Sharp Asymptotics and Simulations for Probabilistic Cellular Automata

Abstract

In this thesis, the problem of metastability for a finite volume Probabilistic Cellular Automata (PCA) in a small external field in the low temperature limit is studied, corresponding to the parallel implementation of the Ising model in a heat bath. In particular, the exit time of the metastable state (the configuration with al pluses), i.e. the time spent nucleating the stable state (the configuration with al pluses) triggered by the formation of a critical droplet, is of interest. The main result of the thesis is the sharp estimate on the mean of the exit times, which is an exponential function of the inverse temperature β times a prefactor that does not scale with β. Additionally finite temperature Markov chain Monte Carlo (MCMC) simulations are performed on PCA that correspond to ferromagnetic and anti-ferromagnetic stochastic lattice spin models.