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 the relevant Fredholm determinants. We establish an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations. Together with some remarkable differential identities for the Hamiltonian, we derive the large gap asymptotics for the thinned case, up to and including the constant term. As an application, we also obtain the asymptotic statistical properties of the counting function for the hard edge Pearcey process. Contents

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

“…There exists also a vast literature on other structured determinants with Fisher-Hartwig singularities, see e.g. [25,26] for Fredholm determinants, [7,35,23] for Toeplitz+Hankel determinants, and [19] for a biorthogonal generalization of Hankel determinants.…”

Exponential moments for disk counting statistics at the hard edge of random normal matrices

“…Remark 1.1. Hard wall ensembles from Hermitian random matrix theory have been well-studied in the literature, see for example [46,41,30,27,62,36,37]; see also [34] for a soft/hard edge. We remark that imposing a hard wall in the interior of a one-dimensional droplet has a well-known global effect on the equilibrium measure, in contrast to (1.5) which just alters the measure locally at the edge.…”

Asymptotics of the deformed Fredholm determinant of the confluent hypergeometric kernel

“…Therefore, one can establish crucial differential identities with respect to parameters (γ in our case). As our subsequent asymptotic analysis is uniform for γ in compact subsets of [0, 1), we can finally evaluate the constant explicitly by integrating with respect to γ; see similar derivations in [21,22,47].…”

“…As only the diagonal entries of Ψ(z; s) are involved in the subsequent asymptotic derivation, we just focus on the diagonal entries of R 1 (z). Substituting the expressions of J R,1 (z) in (6.44)-(6.46) into (6.48), we obtain (6.43) by a direct residue computation and the observation (J R,1 (z)) 11 = −(J R,1 (z)) 22 . This finishes the proof of the proposition.…”

“…In the literature, the undeformed and deformed distributions have been well studied for the classical kernels such as the sine kernel, Bessel kernel and Airy kernel (see e.g. [1,2,8,10,12,14,24,28]), as well as the non-classical ones, such as the Pearcey kernel and the hard edge Pearcey kernel in [21,22]. Although the integral expressions for the Fredholm determinant may look very similar between the undeformed case γ = 1 and the deformed one 0 ≤ γ < 1, the large gap asymptotics are essentially different.…”