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Abstract: We develop a space-time large-deviation point of view on Gibbs-non-Gibbstransitions in spin systems subject to a stochastic spin-flip dynamics. Usingthe general theory for large deviations of functionals of Markov processesoutlined in Feng and Kurtz 11, we show that the trajectory under thespin-flip dynamics of the empirical measure of the spins in a large block inZ^d satisfies a large deviation principle in the limit as the block size tendsto infinity. The associated rate function can be computed as the actionfunctional of a Lagrangian that is the Legendre transform of a certainnon-linear generator, playing a role analogous to the moment-generatingfunction in the Gartner-Ellis theorem of large deviation theory when this isapplied to finite-dimensional Markov processes. This rate function is used todefine the notion of -bad empirical measures-, which are the discontinuitypoints of the optimal trajectories i.e., the trajectories minimizing the ratefunction given the empirical measure at the end of the trajectory. Thedynamical Gibbs-non-Gibbs transitions are linked to the occurrence of badempirical measures: for short times no bad empirical measures occur, while forintermediate and large times bad empirical mea- sures are possible. A futureresearch program is proposed to classify the various possible scenarios behindthis crossover, which we refer to as a -nature-versus-nurture- transition.

Author: Aernout van Enter, Roberto Fernández, Frank den Hollander, Frank Redig

Source: https://arxiv.org/

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