A large-deviation view on dynamical Gibbs-non-Gibbs transitions

TL;DR

This paper introduces a large-deviation framework to analyze Gibbs-non-Gibbs transitions in spin systems under stochastic dynamics, linking the occurrence of bad empirical measures to phase transition phenomena over time.

Contribution

It develops a space-time large deviation approach for Gibbs-non-Gibbs transitions, defining bad empirical measures via a novel rate function and analyzing their role in phase transitions.

Findings

No bad empirical measures at short times

Bad empirical measures appear at intermediate and large times

Framework connects phase transitions to large deviation theory

Abstract

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z^d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gartner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical mea- sures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition.