Probabilistic Operator Algebra Seminar: An Elementary Approach to Free Gibbs States with Convex Potentials

Seminar | January 28 | 2-4 p.m. | 736 Evans Hall

 David Andrew Jekel, UCLA

 Department of Mathematics

We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb C)_{sa}^m$ to prove the following. Suppose $\mu _N$ is a probability measure on $M_N(\mathbb C)_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $\mu _N$ converge to a noncommutative law λ. Moreover, the free entropies $\chi (\lambda )$, $\underline {\chi }(\lambda )$, and $\chi ^*(\lambda )$ agree and equal the limit of the normalized classical entropies of $\mu _N$. We also sketch further applications to conditional expectations, relative entropy, and free transport for these free Gibbs states.