Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as τ-Functions – Spectrum Singularity Case

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 τ-function, in the sense of Okamoto, for the Painlevé system PIII. This then leads to a factorisation of the probability as the product of two τ-functions for the Painlevé system dash. A previous study has given a formula of this type but involving dash systems with different parameters consequently implying an identity between products of τ-functions or equivalently sums of Hamiltonians.