Asymptotic Integral Kernel for Ensembles of Random Normal Matrix with Radial Potentials

TL;DR

This paper analyzes the asymptotic behavior of the integral kernel in ensembles of normal random matrices with radial potentials, providing error estimates and a scaling limit related to the Segal--Bargmann space.

Contribution

It introduces a steepest descents method to derive asymptotic expansions and error bounds for the integral kernel of these matrix ensembles with radial potentials.

Findings

Derived asymptotic expansion with error estimates

Established a scaling limit for the n-point function

Connected the kernel to the Segal--Bargmann space

Abstract

We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{N}(z_{1},...,z_{N}) = Z_{N}^{-1} e^{-N\Sigma_{i=1}^{N}V_{\alpha}(z_{i})} \Pi_{1\leqi<j\leqN}|z_{i}-z_{j}|^{2} where V_{\alpha}(z)=|z|^{\alpha}, z \in C and \alpha \in ]0,\infty[. Asymptotic analysis with error estimates are obtained. A corollary of this expansion is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal--Bargmann space.