Une famille à un paramètre d'ensembles unitaires

Abstract

A one-parameter family of unitary matrix ensembles is studied. We define the ensemble E 2( z) of unitary random matrices, whose eigenvalues are ϵ j = exp 2 πiϕ j , by the following joint probability density: f(ε 1ε 2…εn) dϕ 1 dϕ 2… dϕ n∝ π i<j ε i−ε j ε i−zεε j 2 d ϕ1 d ϕ2… d ϕ1 d n . It realizes a continuous interpolation between the distribution of the eigenvalues in the Dyson unitary ensemble E 2 for z = 0 and the uniform distribution of n random points on the unit circle for z = 1. The thermodynamic analogy with a circular or linear classical repulsive gas at temperature β −1 = 1 2 is developed. The isotherm β = 2 and the corresponding virial series are exactly calculated. All the correlation functions are given in the limit of an infinite linear gas or of an infinite series of levels. This model shows the short-range repulsion effect between eigenvalues but no long-range crystalline order, which is a strong characteristic of all ensembles so far studied.