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

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 Zd 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 Gärtner-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 measures 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. Key words and phrases: Stochastic spin-flip dynamics, Gibbs-non-Gibbs transition, empirical measure, non-linear generator, Nisio control semigroup, large deviation principle, bad configurations, bad empirical measures, nature versus nurture