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

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 (I-γ K(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.