9 research outputs found
Two Phase Transitions for the Contact Process on Small Worlds
In our version of Watts and Strogatz's small world model, space is a
d-dimensional torus in which each individual has in addition exactly one
long-range neighbor chosen at random from the grid. This modification is
natural if one thinks of a town where an individual's interactions at school,
at work, or in social situations introduces long-range connections. However,
this change dramatically alters the behavior of the contact process, producing
two phase transitions. We establish this by relating the small world to an
infinite "big world" graph where the contact process behavior is similar to the
contact process on a tree.Comment: 24 pages, 6 figures. We have rewritten the phase transition in terms
of two parameters and have made improvements to our original result
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
Ecological equilibrium for restrained branching random walks
We study a generalized branching random walk where particles breed at a rate
which depends on the number of neighboring particles. Under general assumptions
on the breeding rates we prove the existence of a phase where the population
survives without exploding. We construct a nontrivial invariant measure for
this case.Comment: Published in at http://dx.doi.org/10.1214/105051607000000203 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes
We study the branching random walk on weighted graphs; site-breeding and
edge-breeding branching random walks on graphs are seen as particular cases. We
describe the strong critical value in terms of a geometrical parameter of the
graph. We characterize the weak critical value and relate it to another
geometrical parameter. We prove that, at the strong critical value, the process
dies out locally almost surely; while, at the weak critical value, global
survival and global extinction are both possible.Comment: 14 pages, corrected some typos and minor mistake
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
Survival, extinction and approximation of discrete-time branching random walks
We consider a general discrete-time branching random walk on a countable set
X. We relate local, strong local and global survival with suitable inequalities
involving the first-moment matrix M of the process. In particular we prove
that, while the local behavior is characterized by M, the global behavior
cannot be completely described in terms of properties involving M alone.
Moreover we show that locally surviving branching random walks can be
approximated by sequences of spatially confined and stochastically dominated
branching random walks which eventually survive locally if the (possibly
finite) state space is large enough. An analogous result can be achieved by
approximating a branching random walk by a sequence of multitype contact
processes and allowing a sufficiently large number of particles per site. We
compare these results with the ones obtained in the continuous-time case and we
give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte