Fast Mixing for the Low Temperature 2D Ising Model Through Irreversible Parallel Dynamics

We study tunneling 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

Fast mixing for the low temperature 2d Ising model through irreversible parallel dynamics

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

Gaussian Mean Fields Lattice Gas

We study rigorously a lattice gas version of the Sherrington-Kirckpatrick spin glass model. In discrete optimization literature this problem is known as Unconstrained Binary Quadratic Programming (UBQP) and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present an heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.Comment: 3 figures, 2 table

Convergent expansions for Random Cluster Model with q>0 on infinite graphs

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.Comment: In this new version the introduction has been revised, some references have been added, and many typos have been corrected. 37 pages, to appear in Communications on Pure and Applied Analysi

Equilibrium and non-equilibrium Ising models by means of PCA

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

Shaken dynamics: an easy way to parallel MCMC

We define a Markovian parallel dynamics for a class of spin systems on general interaction graphs. In this dynamics, beside the usual set of parameters $J_{xy}$, the strength of the interaction between the spins $\sigma_x$ and $\sigma_y$, and $\lambda_x$, the external field at site $x$, there is an inertial parameter $q$ measuring the tendency of the system to remain locally in the same state. This dynamics is reversible with an explicitly defined stationary measure. For suitable choices of parameter this invariant measure concentrates on the ground states of the Hamiltonian. This implies that this dynamics can be used to solve, heuristically, difficult problems in the context of combinatorial optimization. We also study the dynamics on $\mathbb{Z}^2$ with homogeneous interaction and external field and with arbitrary boundary conditions. We prove that for certain values of the parameters the stationary measure is close to the related Gibbs measure. Hence our dynamics may be a good tool to sample from Gibbs measure by means of a parallel algorithm. Moreover we show how the parameter allow to interpolate between spin systems defined on different regular lattices.Comment: 5 figure

On Diffusion Limited Deposition

We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph G_N\times\realmathbb{N}, where the basis has $N$ vertices $G_N:=\{1,\dots,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. Consider there a simple random walk {\it coming from infinity} which {\it deposits} on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally. We prove that there is a critical time scale $N/\log(N)$ for the maximal height of the piles, i.e., there exist constants $\alpha<\beta$ such that the maximal pile height at time $\alpha N/\log(N)$ is of order $\log(N)$, while at time $\beta N/\log(N)$ is larger than $N^\chi$ for some positive $\chi$. This suggests that a \emph{monopolistic regime} starts at such a time and only the highest pile goes on growing. If we rather consider a walk whose height-component goes down deterministically, the resulting \emph{ballistic deposition} has maximal height of order $\log(N)$ at time $N$. These two deposition models, diffusive and ballistic, are also compared with uniform random allocation and Polya's urn

On Lennard-Jones type potentials and hard-core potentials with an attractive tail

We revisit an old tree graph formula, namely the Brydges-Federbush tree identity, and use it to get new bounds for the convergence radius of the Mayer series for gases of continuous particles interacting via non absolutely summable pair potentials with an attractive tail including Lennard-Jones type pair potentials

On the blockage problem and the non-analyticity of the current for the parallel TASEP on a ring

The Totally Asymmetric Simple Exclusion Process (TASEP) is an important example of a particle system driven by an irreversible Markov chain. In this paper we give a simple yet rigorous derivation of the chain stationary measure in the case of parallel updating rule. In this parallel framework we then consider the blockage problem (aka slow bond problem). We find the exact expression of the current for an arbitrary blockage intensity $\varepsilon$ in the case of the so-called rule-184 cellular automaton, i.e. a parallel TASEP where at each step all particles free-to-move are actually moved. Finally, we investigate through numerical experiments the conjecture that for parallel updates other than rule-184 the current may be non-analytic in the blockage intensity around the value $\varepsilon = 0$

Improved bounds on coloring of graphs

Given a graph $G$ with maximum degree $\Delta\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\le\lceil 9.62 (\Delta-1)\rceil$. Moreover we prove that: $a'(G)\le \lceil 6.42(\Delta-1)\rceil$ if $G$ has girth $g\ge 5\,$; a'(G)\le \lceil5.77 (\Delta-1)\rc if $G$ has girth $g\ge 7$; a'(G)\le \lc4.52(\D-1)\rc if $g\ge 53$; a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil. We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the star-chromatic number $\chi_s(G)$ of $G$ is such that \chi_s(G)\le \lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic number \chi^\b(G) of $G$ is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are decreasing functions of \b such that k_1(\b)\in[4, 6] and k_2(\b)\in[2,5]. To obtain these results we use an improved version of the Lov\'asz Local Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of Theorem 2 (items c-f) written in more detail