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

Content Provider	Semantic Scholar
Author	Witte, Nicholas S.
Copyright Year	2001
Abstract	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
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Alternate Webpage(s)	http://arxiv.org/pdf/math-ph/0307063v1.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article