Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as \(\tau\)-Functions -- Spectrum Singularity case

Abstract

The probability for the exclusion of eigenvalues from an interval \((-x,x)\) symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter \( a \) (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a \(\tau\)-function, in the sense of Okamoto, for the Painlev\'e system \PIII. This then leads to a factorisation of the probability as the product of two \(\tau\)-functions for the Painlev\'e system \PIIIdash. A previous study has given a formula of this type but involving \PIIIdash systems with different parameters consequently implying an identity between products of \(\tau\)-functions or equivalently sums of Hamiltonians.