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arXiv · 1809.07074

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

Dan Dai·Shuai-Xia Xu·Lun Zhang
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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.

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Dan Dai, Shuai-Xia Xu, Lun Zhang. 2018-09-19. Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlev\'{e} XXXIV equations. https://doi.org/10.1007/s00023-019-00834-y

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