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

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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 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.

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  1. Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia

    N. S. Witte

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Witte, N.S. Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as τ-Functions – Spectrum Singularity Case. Letters in Mathematical Physics 68, 139–149 (2004). https://doi.org/10.1023/B:MATH.0000045556.53148.02

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