Journal article

Evenness symmetry and inter-relationships between gap probabilities in random matrix theory

Peter J Forrester

Forum Mathematicum | WALTER DE GRUYTER & CO | Published : 2006

Abstract

Our interest is in the generating function E (I;ξ;w ) for the probabilities E (n;I;w ) that in a matrix ensemble with unitary (β = 2) or orthogonal (β = 1) symmetry, characterized by the weight w (λ) and having N eigenvalues, the interval I contains exactly n eigenvalues. Using a determinant formula for E , a general quadratic identity is obtained which relates E in the case I and w (x) even to a product of generating functions E with different I, w (λ) and N, and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum E (2n - 1; I; w ) + E (2n; I; w ) for N even, I = (-t, t) and w an even classical weight is shown to equal E (n; (0, t ); w ) fo..

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