Fast approach to the Tracy-Widom law at the edge of GOE and GUE

Abstract

We study the rate of convergence for the largest eigenvalue distributions in the Gaussian unitary and orthogonal ensembles to their Tracy-Widom limits. We show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the centering and scaling constants. The arguments here also shed light on more complicated cases of Laguerre and Jacobi ensembles, in both unitary and orthogonal versions. Numerical work shows that the suggested constants yield reasonable approximations, even for surprisingly small values of N.

Published in at http://dx.doi.org/10.1214/11-AAP819 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• Approximate null distribution of the largest root in multivariate analysis
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GOE
• Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
• Asymptotics of Orthogonal Polynomials via Recurrence Relations

Cited by in corpus (16)

• Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Limiting distribution of maximal crossing and nesting of Poissonized random matchings
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE
• A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices
• The Asymptotic Distribution of Modularity in Weighted Signed Networks
• Asymptotic expansions relating to the distribution of the length of longest increasing subsequences
• Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Rare-event Analysis for Extremal Eigenvalues of white Wishart matrices
• Tracy-Widom limit for Kendall's tau