Gap Formation Probability of the $\alpha-$ Ensemble

TL;DR

This paper investigates the asymptotic probability of gap formation in the spectrum of Hermitean random matrices with a specific weight function, revealing universality for certain parameter ranges and dependence on parameters for others.

Contribution

It extends Dyson's continuum approximation to analyze gap probabilities for a class of Hermitean matrices with power-law weights, highlighting universality and parameter dependence.

Findings

Probability is universal for α>1/2.

Probability depends on α for 0<α<1/2.

Provides asymptotic analysis of eigenvalue gaps.

Abstract

In this paper we employ the continuum approximation of Dyson to determine the asymptotic gap formation probability in the spectrum of $N\times N$ Hermitean random matrices. The associated orthogonal polynomials has weight function, $w(x)=e^{-u(x)},$ where $u(x)=x^{\alpha},\;\alpha>0,\;0<x<\infty.$ It is shown that the probability that the scaled interval $(0,s)$ contains no eigenvalues is universal for $\alpha>1/2$ and depends on $\alpha$ for $0<\alpha<1/2.$