Free moment measures and laws

Abstract

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on Rn by the map U↦(∇U)#e−Udx. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure e−Udx replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.