A note about Selberg’s integrals in relation with the beta-gamma algebra

Abstract

To prove their formulae for the moments of the characteristic polynomial of the generic matrix of U(N), Keating and Snaith [8] (see also Keating [7]) use Selberg’s integrals as a ‘black box.’ In this note, we point out some identities in law which are equivalent to the expressions of Selberg’s integrals and which involve beta, gamma, and normal variables. However, this is a mere probabilistic translation of Selberg’s results, and does not provide an independent proof of them. An outcome of some of these translations is that certain logarithms of (Vandermonde) random discriminants are self-decomposable, which hinges on the self-decomposability of the logarithms of the beta (a, b) (2a +b = ≥ 1) and gamma (a > 0) variables. Such self-decomposability properties have been of interest in some joint papers with D. Madan.