Asymptotics of the confluent hypergeometric process with a varying external potential in the super-exponential region

Abstract

In this paper, we investigate a determinantal point process on the interval [Formula: see text], associated with the confluent hypergeometric kernel. Let [Formula: see text] denote the trace class integral operator acting on [Formula: see text] with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant [Formula: see text] as [Formula: see text], while simultaneously [Formula: see text] in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues [Formula: see text] of the integral operator [Formula: see text] as [Formula: see text]. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift–Zhou nonlinear steepest descent method to analyze the related Riemann–Hilbert problem.