Asymptotics of the deformed Fredholm determinant of the confluent hypergeometric kernel

Abstract

In this paper, we consider the deformed Fredholm determinant of the confluent hypergeometric kernel. This determinant represents the gap probability of the corresponding determinant point process where each particle is removed independently with probability 1 − γ, 0 ≤ γ < 1. We derive asymptotics of the deformed Fredholm determinant when the gap interval tends to infinity, up to and including the constant term. As an application of our results, we establish a central limit theorem for the eigenvalue counting function and a global rigidity upper bound for its maximum deviation.

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.…”

“…When the weight is supported on the unit circle or on the real line, this problem was studied by many authors, including Lenard, Fisher, Hartwig, Widom, Basor, Böttcher, Silbermann, Ehrhardt, Deift, Its and Krasovsky, see e.g. [17,40,26] for some historical background, [30,27,62,36,37] for structured determinants with discontinuities near a hard edge, and [33,44] for merging discontinuities in the bulk.…”