8,292 research outputs found
Critical percolation on random regular graphs
We show that for all the size of the largest
component of a random -regular graph on vertices around the percolation
threshold is , with high probability. This extends
known results for fixed and for , confirming a prediction of
Nachmias and Peres on a question of Benjamini. As a corollary, for the largest
component of the percolated random -regular graph, we also determine the
diameter and the mixing time of the lazy random walk. In contrast to previous
approaches, our proof is based on a simple application of the switching method.Comment: 10 page
Explicit isoperimetric constants and phase transitions in the random-cluster model
The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model on nonamenable graphs with
cluster parameter . Among these, the main ones are the absence of
percolation for the free random-cluster measure at the critical value, and
examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w
(q) for large enough. The latter follows from considerations of
isoperimetric constants, and we give the first nontrivial explicit calculations
of such constants. Such considerations are also used to prove non-robust phase
transition for the Potts model on nonamenable regular graphs
Interlacement percolation on transient weighted graphs
In this article, we first extend the construction of random interlacements,
introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting
of transient weighted graphs. We prove the Harris-FKG inequality for this model
and analyze some of its properties on specific classes of graphs. For the case
of non-amenable graphs, we prove that the critical value u_* for the
percolation of the vacant set is finite. We also prove that, once G satisfies
the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product
GxZ (where we endow Z with unit weights). When the graph under consideration is
a tree, we are able to characterize the vacant cluster containing some fixed
point in terms of a Bernoulli independent percolation process. For the specific
case of regular trees, we obtain an explicit formula for the critical value
u_*.Comment: 25 pages, 2 figures, accepted for publication in the Elect. Journal
of Pro
Generalized Threshold-Based Epidemics in Random Graphs: the Power of Extreme Values
Bootstrap percolation is a well-known activation process in a graph,
in which a node becomes active when it has at least active neighbors.
Such process, originally studied on regular structures, has been recently
investigated also in the context of random graphs, where it can serve as a simple
model for a wide variety of cascades, such as the
spreading of ideas, trends, viral contents, etc. over large social networks.
In particular, it has been shown that in the final active set
can exhibit a phase transition for a sub-linear number of seeds.
In this paper, we propose a unique framework to study similar
sub-linear phase transitions for a much broader class of graph models
and epidemic processes. Specifically, we consider i) a generalized version
of bootstrap percolation in with random activation thresholds
and random node-to-node influences; ii) different random graph models,
including graphs with given degree sequence and graphs with
community structure (block model). The common thread of our work is to
show the surprising sensitivity of the critical seed set size
to extreme values of distributions, which makes some systems dramatically
vulnerable to large-scale outbreaks. We validate our results running simulation on
both synthetic and real graphs
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
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