2 research outputs found
Phase transitions in probabilistic cellular automata
We investigate the low-noise regime of a large class of probabilistic
cellular automata, including the North-East-Center model of A. Toom. They are
defined as stochastic perturbations of cellular automata with a binary state
space and a monotonic transition function and possessing a property of erosion.
These models were studied by A. Toom, who gave both a criterion for erosion and
a proof of the stability of homogeneous space-time configurations. Basing
ourselves on these major findings, we prove, for a set of initial conditions,
exponential convergence of the induced processes toward the extremal invariant
measure with a highly predominant state. We also show that this invariant
measure presents exponential decay of correlations in space and in time and is
therefore strongly mixing. This result is due to joint work with A. de Maere.
For the two-dimensional probabilistic cellular automata in the same class and
for the same extremal invariant measure, we give an upper bound to the
probability of a block of cells with the opposite state. The upper bound
decreases exponentially fast as the diameter of the block increases. This upper
bound complements, for dimension 2, a lower bound of the same form obtained for
any dimension greater than 1 by R. Fern\'andez and A. Toom. In order to prove
these results, we use graphical objects that were introduced by A. Toom and we
give a review of their construction.Comment: PhD thesis, 229 pages. The author was supported by a grant from the
Belgian F.R.S.-FNRS (Fonds de la Recherche Scientifique) as Aspirant FNR
Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata
We investigate the low-noise regime of a large class of probabilistic
cellular automata, including the North-East-Center model of Toom. They are
defined as stochastic perturbations of cellular automata belonging to the
category of monotonic binary tessellations and possessing a property of
erosion. We prove, for a set of initial conditions, exponential convergence of
the induced processes toward an extremal invariant measure with a highly
predominant spin value. We also show that this invariant measure presents
exponential decay of correlations in space and in time and is therefore
strongly mixing.Comment: 21 pages, 0 figure, revised version including a generalization to a
larger class of models, structure of the arguments unchanged, minor changes
suggested by reviewers, added reference