Abstract:
We propose a unified approach to reversible and irreversible PCA dynamics, and we show that in the case of 1D and 2D nearest neighbour Ising systems with periodic boundary conditions we are able to compute the stationary measure of the dynamics also when the latter is irreversible. We also show how, according to [DPSS12], the stationary measure is very close to the Gibbs for a suitable choice of the parameters of the PCA dynamics, both in the reversible and in the irreversible cases. We discuss some numerical aspects regarding this topic, including a possible parallel implementation.

Abstract:
In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\Z^d$, to a much wider class of infinite graphs. In particular, concerning the subcritical regime, we show that the connectivity functions are analytic and decay exponentially in any bounded degree graph. In the supercritical phase, we are able to prove the analyticity of finite connectivity functions in a smaller class of graphs, namely, bounded degree graphs with the so called minimal cut-set property and satisfying a (very mild) isoperimetric inequality. On the other hand we show that the large distances decay of finite connectivity in the supercritical regime can be polynomially slow depending on the topological structure of the graph. Analogous analyticity results are obtained for the pressure of the Random Cluster Model on an infinite graph, but with the further assumptions of amenability and quasi-transitivity of the graph.

Abstract:
We consider the problem of approximate sampling from the finite volume Gibbs measure with a general pair interaction. We exhibit a parallel dynamics (Probabilistic Cellular Automaton) which efficiently implements the sampling. In this dynamics the product measure that gives the new configuration in each site contains a term that tends to favour the original value of each spin. This is the main ingredient that allows to prove that the stationary distribution of the PCA is close in total variation to the Gibbs measure. The presence of the parameter that drives the "inertial" term mentioned above gives the possibility to control the degree of parallelism of the numerical implementation of the dynamics.

Abstract:
In this paper we introduce a new algorithm to study some NP-complete problems. This algorithm is a Markov Chain Monte Carlo (MCMC) inspired by the cavity method developed in the study of spin glass. We will focus on the maximum clique problem and we will compare this new algorithm with several standard algorithms on some DIMACS benchmark graphs and on random graphs. The performances of the new algorithm are quite surprising. Our effort in this paper is to be clear as well to those readers who are not in the field.

Abstract:
We study metastability and mixing time for a non-reversible probabilistic cellular automaton. With a suitable choice of the parameters, we first show that the stationary distribution is close in total variation to a low temperature Ising model. Then we prove that both the mixing time and the time to exit a metastable state grow polynomially in the size of the system, while this growth is exponential in reversible dynamics. In this model, non-reversibility, parallel updatings and a suitable choice of boundary conditions combine to produce an efficient dynamical stability.

Abstract:
We give a rigorous proof of two phase transitions for a disordered system designed to find large cliques inside Erdos random graphs. Such a system is associated with a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.

Abstract:
We find an improved estimate of the radius of analyticity of the pressure of the hard-sphere gas in $d$ dimensions. The estimates are determined by the volume of multidimensional regions that can be numerically computed. For $d=2$, for instance, our estimate is about 40% larger than the classical one.

Abstract:
Given an infinite graph $\GI$ quasi-transitive and amenable with maximum degree $\D$, we show that reduced ground state degeneracy per site $W_r(\GI,q)$ of the q-state antiferromagnetic Potts model at zero temperature on $\GI$ is analytic in the variable 1/q, whenever $|2\D e^3/q|< 1$. This result proves, in an even stronger formulation, a conjecture originally sketched in [KE] (Kim and Enting, 1979) and explicitly formulated in [ST] (Shrock and Tsai,1997), based on which a sufficient condition for $W_r(\GI,q)$ to be analytic at 1/q=0 is that $\GI$ is a regular lattice.

Abstract:
In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a chain which is non-reversible w.r.t. its stationary measure. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.