On the deformed Pearcey determinant

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Article number108291
Journal / PublicationAdvances in Mathematics
Volume400
Online published24 Feb 2022
Publication statusPublished - 14 May 2022

Abstract

In this paper, we are concerned with the deformed Pearcey determinant det⁡ (I−γKs,ρPe), where  0 ≤ γ < 1 and Ks,ρPe stands for the trace class operator acting on L(−s,s) 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−γ. 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., γ=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 γ 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.

Research Area(s)

  • Asymptotic analysis, Pearcey determinant, Random matrix theory, Riemann-Hilbert problems

Citation Format(s)

On the deformed Pearcey determinant. / Dai, Dan; Xu, Shuai-Xia; Zhang, Lun.

In: Advances in Mathematics, Vol. 400, 108291, 14.05.2022.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review