Relationships between $τ$-function and Fredholm determinant expressions for gap probabilities in random matrix theory

Details Relationships between $τ$-function and Fredholm determinant expressions for gap probabilities in random matrix theory Relationships between $τ$-function and Fredholm determinant expressions for gap probabilities in random matrix theory

2006-04-13

arxiv.org/abs/math-ph/0604027v2

2006 Nonlinearity 19 1643-1656

Physics

Mathematical Physics

16 pages, amstex; (v2) published version, some equations corrected

Scientific paper

10.1088/0951-7715/19/7/012

The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of $\tau$-functions. Extending recent work relating to the soft edge, it is shown that these $\tau$-functions, and their generalizations to contain a generating function parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry.

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