On the deformed Pearcey determinant

TL;DR

This paper analyzes the deformed Pearcey determinant related to the Pearcey process with thinning, deriving its large gap asymptotics, and revealing a transition in behavior as the thinning parameter varies.

Contribution

We derive an integral representation and asymptotic formulas for the deformed Pearcey determinant, including the constant term, highlighting differences from the undeformed case.

Findings

Large gap asymptotics with exact constant term

Distinct asymptotic behavior from the undeformed case

Central limit theorem for the Pearcey process

Abstract

In this paper, we are concerned with the deformed Pearcey determinant $\det\left(I-\gamma K^{\mathrm{Pe}}_{s,\rho}\right)$, where $0 \leq \gamma<1$ and $K^{\mathrm{Pe}}_{s,\rho}$ stands for the trace class operator acting on $L^2\left(-s, s\right)$ with the classical Pearcey kernel arising from random matrix theory. This determinant corresponds to the gap probability for the Pearcey process after thinning, which means each particle in the Pearcey process is removed independently with probability $1-\gamma$. We establish an integral representation of the deformed Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. Together with some remarkable differential identities for the Hamiltonian, this allows us to obtain the large gap asymptotics, including the exact calculation of the constant term, which complements our previous work on the undeformed case (i.e., $\gamma=1$). It comes out that the deformed Pearcey determinant exhibits a significantly different asymptotic behavior from the undeformed case, which suggests a transition will occur as the parameter $\gamma$ varies. As an application of our results, we obtain the asymptotics for the expectation and variance of the counting function for the Pearcey process, and a central limit theorem as well.