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abstract: 'The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. We consider the Fredholm determinant of a trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel. Based on a steepest descent analysis for a $3\times 3$ matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as $s\to +\infty$, which is also interpreted as large gap asymptotics in the context of random matrix theory.'
author:
- 'Dan Dai,  Shuai-Xia Xu and Lun Zhang'
title: Asymptotics of Fredholm determinant associated with the Pearcey kernel
---

Introduction and statement of the results
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In a series of papers [@BH]–[@BH96a], Brézin and Hikami initiated the studies of deformed complex Gaussian unitary ensemble (GUE) of the form $$\frac{1}{Z_n}e^ {-n \textrm{Tr}\left(\frac{M^2}{2}-AM \right)}{\,\mathrm{d}}M,$$ defined on the space of $n \times n$ Hermitian matrices, where $Z_n$ is a normalization constant and $A$ is a deterministic matrix also known as the external source. An interesting feature of this matrix ensemble is that it provides a simple model to create 