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

摘要

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