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_s^(α,β) denote the trace class integral operator acting on L²(−s, s) with the confluent hypergeometric kernel. Our fo-cus is on deriving the asymptotics of the Fredholm determinant det(I − γK_s^(α,β) ) 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_k^(α,β) (s) of the integral operator K_s^(α,β) 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. © World Scientific Publishing Company