The log-Sobolev inequality for spin systems of higher order interactions
Takis Konstantopoulos, Ioannis Papageorgiou
TL;DR
This work establishes a log-Sobolev inequality for infinite-dimensional Gibbs measures of spin systems on $\mathbb{Z}^d$ with higher-order interactions, assuming the single-site phase measure $\mu(dx)=\frac{e^{-\varphi(x)}dx}{\int e^{-\varphi(x)}dx}$ satisfies LS. The authors develop a framework based on coercive single-site inequalities and sweeping-out (conditioning) techniques to transfer LS from the one-site level to the full Gibbs measure, under small coupling, without requiring uniform LS on the site measures. They formulate a general setting with spin spaces modeled as nilpotent Lie groups (notably the Heisenberg group $\mathbb H_1$) and prove LS for the Gibbs measure in this context, yielding spectral-gap and concentration-type consequences. As a concrete application, they verify the hypotheses for a class of non-quadratic potentials on infinite products of Heisenberg groups, demonstrating exponential convergence to equilibrium for these high-order interaction spin systems. The results broaden the scope of LS-transport techniques to nonlinear, non-quadratic interactions and provide a concrete nontrivial class of models with guaranteed ergodic convergence properties.
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 $e^{-φ(x)}dx/Z$ 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.