213 research outputs found
Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation
Consider a cellular automaton with state space
where the initial configuration is chosen according to a Bernoulli
product measure, 1's are stable, and 0's become 1's if they are surrounded by
at least three neighboring 1's. In this paper we show that the configuration
at time n converges exponentially fast to a final configuration
, and that the limiting measure corresponding to is in
the universality class of Bernoulli (independent) percolation.
More precisely, assuming the existence of the critical exponents ,
, and , and of the continuum scaling limit of crossing
probabilities for independent site percolation on the close-packed version of
(i.e., for independent -percolation on ), we
prove that the bootstrapped percolation model has the same scaling limit and
critical exponents.
This type of bootstrap percolation can be seen as a paradigm for a class of
cellular automata whose evolution is given, at each time step, by a monotonic
and nonessential enhancement.Comment: 15 page
Spiral Model: a cellular automaton with a discontinuous glass transition
We introduce a new class of two-dimensional cellular automata with a
bootstrap percolation-like dynamics. Each site can be either empty or occupied
by a single particle and the dynamics follows a deterministic updating rule at
discrete times which allows only emptying sites. We prove that the threshold
density for convergence to a completely empty configuration is non
trivial, , contrary to standard bootstrap percolation. Furthermore
we prove that in the subcritical regime, , emptying always occurs
exponentially fast and that coincides with the critical density for
two-dimensional oriented site percolation on \bZ^2. This is known to occur
also for some cellular automata with oriented rules for which the transition is
continuous in the value of the asymptotic density and the crossover length
determining finite size effects diverges as a power law when the critical
density is approached from below. Instead for our model we prove that the
transition is {\it discontinuous} and at the same time the crossover length
diverges {\it faster than any power law}. The proofs of the discontinuity and
the lower bound on the crossover length use a conjecture on the critical
behaviour for oriented percolation. The latter is supported by several
numerical simulations and by analytical (though non rigorous) works through
renormalization techniques. Finally, we will discuss why, due to the peculiar
{\it mixed critical/first order character} of this transition, the model is
particularly relevant to study glassy and jamming transitions. Indeed, we will
show that it leads to a dynamical glass transition for a Kinetically
Constrained Spin Model. Most of the results that we present are the rigorous
proofs of physical arguments developed in a joint work with D.S.Fisher.Comment: 42 pages, 11 figure
Higher order corrections for anisotropic bootstrap percolation
We study the critical probability for the metastable phase transition of the
two-dimensional anisotropic bootstrap percolation model with
-neighbourhood and threshold . The first order asymptotics for
the critical probability were recently determined by the first and second
authors. Here we determine the following sharp second and third order
asymptotics:
We note that the second and third order terms are so large that the first order
asymptotics fail to approximate even for lattices of size well beyond
.Comment: 46 page
The time of graph bootstrap percolation
Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular
automaton defined as follows. Given a "small" graph and a "large" graph , in consecutive steps we obtain from by
adding to it all new edges such that contains a new copy of
. We say that percolates if for some , we have .
For , the question about the size of the smallest percolating graphs
was independently answered by Alon, Frankl and Kalai in the 1980's. Recently,
Balogh, Bollob\'as and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that
the graph percolates with high probability. In this paper, using the same
setup, we determine, up to a logarithmic factor, the critical probability for
percolation by time for all .Comment: 18 pages, 3 figure
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