Ewens measures on compact groups and hypergeometric kernels

Abstract

The general topic of the article is the study of probability measures on the unitary group U(N) and its closed subgroups. The authors start from the Haar measure on a compact subgroup on U(N) and show that such distributed random matrix is a product of independent random reflections. The authors also decompose certain determinants of random matrices into products of independent simply-distributed random variables. More precisely, they discuss the decomposition of det(Id-U), where U is either orthogonal, or unitary, or symplectic random matrix sampled from the Haar measure. In the second part of the article the deformations of the Haar measure on U(N) (and on some of its subgroups) are discussed. The authors consider the measure assigning to a matrix U the density proportional to det(Id-U) z ¯ det(Id-U ¯) z (where z is a complex parameter) and explain that this measure is a natural analogue of the Ewens measure on the symmetric group S(n). Recall that the Ewens measure is a probability distribution on S(n) with weight of a permutation g being proportional to θ c(g) , where c(g) stays for the number of cycles in g and θ is a real parameter. The similarity between the Ewens measures and the above measures on U(N) has been known before, especially in the context of harmonic analysis on the infinite symmetric group and the infinite-dimensional unitary group [see S. Kerov, G. Olshanski and A. Vershik, Invent. Math 158, No. 3, 551–642 (2004; Zbl 1057.43005) and G. Olshanski, Funct. Anal. 205, No. 2, 464–524 (2003; Zbl 1036.43002)]. The authors give some new explanations for the relations between these two families of measures. Finally, the distribution of the eigenvalues of the random matrices sampled from the above deformations of the Haar measure is discussed in the last part of the article.