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

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

Economies of Scale and Endogenous Market Structures in International Grain Trade

A large body of literature has examined international grain trade in imperfectly competitive models. The model developed in this paper differs from previous ones, since it specifically considers the export strategies used by multinational firms, which maintain a direct control over all importing/exporting functions; this strategy requires high fixed cost but allows firms to avoid the transactions costs arising from negotiating with downstream operators. The model considers a multinational firm and a state trading enterprises competing on a foreign market in a two-stage duopoly framework; while the Ste is assumed to export only indirectly, the multinational firm chooses between indirect and direct exports, according to the relative values of transaction and fixed costs. The results of the game are examined by using numerical examples. The examples show that external shocks on export markets affecting the relative values of transaction and fixed costs of the international grain trading, may result in a change of market structures and, consequently, of market shares. These effects depend on the initial market structure. The assumption that there is a gap between firms' domestic costs does not significantly change the outcome of the game. The effect of a worsening of the Ste's competitive position, in relation to that of the multinational, depends again on the starting market structure. On the whole, the analytical framework developed herein has a number of interesting policy implications for the ongoing WTO negotiations in the DDA round about the possible effects of regulating Ste.International Relations/Trade,

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

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