267 research outputs found
Rumor processes in random environment on N and on Galton-Watson trees
The aim of this paper is to study rumor processes in random environment. In a
rumor process a signal starts from the stations of a fixed vertex (the root)
and travels on a graph from vertex to vertex. We consider two rumor processes.
In the firework process each station, when reached by the signal, transmits it
up to a random distance. In the reverse firework process, on the other hand,
stations do not send any signal but they "listen" for it up to a random
distance. The first random environment that we consider is the deterministic
1-dimensional tree N with a random number of stations on each vertex; in this
case the root is the origin of N. We give conditions for the
survival/extinction on almost every realization of the sequence of stations.
Later on, we study the processes on Galton-Watson trees with random number of
stations on each vertex. We show that if the probability of survival is
positive, then there is survival on almost every realization of the infinite
tree such that there is at least one station at the root. We characterize the
survival of the process in some cases and we give sufficient conditions for
survival/extinction.Comment: 28 page
The small world effect on the coalescing time of random walks
A small world is obtained from the -dimensional torus of size 2L adding
randomly chosen connections between sites, in a way such that each site has
exactly one random neighbour in addition to its deterministic neighbours. We
study the asymptotic behaviour of the meeting time of two random walks
moving on this small world and compare it with the result on the torus. On the
torus, in order to have convergence, we have to rescale by a factor
if , by if and if . We prove
that on the small world the rescaling factor is and identify
the constant , proving that the walks always meet faster on the
small world than on the torus if , while if this depends on the
probability of moving along the random connection. As an application, we obtain
results on the hitting time to the origin of a single walk and on the
convergence of coalescing random walk systems on the small world.Comment: 33 pages, 2 figures, revised arguments, results unchange
A generating function approach to branching random walks
It is well known that the behaviour of a branching process is completely
described by the generating function of the offspring law and its fixed points.
Branching random walks are a natural generalization of branching processes: a
branching process can be seen as a one-dimensional branching random walk. We
define a multidimensional generating function associated to a given branching
random walk. The present paper investigates the similarities and the
differences of the generating functions, their fixed points and the
implications on the underlying stochastic process, between the one-dimensional
(branching process) and the multidimensional case (branching random walk). In
particular, we show that the generating function of a branching random walk can
have uncountably many fixed points and a fixed point may not be an extinction
probability, even in the irreducible case (extinction probabilities are always
fixed points). Moreover, the generating function might not be a convex
function. We also study how the behaviour of a branching random walk is
affected by local modifications of the process. As a corollary, we describe a
general procedure with which we can modify a continuous-time branching random
walk which has a weak phase and turn it into a continuous-time branching random
walk which has strong local survival for large or small values of the parameter
and non-strong local survival for intermediate values of the parameter.Comment: 17 pages, 5 figures, a few minor misprints have been fixe
Strong local survival of branching random walks is not monotone
The aim of this paper is the study of the strong local survival property for
discrete-time and continuous-time branching random walks. We study this
property by means of an infinite dimensional generating function G and a
maximum principle which, we prove, is satisfied by every fixed point of G. We
give results about the existence of a strong local survival regime and we prove
that, unlike local and global survival, in continuous time, strong local
survival is not a monotone property in the general case (though it is monotone
if the branching random walk is quasi transitive). We provide an example of an
irreducible branching random walk where the strong local property depends on
the starting site of the process. By means of other counterexamples we show
that the existence of a pure global phase is not equivalent to nonamenability
of the process, and that even an irreducible branching random walk with the
same branching law at each site may exhibit non-strong local survival. Finally
we show that the generating function of a irreducible BRW can have more than
two fixed points; this disproves a previously known result.Comment: 19 pages. The paper has been deeply reorganized and two pictures have
been added. arXiv admin note: substantial text overlap with arXiv:1104.508
Branching random walks and multi-type contact-processes on the percolation cluster of
In this paper we prove that, under the assumption of quasi-transitivity, if a
branching random walk on survives locally (at arbitrarily
large times there are individuals alive at the origin), then so does the same
process when restricted to the infinite percolation cluster
of a supercritical Bernoulli percolation. When no
more than individuals per site are allowed, we obtain the -type contact
process, which can be derived from the branching random walk by killing all
particles that are born at a site where already individuals are present. We
prove that local survival of the branching random walk on
also implies that for sufficiently large the associated -type contact
process survives on . This implies that the strong
critical parameters of the branching random walk on and on
coincide and that their common value is the limit of
the sequence of strong critical parameters of the associated -type contact
processes. These results are extended to a family of restrained branching
random walks, that is, branching random walks where the success of the
reproduction trials decreases with the size of the population in the target
site.Comment: Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …