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Low-Temperature Asymptotics of the Poincaré and the log-Sobolev Constants for Łojasiewicz Potentials

Aziz Ben Nejma

Abstract

In this paper, we establish the low-temperature asymptotics of the Poincaré inequality constant for a class of convex potentials satisfying a Łojasiewicz 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.

Low-Temperature Asymptotics of the Poincaré and the log-Sobolev Constants for Łojasiewicz Potentials

Abstract

In this paper, we establish the low-temperature asymptotics of the Poincaré inequality constant for a class of convex potentials satisfying a Łojasiewicz 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.

Paper Structure

This paper contains 13 sections, 17 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries on functional inequalities
  4. The Poincaré inequality as a spectral gap.
  5. The Bakry-Émery criterion and the $\Gamma$-calculus.
  6. Comparing log-Sobolev and Poincaré constants
  7. About the assumptions and the Polyak-Łojasiewicz inequality.
  8. Low-temperature asymptotics of Poincaré constants
  9. A refutation of Conjecture \ref{['conj']}
  10. General heuristics.
  11. The one-dimensional case
  12. The general case
  13. Low-temperature asymptotics of log-Sobolev constants in dimension $1$

Key Result

Theorem 1.1

Let $V \in C^2(\mathbb{R}^n)$ be a potential having a unique minimizer $x_0$ and satisfying a Polyak-Łojasiewicz inequality, namely where $C_{PL}$ is the best constant in the last inequality. Assume $\dfrac{\Delta V}{1+\lVert \nabla V\rVert^2}$ is bounded above, then and $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Theorem 1.1: Chewi-Stromme
  • Conjecture 1.2: Chewi-Stromme
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • Remark 1.11
  • ...and 23 more