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

Abstract

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.

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

References in corpus (3)

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• A determinantal formula for the GOE Tracy-Widom distribution
• Simple strong glass forming models: mean-field solution with activation

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