We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDEs, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on to prove the following. Suppose is a probability measure on given by uniformly convex and semiconcave potentials , and suppose that the sequence is asymptotically approximable by trace polynomials. Then the moments of converge to a noncommutative law . Moreover, the free entropies , , and agree and equal the limit of the normalized classical entropies of .
"An elementary approach to free entropy theory for convex potentials." Anal. PDE 13 (8) 2289 - 2374, 2020. https://doi.org/10.2140/apde.2020.13.2289