Universality for random matrices with an edge spectrum singularity

TL;DR

This paper investigates the universal behavior of eigenvalue distributions near the spectral edge in invariant random matrix ensembles, extending previous results using advanced analytical techniques involving Painlevé equations.

Contribution

It provides a new asymptotic analysis of eigenvalue statistics at the spectral edge, employing Riemann-Hilbert problems and Painlevé-XXXIV solutions, extending prior work by Forrester and Witte.

Findings

Derived large-$n$ asymptotics for edge eigenvalue distributions.

Extended universality results to broader parameter ranges.

Connected eigenvalue statistics to Painlevé-XXXIV solutions.

Abstract

We study invariant random matrix ensembles \begin{equation*} \mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the underlying density of states is one-cut regular. Considering the average \begin{equation*} E_n[\phi;\lambda,\alpha,\beta]:=\mathbb{E}_n\bigg(\prod_{\ell=1}^n\big(1-\phi(\lambda_{\ell}(M))\big)\omega_{\alpha\beta}(\lambda_{\ell}(M)-\lambda)\bigg),\ \ \ \ \ \omega_{\alpha\beta}(x):=|x|^{\alpha}\begin{cases}1,&x<0\\ \beta,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $\phi$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $\lambda$ lies within the soft edge boundary layer, and $(\alpha,\beta)\in\mathbb{R}\times\mathbb{C}$ satisfy $\alpha>-1,\beta\notin(-\infty,0)$. Our results are obtained by using Riemann-Hilbert problems for orthogonal polynomials and integrable operators and they extend previous results of Forrester and Witte \cite{FW} that were obtained by an application of Okamoto's $\tau$-function theory. A key role throughout is played by distinguished solutions to the Painlev\'e-XXXIV equation.