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We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure µ = ν. Both ν and µ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:(1) For all ν and µ, νS(t) is Gibbs for small t.(2) If both ν and µ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.(3) If ν has a low non-zero temperature and a zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.(4) If ν has a low non-zero temperature and a non-zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where µ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

We prove that various SO(n)-invariant n-vector models with interactions which have a deep and narrow enough minimum have a first-order transition in the temperature. The result holds in dimension two or more, and is independent on the nature of the low-temperature phase.Recently Blöte, Guo and Hilhorst [2], extending earlier work by Domany, Schick and Swendsen [4] on 2-dimensional classical XY-models, performed a numerical study of 2-dimensional n-vector models with non-linear interactions. For sufficiently strong values of the non-linearity, they found the presence of a first-order transition in temperature. In [4] a heuristic explanation of this first-order behavior, based on a similarity with the high-q Potts model, was suggested, explaining the numerical results. A further confirmation of this transition was found by Caracciolo and Pellisetto [1], who 1

We consider various sufficiently nonlinear sigma models for nematic liquid crystal ordering of RP N −1 type and of lattice gauge type with continuous symmetries. We rigorously show that they exhibit a first-order transition in the temperature. The result holds in dimension 2 or more for the RP N −1 models and in dimension 3 or more for the lattice gauge models. In the twodimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.

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Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set [{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)}.] At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

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.

We study the Gibbsian character of time-evolved planar rotor systems on Z d , d ≥ 2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure ν. We model the system by interacting Brownian diffusions X = (X i (t)) t≥0,i∈Z d moving on circles. We prove that for small times t and arbitrary initial Gibbs measures ν, or for long times and both high-or infinite-temperature initial measure and dynamics, the evolved measure ν t stays Gibbsian. Furthermore, we show that for a low-temperature initial measure ν evolving under infinite-temperature dynamics there is a time interval (t 0 , t 1 ) such that ν t fails to be Gibbsian in d = 2.