The largest eigenvalue distribution of the Laguerre unitary ensemble

TL;DR

This paper analyzes the distribution of the largest eigenvalue in the Laguerre unitary ensemble, deriving differential equations and asymptotic behaviors, including connections to Painlevé equations, for finite and large matrix sizes.

Contribution

It introduces new differential equations and asymptotic formulas for the largest eigenvalue distribution, linking orthogonal polynomial theory with Painlevé equations.

Findings

Derived second order differential equations for beta n(t) and rn(t).

Established asymptotic solutions near singular points for large n.

Connected the distribution to Chazy and Painlevé equations.

Abstract

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are between 0 and t, i.e., the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely {\alpha}n(t) and \b{eta}n(t), as well as three auxiliary quantities, denoted by rn(t), Rn(t) and sigma n(t). We establish the second order differential equations for both beta n(t) and rn(t). By investigating the soft edge scaling limit when alpha = O(n) as n ! 1 or alpha is finite, we derive a PII , the sigma-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy equation for rho(t) = d/dt(capsigma(t) with capsigma(t) satisfying the sigma-form of PIV or PV . 1 I