Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

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

Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter $m$, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP$(m)$ and the KMP, and a nonlinear heat equation for the GBEP($a$). We prove the hydrodynamic limit rigorously for the BEP$(m)$, and give a formal derivation for the GBEP($a$). We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form $-\log \rho$; they involve dissipation or mobility terms of order $\rho^2$ for the linear heat equation, and a nonlinear function of $\rho$ for the nonlinear heat equation.Comment: 29 page

Stretched Exponential Relaxation in the Biased Random Voter Model

We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild condions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent $d/(d+\alpha)$, where $0<\alpha\le 2$ depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe

Limiting shapes for deterministic centrally seeded growth models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d \geq 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit $h \to -\infty$. Finally, we prove that the rotor router shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation added, to appear in Journal of Statistical Physic

A probabilistic approach to Zhang's sandpile model

The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation tends to $\sqrt{1/2}$.Comment: 47 pages, 3 figure

Gibbs-non-Gibbs transitions via large deviations: computable examples

We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation approach introduced in [4]. These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birth and death processes. We show for a large class of initial measures and diffusive dynamics both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs transitions

Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics ("infinite-temperature" dynamics). We show that, in this setup, the program outlined in van Enter, Fern\'andez, den Hollander and Redig can be fully completed, namely that Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs -and vice versa- with sharp transition times (under the dynamics Gibbsianness can be lost and can be recovered). Our analysis expands the work of Ermolaev and Kulske who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.Comment: Key words and phrases: Curie-Weiss model, spin-flip dynamics, Gibbs vs. non-Gibbs, dynamical transition, large deviations, action integral, bifurcation of rate functio

Infinite volume limit of the Abelian sandpile model in dimensions d >= 3

We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit mu of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure mu, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning tree measure on Z^d.Comment: First version: LaTeX; 29 pages. Revised version: LaTeX; 29 pages. The main result of the paper has been extended to all dimensions at least 3, with a new and simplyfied proof of finiteness of the two-component spanning tree. Second revision: LaTeX; 32 page