A Large-Deviation View on Dynamical Gibbs-Non-Gibbs Transitions

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 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.MSC2010: Primary 60F10, 60G60, 60K35; Secondary 82B26, 82C22.

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

“…3) Definition 2.2 assigns the notion of Gibbsianness to a sequence of probability measures that live on different spaces. This is different from the notion of Gibbsianness used for example in lattice systems [14,15,16,17], but in that respect similar to the definition of Gibbsianness used in the mean-field setting [20,22]. Since there is spatial dependence in our case it makes sense to call the quantity in (11) a specification kernel and α a boundary condition.…”

“…Remarks: 1) In case (i) the limiting kernels (16) are continuous functions of the conditioning α, as it is explicit from the given expression. 2) In the mean-field setting, by the fact that for the Ising model phase transitions are of second order, the Ising classes r i = 2 can never be a source of discontinuities.…”

“…It has been noted in some cases for mean-field models [24,22,21] when the appropriate notion of mean-field Gibbsianness is employed, the question of continuity can be reduced to variational problems. For systems for which lattice results and mean-field results are available it turns out that these results are often in a striking parallel [25,16]. It is an open challenge to understand this relation better.…”

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

“…In [4] a scheme is given to compute the "Lagrangian" L, see also [13] for an illustration of this scheme in the large-deviation view on Gibbs-non-Gibbs transitions. First one computes the "Hamiltonian"…”

“…These special conditionings leading to multiple histories are the analogue of "bad configurations" (essential points of discontinuity of conditional probabilities of the measure at time t ) in the (lattice) Gibbs-non-Gibbs transition scenario. This "trajectory-large-deviation approach" has then been studied in more generality, including the lattice case, in [13].…”