5,657 research outputs found
Transience and recurrence of random walks on percolation clusters in an ultrametric space
We study existence of percolation in the hierarchical group of order ,
which is an ultrametric space, and transience and recurrence of random walks on
the percolation clusters. The connection probability on the hierarchical group
for two points separated by distance is of the form , with , non-negative constants , and . Percolation was proved in Dawson and Gorostiza
(2013) for , with
. In this paper we improve the result for the critical case by
showing percolation for . We use a renormalization method of the type
in the previous paper in a new way which is more intrinsic to the model. The
proof involves ultrametric random graphs (described in the Introduction). The
results for simple (nearest neighbour) random walks on the percolation clusters
are: in the case the walk is transient, and in the critical case
, there exists a critical
such that the walk is recurrent for and transient for
. The proofs involve graph diameters, path lengths, and
electric circuit theory. Some comparisons are made with behaviours of random
walks on long-range percolation clusters in the one-dimensional Euclidean
lattice.Comment: 27 page
Surprise probabilities in Markov chains
In a Markov chain started at a state , the hitting time is the
first time that the chain reaches another state . We study the probability
that the first visit to occurs precisely at a
given time . Informally speaking, the event that a new state is visited at a
large time may be considered a "surprise". We prove the following three
bounds:
1) In any Markov chain with states, .
2) In a reversible chain with states, for .
3) For random walk on a simple graph with vertices,
.
We construct examples showing that these bounds are close to optimal. The
main feature of our bounds is that they require very little knowledge of the
structure of the Markov chain.
To prove the bound for random walk on graphs, we establish the following
estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication):
For random walk on an -vertex graph, for every initial vertex ,
\[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \
Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions
We introduce spatially explicit stochastic processes to model multispecies
host-symbiont interactions. The host environment is static, modeled by the
infinite percolation cluster of site percolation. Symbionts evolve on the
infinite cluster through contact or voter type interactions, where each host
may be infected by a colony of symbionts. In the presence of a single symbiont
species, the condition for invasion as a function of the density of the habitat
of hosts and the maximal size of the colonies is investigated in details. In
the presence of multiple symbiont species, it is proved that the community of
symbionts clusters in two dimensions whereas symbiont species may coexist in
higher dimensions.Comment: Published in at http://dx.doi.org/10.1214/10-AAP734 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Giant vacant component left by a random walk in a random d-regular graph
We study the trajectory of a simple random walk on a d-regular graph with d>2
and locally tree-like structure as the number n of vertices grows. Examples of
such graphs include random d-regular graphs and large girth expanders. For
these graphs, we investigate percolative properties of the set of vertices not
visited by the walk until time un, where u>0 is a fixed positive parameter. We
show that this so-called vacant set exhibits a phase transition in u in the
following sense: there exists an explicitly computable threshold u* such that,
with high probability as n grows, if u<u*, then the largest component of the
vacant set has a volume of order n, and if u>u*, then it has a volume of order
log(n). The critical value u* coincides with the critical intensity of a random
interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular
tree. We also show that the random interlacement model describes the structure
of the vacant set in local neighbourhoods
Universality of trap models in the ergodic time scale
Consider a sequence of possibly random graphs , ,
whose vertices's have i.i.d. weights with a distribution
belonging to the basin of attraction of an -stable law, .
Let , , be a continuous time simple random walk on which
waits a \emph{mean} exponential time at each vertex . Under
considerably general hypotheses, we prove that in the ergodic time scale this
trap model converges in an appropriate topology to a -process. We apply this
result to a class of graphs which includes the hypercube, the -dimensional
torus, , random -regular graphs and the largest component of
super-critical Erd\"os-R\'enyi random graphs
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