Log-Sobolev inequality for near critical Ising models

TL;DR

This paper establishes bounds on the log-Sobolev constant for ferromagnetic Ising models near criticality, linking it to susceptibility and providing insights into system behavior close to phase transitions.

Contribution

It introduces a new bound on the log-Sobolev constant based solely on susceptibility, applicable uniformly up to the critical point without mixing conditions.

Findings

Log-Sobolev constant is uniform up to the critical point.

Bound depends polynomially on the distance to criticality when susceptibility satisfies mean-field bounds.

Results apply to high-dimensional Ising models on subsets of bZ^d for d>4.

Abstract

For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean-field bound as the critical point is approached, our bound implies that the log-Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of $\mathbb{Z}^d$ when $d>4$. The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron--Frobenius theorem, and the log-Sobolev inequality for product Bernoulli measures.