Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

TL;DR

This paper develops transfer principles for log-Sobolev and spectral-gap inequalities under curvature bounds, applying them to conservative spin systems to derive explicit spectral-gap estimates, including for non-strongly convex interactions.

Contribution

It introduces new transfer principles for inequalities under curvature bounds and applies them to obtain explicit estimates for conservative spin systems, extending previous results.

Findings

Explicit log-Sobolev and spectral-gap estimates for spin systems.

Spectral-gap bounds independent of system size for non-strongly convex cases.

Extension of previous results to broader interaction potentials.

Abstract

We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below. As our main application, we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of various conservative spin system models, consisting of non-interacting and weakly-interacting particles, constrained to conserve the mean-spin. When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg and Yau. When the self-interaction is only assumed to be (non-strongly) convex, as in the case of the two-sided exponential measure, we obtain sharp estimates on the system's spectral-gap as a function of the mean-spin, independently of the size of the system.