Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

TL;DR

This paper establishes a uniform logarithmic Sobolev inequality for a class of noninteracting spin systems with super-quadratic potentials, extending previous quadratic cases and optimizing the system size scaling.

Contribution

It generalizes the uniform LSI to super-quadratic potentials using a two-scale approach and covariance inequalities, improving understanding of spin system behavior.

Findings

Uniform LSI holds for super-quadratic potentials

LSI constant scaling is optimal in system size

Method extends quadratic case techniques to more general potentials

Abstract

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cram\'{e}r theorem.