Extending multimatrix model theory to non-convex potentials

Determine which results known for multimatrix models with convex potentials extend to the non-convex case V, specifically for random matrix tuples Y^N with joint density proportional to exp(-N^2 tr_N(V)). In particular, ascertain whether the large-N limits of normalized traces exist for noncommutative polynomial test functions and whether the resulting limiting functional exhibits the standard free Gibbs law properties such as the Schwinger–Dyson equations and stationarity for the free Langevin stochastic differential equation when V is non-convex.

Background

The paper develops asymptotic expansions, transport maps, and strong convergence for multimatrix models under convexity assumptions on the potential V. For convex V, the authors recall several fundamental consequences, including Schwinger–Dyson equations, Langevin SDE stationarity, and a free-entropy characterization. They then note that analogous questions for non-convex V remain unresolved, despite some progress in the literature and connections to large deviations.

Clarifying which parts of the convex theory survive in the non-convex setting is important both for the mathematical structure of free Gibbs laws and for applications in random matrix theory, where many natural models involve non-convex interactions.