886 research outputs found
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 , the strength of the interaction between the spins
and , and , the external field at site ,
there is an inertial parameter 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 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 vertices
, and two vertices and
are adjacent if . 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
for the maximal height of the piles, i.e., there exist
constants such that the maximal pile height
at time is of order , while at time
is larger than for some positive .
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 at time
.
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
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