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

Abstract

Our interest is in the generating function E N,β(I;ξ;wβ) for the probabilities E N,β(n;I;wβ) that in a matrix ensemble with unitary (β = 2) or orthogonal (β = 1) symmetry, characterized by the weight wβ(λ) and having N eigenvalues, the interval I contains exactly n eigenvalues. Using a determinant formula for EN,2, a general quadratic identity is obtained which relates EN,2 in the case I and w2(x) even to a product of generating functions E N,2 with different I, w2(λ) and N, and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum EN,1 (2n - 1; I; w1) + EN,1 (2n; I; w 1) for N even, I = (-t, t) and w1 an even classical weight is shown to equal EN/2,2(..