Asymptotics of the deformed Fredholm determinant of the confluent hypergeometric kernel

TL;DR

This paper derives the asymptotic behavior of the deformed Fredholm determinant associated with the confluent hypergeometric kernel, revealing insights into gap probabilities, eigenvalue fluctuations, and rigidity in determinantal point processes.

Contribution

It provides the first detailed asymptotic analysis of the deformed Fredholm determinant for this kernel, including the constant term, and applies these results to eigenvalue statistics.

Findings

Asymptotics of the deformed Fredholm determinant including the constant term.

Establishment of a central limit theorem for eigenvalue counting.

Global rigidity bounds for eigenvalue deviations.

Abstract

In this paper, we consider the deformed Fredholm determinant of the confluent hypergeometric kernel. This determinant represents the gap probability of the corresponding determinantal point process where each particle is removed independently with probability $1- \gamma$, $0 \leq \gamma <1$. 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.