In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration -1 is the only metastable state and we estimate the mean exit time. Moreover, w
e illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet can be macroscopic or mesoscopic, according to the value of the external magnetic field.
In cite{Cipriani2016}, the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study $alpha$-long-range divisible sandpiles, si
milar to those introduced in cite{Frometa2018}. We show that, for $alpha in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $alpha/2$. However, for $alpha in [2,infty)$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Levy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter $gamma=min {alpha,2}$ and $alpha in mathbb{R}_+backslash{2}$ on a finite grid.
In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every stationary
measure $ u$ of the PCA we can associate a space-time Gibbs measure $mu_{ u}$ on $mathbb{Z} times mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true. A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $din { 1,2,3}$ and characterizing the invariant product Bernoulli measures.
We characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in
the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition.
In this note we consider non-equilibrium steady states of one-dimensional models of heat conduction (wealth exchange) which are coupled to some reservoirs creating currents. In particular we will give sufficient and necessary conditions which will de
pend only on the first two moments of the reservoir measures and the redistribution parameter under which the two-point functions are multilinear. This presents the first example of multilinear two-point functions in the absence of product stationary measures.
We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a stron
g summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.
We analyze a class of energy and wealth redistribution models. We characterize their stationary measures and show that they have a discrete dual process. In particular we show that the wealth distribution model with non-zero propensity can never have
invariant product measures. We also introduce diffusion processes associated to the wealth distribution models by instantaneous thermalization.
We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar & Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives w
ith positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution; (2) quenched and annealed estimates of the eigenvalues of a product of $n$ random transfer matrices.
