803 research outputs found
Coexistence of competing first passage percolation on hyperbolic graphs
We study a natural growth process with competition, which was recently
introduced to analyze MDLA, a challenging model for the growth of an aggregate
by diffusing particles. The growth process consists of two first-passage
percolation processes and , spreading with
rates and respectively, on a graph . starts
from a single vertex at the origin , while the initial configuration of
consists of infinitely many \emph{seeds} distributed
according to a product of Bernoulli measures of parameter on
. starts spreading from time 0, while each
seed of only starts spreading after it has been reached by
either or . A fundamental question in this
model, and in growth processes with competition in general, is whether the two
processes coexist (i.e., both produce infinite clusters) with positive
probability. We show that this is the case when is vertex transitive,
non-amenable and hyperbolic, in particular, for any there is a
such that for all the two
processes coexist with positive probability. This is the first non-trivial
instance where coexistence is established for this model. We also show that
produces an infinite cluster almost surely for any
positive , establishing fundamental differences with the behavior
of such processes on .Comment: 53 pages, 13 figure
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur
The quantifier semigroup for bipartite graphs
In a bipartite graph there are two widely encountered monotone mappings from subsets of one side of the graph to subsets of the other side: one corresponds to the quantifier "there exists a neighbor in the subset" and the other to the quantifier "all neighbors are in the subset." These mappings generate a partially ordered semigroup which we characterize in terms of "run-unimodal" words
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