Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations

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;λ,\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 $\vec{t}=(t_1,t_2,\dots ,t_{2m})\in {\mathbb{R}}^{2m-1}\times (0,\infty )$, 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 Painlevé system generalizing the Painlevé 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 Painlevé system.

51 pages, 7 figures