Surface order large deviations for 2d FK-percolation and Potts models

By adapting the renormalization techniques of Pisztora, we establish surface order large deviations estimates for FK-percolation on $\Z^2$ with parameter $q\geq 1$ and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.Comment: 18 pages, 4 figure

Large Deviations Principle for Stochastic Scalar Conservation Laws

We investigate large deviations for a family of conservative stochastic PDEs (conservation laws) in the asymptotic of jointly vanishing noise and viscosity. We obtain a first large deviations principle in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. We therefore investigate a second order large deviations principle, thus providing a quantitative characterization of non-entropic solutions to the conservation law.Comment: 40 page

Branching random walks and multi-type contact-processes on the percolation cluster of Zd{\mathbb{Z}}^{d}

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{\mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${{\mathcal{C}}_{\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{\mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{\mathcal{C}}_{\infty}}$. This implies that the strong critical parameters of the branching random walk on ${{\mathbb{Z}}^d}$ and on ${{\mathcal{C}}_{\infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.Comment: Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the 2D Ising Wulff crystal near criticality

We study the behavior of the two-dimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal.Comment: Published in at http://dx.doi.org/10.1214/08-AOP449 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org