45 research outputs found
Nonconcentration of return times
We show that the distribution of the first return time to the origin,
v, of a simple random walk on an infinite recurrent graph is heavy tailed and
nonconcentrated. More precisely, if is the degree of v, then for any
we have and
for some
universal constants and . The first bound is attained for all t
when the underlying graph is , and as for the second bound, we
construct an example of a recurrent graph G for which it is attained for
infinitely many t's. Furthermore, we show that in the comb product of that
graph G with , two independent random walks collide infinitely many
times almost surely. This answers negatively a question of Krishnapur and Peres
[Electron. Commun. Probab. 9 (2004) 72-81] who asked whether every comb product
of two infinite recurrent graphs has the finite collision property.Comment: Published in at http://dx.doi.org/10.1214/12-AOP785 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On Fixation of Activated Random Walks
We prove that for the Activated Random Walks model on transitive unimodular
graphs, if there is fixation, then every particle eventually fixates, almost
surely. We deduce that the critical density is at most 1.
Our methods apply for much more general processes on unimodular graphs.
Roughly put, our result apply whenever the path of each particle has an
automorphism invariant distribution and is independent of other particles'
paths, and the interaction between particles is automorphism invariant and
local. This allows us to answer a question of Rolla and Sidoravicius, in a more
general setting then had been previously known (by Shellef).Comment: 5 page