abstract:
We consider the problem of metastability for a stochastic dynamics with a parallel updating rule. In particular we compare the behaviour of two Probabilistic Cellular Automata, the first in which the updating rule has single spin rates equal to those of the heat bath for the Ising nearest neighbors interaction. The second one has an updating rule with single spin flip rate which depends on the value of the spin itself and the spins on its four nearest neighbour sites.
We study the exit from the metastable phase, we describe the typical exit path and evaluate the exit time. We prove that the phenomenology of metastability is different from the one observed in the case of the serial implementation of the heat bath dynamics. In particular we prove that an intermediate chessboard phase appears during the excursion from the minus metastable phase toward the plus stable phase.
abstract:
The ordinary contact process is a model used to study the spread of an infection in a population, for instance, Z^d. A healthy individual gets infected at a rate which is proportional to the number of infected neighbors while an infected individual gets healthy at rate 1. In this talk I will introduce a variant of this model in which the rate at which individuals gets healthy changes through time in a simple way.
Starting with only one infected
individual, a natural question to ask is whether with probability 1, the
infection dies out within finite time.
In order to analyze this question we determine certain sharp stochastic
domination results for a discrete time hidden Markov chain and its
corresponding continuous version.