3,187 research outputs found
Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections
In this note we analyze an anisotropic, two-dimensional bootstrap percolation
model introduced by Gravner and Griffeath. We present upper and lower bounds on
the finite-size effects. We discuss the similarities with the semi-oriented
model introduced by Duarte.Comment: Key words: Bootstrap percolation, anisotropy, finite-size effect
Dynamics of bootstrap percolation
Bootstrap percolation transition may be first order or second order, or it
may have a mixed character where a first order drop in the order parameter is
preceded by critical fluctuations. Recent studies have indicated that the mixed
transition is characterized by power law avalanches, while the continuous
transition is characterized by truncated avalanches in a related sequential
bootstrap process. We explain this behavior on the basis of a through
analytical and numerical study of the avalanche distributions on a Bethe
lattice.Comment: Proceedings of the International Workshop and Conference on
Statistical Physics Approaches to Multidisciplinary Problems, IIT Guwahati,
India, 7-13 January 200
Linear algebra and bootstrap percolation
In \HH-bootstrap percolation, a set A \subset V(\HH) of initially
'infected' vertices spreads by infecting vertices which are the only uninfected
vertex in an edge of the hypergraph \HH. A particular case of this is the
-bootstrap process, in which \HH encodes copies of in a graph . We
find the minimum size of a set that leads to complete infection when
and are powers of complete graphs and \HH encodes induced copies of
in . The proof uses linear algebra, a technique that is new in bootstrap
percolation, although standard in the study of weakly saturated graphs, which
are equivalent to (edge) -bootstrap percolation on a complete graph.Comment: 10 page
Noise sensitivity in bootstrap percolation
Answering questions of Itai Benjamini, we show that the event of complete
occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d,
for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while
in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for
2\leq k\leq d-2, it is insensitive. Many open problems remain.Comment: 16 page
Monotone cellular automata in a random environment
In this paper we study in complete generality the family of two-state,
deterministic, monotone, local, homogeneous cellular automata in
with random initial configurations. Formally, we are given a set
of finite subsets of
, and an initial set
of `infected' sites, which we take to be random
according to the product measure with density . At time ,
the set of infected sites is the union of and the set of all
such that for some . Our
model may alternatively be thought of as bootstrap percolation on
with arbitrary update rules, and for this reason we call it
-bootstrap percolation.
In two dimensions, we give a classification of -bootstrap
percolation models into three classes -- supercritical, critical and
subcritical -- and we prove results about the phase transitions of all models
belonging to the first two of these classes. More precisely, we show that the
critical probability for percolation on is for all models in the critical class, and that it is
for all models in the supercritical class.
The results in this paper are the first of any kind on bootstrap percolation
considered in this level of generality, and in particular they are the first
that make no assumptions of symmetry. It is the hope of the authors that this
work will initiate a new, unified theory of bootstrap percolation on
.Comment: 33 pages, 7 figure
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