687 research outputs found
Bootstrap percolation on the stochastic block model
We analyze the bootstrap percolation process on the stochastic block model
(SBM), a natural extension of the Erd\"{o}s--R\'{e}nyi random graph that allows
representing the "community structure" observed in many real systems. In the
SBM, nodes are partitioned into subsets, which represent different communities,
and pairs of nodes are independently connected with a probability that depends
on the communities they belong to. Under mild assumptions on system parameters,
we prove the existence of a sharp phase transition for the final number of
active nodes and characterize sub-critical and super-critical regimes in terms
of the number of initially active nodes, which are selected uniformly at random
in each community.Comment: 53 pages 3 figure
Bootstrap percolation on the stochastic block model
We analyze the bootstrap percolation process on the stochastic
block model (SBM), a natural extension of the ErdH{o}s--R'{e}nyi random graph
that incorporates the community structure observed in many real systems.
In the SBM, nodes are partitioned into two subsets, which represent different communities,
and pairs of nodes are independently connected with a probability that depends on the communities they belong to.
Under mild assumptions on the system parameters, we prove the existence of a sharp phase transition
for the final number of active nodes and characterize the sub-critical and the super-critical regimes
in terms of the number of initially active nodes, which are selected uniformly at random in each community
Kinetically constrained spin models on trees
We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and
unrooted trees of finite connectivity. We focus in particular on the class of
Friedrickson-Andersen models FA-jf and on an oriented version of them. These
tree models are particularly relevant in physics literature since some of them
undergo an ergodicity breaking transition with the mixed first-second order
character of the glass transition. Here we first identify the ergodicity regime
and prove that the critical density for FA-jf and OFA-jf models coincide with
that of a suitable bootstrap percolation model. Next we prove for the first
time positivity of the spectral gap in the whole ergodic regime via a novel
argument based on martingales ideas. Finally, we discuss how this new technique
can be generalized to analyze KCSM on the regular lattice .Comment: Published in at http://dx.doi.org/10.1214/12-AAP891 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Kinetically constrained spin models
We analyze the density and size dependence of the relaxation time for
kinetically constrained spin models (KCSM) intensively studied in the physical
literature as simple models sharing some of the features of a glass transition.
KCSM are interacting particle systems on with Glauber-like dynamics,
reversible w.r.t. a simple product i.i.d Bernoulli() measure. The essential
feature of a KCSM is that the creation/destruction of a particle at a given
site can occur only if the current configuration of empty sites around it
satisfies certain constraints which completely define each specific model. No
other interaction is present in the model. From the mathematical point of view,
the basic issues concerning positivity of the spectral gap inside the
ergodicity region and its scaling with the particle density remained open
for most KCSM (with the notably exception of the East model in
\cite{Aldous-Diaconis}). Here for the first time we: i) identify the ergodicity
region by establishing a connection with an associated bootstrap percolation
model; ii) develop a novel multi-scale approach which proves positivity of the
spectral gap in the whole ergodic region; iii) establish, sometimes optimal,
bounds on the behavior of the spectral gap near the boundary of the ergodicity
region and iv) establish pure exponential decay for the persistence function.
Our techniques are flexible enough to allow a variety of constraints and our
findings disprove certain conjectures which appeared in the physical literature
on the basis of numerical simulations
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