Asymptotic Study of Deformed Fredholm Determinants Associated with Confluent Hypergeometric Kernels in Random Matrix Theory


Student thesis: Doctoral Thesis

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Award date23 Aug 2022


In this thesis, we obtain the asymptotic behavior of the deformed confluent hypergeometric kernel determinant det(I - γ K(α,β), where 0 ≤ γ < 1 and  K(α,β) is the trace class operator acting on L2(-s,s) with the confluent hypergeometric kernel. Here parameter α > -½ while β is a pure imaginary parameter. The confluent hypergeometric kernel appears as the limiting eigenvalue correlation kernel of unitary random matrix ensembles with a Fisher-Hartwig singularity in the bulk. It reduces to a type-I Bessel kernel when β = 0 and a sine kernel when α = β = 0. This determinant gives the gap probability of a determinantal point process with the confluent geometric kernel after thinning, that is, each particle in the point process is removed independently with a removal probability 1-γ.

When 0 ≤ γ < 1 is fixed, the large gap asymptotics of the deformed determinant are derived including the explicit constant term. Our derivation is based on its explicit integral representation involving the Hamiltonian of the coupled Painlevé V system.

It is well-known that there is a non-trivial transition between the deformed (when γ < 1) and undeformed (when γ = 1) Fredholm determinants. Then, we consider the scenario when s → ∞ and γ → 1- simultaneously in a Stokes type scaling regime. Asymptotic expansions are given explicitly, which is significantly different from the case when γ is fixed. The Deift-Zhou nonlinear steepest method for Riemann-Hilbert problems plays an important role in the derivation for both cases.