BESSIS–MOUSSA–VILLANI CONJECTURE AND GENERALIZED GAUSSIAN RANDOM VARIABLES

Abstract

In this paper we give the solution of Bessis–Moussa–Villani (BMV) conjecture for the generalized Gaussian random variables [Formula: see text] where f is in the real Hilbert space [Formula: see text]. The main examples of generalized Gaussian random variables are q-Gaussian random variables, (-1 ≤ q ≤ 1), related to q-CCR relation and other commutation relations. We will prove that BMV conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function [Formula: see text] is positive-definite function on the real line. The case q = 0, i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by Fannes and Petz.23