Convergence rates of Gibbs measures with degenerate minimum

Abstract

We study convergence rates of Gibbs measures, with density proportional to exp(−f(x)∕t), as t→0 where f: Rd→R admits a unique global minimum at x⋆. We focus on the case where the Hessian is not definite at x⋆. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of f at x⋆, which depends on every coordinate. We give an algorithm yielding such an expansion if the polynomial order of x⋆ is no more than 8, in connection with Hilbert’s 17th problem. However, we prove that the case where the order is 10 or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.