# The log-Sobolev inequality for spin systems of higher order interactions

## TL;DR

This paper establishes conditions under which infinite-dimensional spin systems with higher-order interactions satisfy the log-Sobolev inequality, extending known results to non-quadratic potentials and specific non-abelian group structures.

## Contribution

It provides new criteria for the log-Sobolev inequality in spin systems with higher-order interactions and applies these to non-quadratic potentials on Heisenberg groups.

## Key findings

- Infinite-dimensional Gibbs measures with higher-order interactions satisfy the log-Sobolev inequality.
- Conditions are identified for non-quadratic interaction potentials.
- Certain nontrivial Gibbs measures on Heisenberg groups meet the inequality.

## Abstract

We study the infinite-dimensional log-Sobolev inequality for spin systems on $\mathbb{Z}^d$ with interactions of power higher than quadratic. We assume that the one site measure without a boundary $e^{-\phi(x)}dx/Z$ satisfies a log-Sobolev inequality and we determine conditions so that the infinite-dimensional Gibbs measure also satisfies the inequality. As a concrete application, we prove that a certain class of nontrivial Gibbs measures with non-quadratic interaction potentials on an infinite product of Heisenberg groups satisfy the log-Sobolev inequality.