Bernoulli

  • Bernoulli
  • Volume 26, Number 3 (2020), 1863-1890.

Logarithmic Sobolev inequalities for finite spin systems and applications

Holger Sambale and Arthur Sinulis

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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 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.

Article information

Source
Bernoulli, Volume 26, Number 3 (2020), 1863-1890.

Dates
Received: February 2019
Revised: October 2019
First available in Project Euclid: 27 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.bj/1587974526

Digital Object Identifier
doi:10.3150/19-BEJ1172

Mathematical Reviews number (MathSciNet)
MR4091094

Zentralblatt MATH identifier
07193945

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

Citation

Sambale, Holger; Sinulis, Arthur. Logarithmic Sobolev inequalities for finite spin systems and applications. Bernoulli 26 (2020), no. 3, 1863--1890. doi:10.3150/19-BEJ1172. https://projecteuclid.org/euclid.bj/1587974526


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