Universality for random matrices with an edge spectrum singularity

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[ϕ;λ,α,β]:=\mathbb{E}_n\bigg(\prod_{\ell=1}^n\big(1-ϕ(λ_{\ell}(M))\big)ω_{αβ}(λ_{\ell}(M)-λ)\bigg),\ \ \ \ \ ω_{αβ}(x):=|x|^α\begin{cases}1,&x<0\\ β,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $\ddot{I}•$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $\hat{I}»$ lies within the soft edge boundary layer, and $(\hat{I}\pm ,{\hat{I}}^2)\in \mathbb{R}\times \mathbb{C}$ satisfy $\hat{I}\pm >-1,{\hat{I}}^2\notin (-\infty ,0)$. 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 $\ddot{I}„$-function theory. A key role throughout is played by distinguished solutions to the Painlevé-XXXIV equation.

47 pages, 3 figures; to appear in Nonlinearity; Version 2 adds Appendix C and updates literature