #Eigenvalues and eigenvectors #Fredholm determinant #Gaussian #Generating function #Laguerre polynomials #Limit (mathematics) #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Molecular spectroscopy and chirality #Operator (biology) #Orthogonal polynomials #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Statistics #Wishart distribution #cond-mat.stat-mech #math-ph #math.MP #math.PR

paper · pdf · doi:10.1142/s2010326316500015

published as Random Matrices: Theory Appl. 05, 1650001 (2016) · 28 pages, 2 Figures

arxiv created 2015/06/08 · openalex publication_date 2015/10/06 · arxiv updated 2016/04/15 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05

Abstract

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices [Formula: see text] where [Formula: see text] is a random [Formula: see text] ([Formula: see text]) 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 [Formula: see text], large [Formula: see text] with [Formula: see text] — i.e. for quasi-square large matrices [Formula: see text] — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlevé III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first [Formula: see text] corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Péché. We also study the soft edge limit, when [Formula: see text], for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

Citations

• Exact asymptotic results for the Bernoulli matching model of sequence alignment
• On the distribution of the largest eigenvalue in principal components analysis
• A rate of convergence result for the largest eigenvalue of complex white Wishart matrices
• Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
• Orthogonal polynomials with discontinuous weights
• Limiting distribution of maximal crossing and nesting of Poissonized random matchings
• Fredholm determinants, differential equations and matrix models
• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Cited by

• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble