Logarithmic Sobolev inequalities for finite spin systems and applications

TL;DR

This paper establishes conditions under which finite spin systems satisfy logarithmic Sobolev inequalities, leading to insights on mixing times, entropy decay, and measure concentration, with applications to random graph models and subgraph counts.

Contribution

It provides new sufficient conditions for logarithmic Sobolev inequalities in spin systems and applies these to analyze mixing times and measure concentration in various models.

Findings

Exponential decay of relative entropy along Glauber dynamics.

Central limit theorems for subgraph counts in random graph models.

Quantitative bounds on mixing times for specific spin systems.

Abstract

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erd\"os-R\'enyi model the first order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.