Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlev\'{e} XXXIV equations

TL;DR

This paper investigates Gaussian unitary ensembles with pole singularities near the soft edge, deriving asymptotics of the partition function and correlation kernel through a new coupled Painlevé system, revealing a novel universal class.

Contribution

It introduces a new coupled Painlevé system generalizing Painlevé XXXIV and connects it to the asymptotics of singularly perturbed Gaussian ensembles.

Findings

Asymptotic formulas for the partition function involving the coupled Painlevé system.

Derivation of the large n limit of the correlation kernel.

Identification of a new universal class based on the associated -function.

Abstract

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure \begin{equation*} \frac{1}{C_n} e^{- n\textrm{tr}\, V(M;\lambda,\vec{t}\;)}dM, \end{equation*} over the space of $n \times n$ Hermitian matrices $M$, where $V(x;\lambda,\vec{t}\;):= 2x^2 + \sum_{k=1}^{2m}t_k(x-\lambda)^{-k}$ with $\vec{t}= (t_1, t_2, \ldots, t_{2m})\in \mathbb{R}^{2m-1} \times (0,\infty)$, in the multiple scaling limit where $\lambda\to 1$ together with $\vec{t} \to \vec{0}$ as $n\to \infty$ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlev\'e system generalizing the Painlev\'e XXXIV equation. The large $n$ limit of the correlation kernel is also derived, which leads to a new universal class built out of the $\Psi$-function associated with the coupled Painlev\'e system.