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

2024/03/25 by Dan Dai, Dai, Dan, Luming Yao +2
Engineering · Mathematics · #33C10 #34M50 #45C05 #82B26 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Heat Transfer and Mathematical Modeling #Material Science and Thermodynamics #Mathematical Physics (math-ph) #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2403.16475

openalex publication_date 2024/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate a determinantal point process on the interval (-s,s), associated with the confluent hypergeometric kernel. Let K^(α,β)_s denote the trace class integral operator acting on L²(-s, s) with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant det(I-γK^(α,β)_s) as s → +∞, while simultaneously γ→ 1^- in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues λ^{(α, β)}_k(s) of the integral operator K^(α,β)_s as s → +∞. 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.

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