Formulae for the Derivative of the Poincar\'e Constant of Gibbs Measures

TL;DR

This paper derives formulas for how the Poincaré constant of Gibbs measures changes with temperature, revealing monotonicity properties and providing sharper bounds for specific models like the $O(2)$ model.

Contribution

It introduces new formulas for the derivative of the Poincaré constant of Gibbs measures and applies them to analyze temperature dependence and bounds in specific models.

Findings

Poincaré constant decreases with inverse temperature for increasing Hamiltonians.

Sharpened upper bounds on the Poincaré constant for the $O(2)$ model.

Different zero-temperature behaviors of Poincaré and Log-Sobolev constants.

Abstract

We establish formulae for the derivative of the Poincar\'e constant of Gibbs measures on both compact domains and all of $\R^d$. As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the Poincar\'e constant is strictly decreasing in the inverse temperature, and vice versa. Applying this result to the $O(2)$ model allows us to give a sharpened upper bound on its Poincar\'e constant. We further show that this model exhibits a qualitatively different zero-temperature behavior of the Poincar\'e and Log-Sobolev constants.