Gap probabilities for double intervals in hermitian random matrix ensembles as τ-functions - the Bessel kernel case

The probability for the exclusion of eigenvalues from an interval (−x, x) symmetrical about the origin for a scaled ensemble of Her-mitian 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 P III. This then leads to a factorisation of the probability as the product of two τ-functions for the Painlevé system P III ′. A previous study has given a formula of this type but involving P III ′ systems with different parameters consequently implying an identity between products of τ-functions or equivalently sums of Hamiltonians. The probability E β (0; J; g(x); N) that a subset of the real line J is free of eigen-values for an ensemble of N × N random matrices with eigenvalue probability density function proportional to (1) N l=1 g(x l) 1≤j<k≤N |x j − x k | β , (β = 1, 2 or 4 according to the ensemble exhibiting orthogonal, unitary or symplectic symmetry respectively) is a fundamental statistic in the study of these ensembles. Most effort has focused on the case where J is a single interval, one endpoint fixed at the edge of the support of the measure defining the ensemble whilst the other is free, and taken to be the independent variable in the system of equations determining the gap probability. There is, however, another interesting case where the set J consists of two disconnected intervals, but related to each other so that there is still only one free variable. For instance there is the result for unitary ensembles of Hermitian matrices that the gap probability for an interval symmetrical about the origin J and an even weight function g 2 (x) (where the integrable examples include Gaussian, symmetric Jacobi and Cauchy weights) factorises [4]