Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System

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 Painlevé V system in dimension four. This leads to a Painlevé-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 and the sine-kernel determinant, including the constant term conjectured earlier by Dyson.

Asymptotics for averages over classical orthogonal ensembles

“…In this case, like in the case m = 1 of Theorem 2.2, it is also possible to evaluate the multiplicative constant in the asymptotic expansion in terms of solutions to the Painlevé V equation. Yet another example consists of symbols with a gap, but with an additional Fisher-Hartwig singularity inside the gap, as considered in [43]. This situation is related to a system of coupled Painlevé V equations.…”

Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems

“…5 where 1 is used instead of 1 . As was pointed out there, (26) can be transformed into a Painlevé IV equation satisfied by ( 1 ) ∶= (− 1 ).…”

“…In Ref. 26, the symmetric gap probability of a circular unitary ensemble with Fisher‐Hartwig singularities was represented by the integral of the Hamiltonian for a coupled Painlevé V system. In Ref.…”

The smallest eigenvalue distribution of the Jacobi unitary ensembles

“…In addition to being applied to derive the constant term in the asymptotics of the gap probability of unitary ensembles, RH method is a powerful tool to study many other problems in large‐dimensional unitary ensembles, for example, the partition function, 14,15 the gap probability distribution, 16,17 the correlation kernel, 18 and orthogonal polynomials 19 . For finite n analysis, the ladder operators adapted to monic orthogonal polynomials are usually used.…”