Low-Temperature Asymptotics of the Poincar\'e and the log-Sobolev Constants for {\L}ojasiewicz Potentials

TL;DR

This paper analyzes the low-temperature behavior of Poincaré and log-Sobolev constants for convex potentials with Lojasiewicz inequality, correcting previous conjectures and providing precise asymptotics in one dimension.

Contribution

It establishes the asymptotics of the Poincaré constant for certain convex potentials and refutes a prior conjecture on the log-Sobolev constant's behavior.

Findings

Asymptotics of Poincaré constant for convex potentials are derived.

The conjecture on log-Sobolev constant asymptotics is disproved.

The correct low-temperature asymptotics for the log-Sobolev constant are identified in one dimension.

Abstract

In this paper, we establish the low-temperature asymptotics of the Poincar\'e inequality constant for a class of convex potentials satisfying a {\L}ojasiewicz inequality. In addition, we disprove a conjecture previously posed by Chewi and Stromme on the low-temperature asymptotics of the log-Sobolev constant and determine the correct asymptotic behavior in dimension one.