Evolution of concentration under lattice spin-flip dynamics

TL;DR

This paper studies how concentration inequalities evolve over time in lattice spin-flip dynamics, showing preservation of Gaussian concentration bounds under certain conditions and their impossibility to be achieved from low-temperature states in finite time.

Contribution

It demonstrates the preservation of Gaussian concentration bounds in weakly interacting dynamics and the impossibility of reaching such bounds from low-temperature Gibbs states in finite time.

Findings

Gaussian concentration bound is conserved in weakly interacting dynamics

It is impossible to reach Gaussian concentration bounds from low-temperature Gibbs states in finite time

Uniform variance bounds are conserved under general spin-flip dynamics

Abstract

We consider spin-flip dynamics of configurations in $\{-1,1\}^{\mathbb{Z}^d}$, and study the time evolution of concentration inequalities. For "weakly interacting" dynamics we show that the Gaussian concentration bound is conserved in the course of time and it is satisfied by the unique stationary Gibbs measure. Next we show that, for a general class of translation-invariant spin-flip dynamics, it is impossible to evolve in finite time from a low-temperature Gibbs state towards a measure satisfying the Gaussian concentration bound. Finally, we consider the time evolution of the weaker uniform variance bound, and show that this bound is conserved under a general class of spin-flip dynamics.