A Variance Inequality for Glauber Dynamics applicable to High and Low Temperature Regimes

TL;DR

This paper introduces a new variance inequality for spin-flip systems that bridges existing inequalities and applies to both high and low temperature regimes, enhancing understanding of relaxation to equilibrium.

Contribution

It develops a variance inequality interpolating between Poincaré and uniform inequalities, applicable to monotone dynamics and the low temperature Ising model.

Findings

Variance inequalities interpolate between known bounds.

Decay of autocorrelation implies decay for general functions.

Extended decay results to the low temperature Ising model.

Abstract

A variance inequality for spin-flip systems is obtained using comparatively weaker knowledge of relaxation to equilibrium based on coupling estimates for single site disturbances. We obtain variance inequalities interpolating between the Poincar\'e inequality and the uniform variance inequality, and a general weak Poincar\'e inequality. For monotone dynamics the variance inequality can be obtained from decay of the autocorrelation of the spin at the origin, i.e., from that decay we conclude decay for general functions. This method is then applied to the low temperature Ising model, where the time-decay of the autocorrelation of the origin is extended to arbitrary quasi-local functions.