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 $\mathcal{K}^{(\alpha,\beta)}_s$ denote the trace class integral operator acting on $L^2(-s, s)$ with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant $\det(I-\gamma \mathcal{K}^{(\alpha,\beta)}_s)$ as $s \to +\infty$, while simultaneously $\gamma \to 1^-$ in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues $\lambda^{(\alpha, \beta)}_k(s)$ of the integral operator $\mathcal{K}^{(\alpha,\beta)}_s$ as $s \to +\infty$. 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.