Gap Probability for the Hard Edge Pearcey Process

Abstract

The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. This paper deals with the gap probability for the thinned/unthinned hard edge Pearcey process over the interval (0, s) by working on a \(3\times 3\) matrix-valued Riemann–Hilbert problem for the relevant Fredholm determinants. We establish an integral representation of the gap probability via the Hamiltonian related to a new system of coupled differential equations. Together with some remarkable differential identities for the Hamiltonian, we derive the large gap asymptotics for the thinned hard edge Pearcey process, including the explicitly evaluation of the constant factor in terms of the Barnes G-function. As an application, we also obtain the asymptotic statistical properties of the counting function for the hard edge Pearcey process.