A very simple proof of the LSI for high temperature spin systems

TL;DR

This paper provides a simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality under certain spectral conditions, leading to new insights into the rapid relaxation of high-temperature spin systems like the SK model.

Contribution

It introduces a straightforward proof technique for LSI in high-temperature spin systems, including the SK model, using zero range renormalisation and Bakry--Emery theory.

Findings

Proves LSI for the $O(n)$ model under spectral conditions.

First rapid relaxation result for the SK spin glass model.

Applicable to a broad class of bounded and unbounded spin systems.

Abstract

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the positive definite coupling matrix has largest eigenvalue less than $n$. This condition applies in particular to the SK spin glass model at inverse temperature $\beta < 1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry--Emery theory for the renormalised measure.