# Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlev\'{e} V system

## TL;DR

This paper analyzes the gap probability in the circular unitary ensemble with a Fisher-Hartwig singularity, connecting it to Painlevé V systems and deriving asymptotics for the associated determinants.

## Contribution

It introduces a Painlevé V system representation for the gap probability in the ensemble with Fisher-Hartwig singularity, including large gap asymptotics and constant terms.

## Key findings

- Explicit evaluation of gap probability via Painlevé V Hamiltonian
- Derivation of large gap asymptotics including constants
- Reproduction of known results for sine-kernel and Fisher-Hartwig cases

## Abstract

We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlev\'{e} V system in dimension four. This leads to a Painlev\'e-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson.