Open Access
August 2020 Logarithmic Sobolev inequalities for finite spin systems and applications
Holger Sambale, Arthur Sinulis
Bernoulli 26(3): 1863-1890 (August 2020). DOI: 10.3150/19-BEJ1172


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 around 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 Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.


Download Citation

Holger Sambale. Arthur Sinulis. "Logarithmic Sobolev inequalities for finite spin systems and applications." Bernoulli 26 (3) 1863 - 1890, August 2020.


Received: 1 February 2019; Revised: 1 October 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193945
MathSciNet: MR4091094
Digital Object Identifier: 10.3150/19-BEJ1172

Keywords: central limit theorem , concentration of measure , exponential random graph model , finite product spaces , Logarithmic Sobolev inequality , mixing time , spin systems

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 3 • August 2020
Back to Top