Phase transitions in probabilistic cellular automata
L. Ponselet. (2013)cite arxiv:1312.3612Comment: 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 FNRS.
Abstract
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. Fernández 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.
Description
Phase transitions in probabilistic cellular automata
cite arxiv:1312.3612Comment: 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 FNRS
%0 Journal Article
%1 ponselet2013phase
%A Ponselet, Lise
%D 2013
%K PCA
%T Phase transitions in probabilistic cellular automata
%U http://arxiv.org/abs/1312.3612
%X 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. Fernández 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.
@article{ponselet2013phase,
abstract = {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.},
added-at = {2018-11-26T17:56:57.000+0100},
author = {Ponselet, Lise},
biburl = {https://www.bibsonomy.org/bibtex/22d97e429cf8eb9ff0f073bd073c89942/joecolvin},
description = {Phase transitions in probabilistic cellular automata},
interhash = {858d3a235018082aae48fc9ba6dc70e0},
intrahash = {2d97e429cf8eb9ff0f073bd073c89942},
keywords = {PCA},
note = {cite arxiv:1312.3612Comment: 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 FNRS},
timestamp = {2018-11-26T17:56:57.000+0100},
title = {Phase transitions in probabilistic cellular automata},
url = {http://arxiv.org/abs/1312.3612},
year = 2013
}