Convergence rates of Gibbs measures with degenerate minimum

TL;DR

This paper investigates the convergence rates of Gibbs measures near degenerate minima, providing a polynomial expansion approach and revealing fundamental differences in behavior depending on the polynomial order.

Contribution

It introduces a higher order nested expansion of the potential function at degenerate minima and analyzes convergence rates, highlighting a phase transition at polynomial order 10.

Findings

Convergence rates depend on the polynomial order of the minimum.

A new algorithm for polynomial decomposition up to order 8.

Fundamental change in behavior occurs at polynomial order 10.

Abstract

We study convergence rates for Gibbs measures, with density proportional to $e^{-f(x)/t}$, as $t \rightarrow 0$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ admits a unique global minimum at $x^\star$. We focus on the case where the Hessian is not definite at $x^\star$. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of $f$ at $x^\star$, which depends on every coordinate. We give an algorithm yielding such a decomposition if the polynomial order of $x^\star$ is no more than $8$, in connection with Hilbert's $17^{\text{th}}$ 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.