7 research outputs found
Monotonicity and phase diagram for multirange percolation on oriented trees
We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds
Limiting geodesics for first-passage percolation on subsets of
It is an open problem to show that in two-dimensional first-passage
percolation, the sequence of finite geodesics from any point to has a
limit in . In this paper, we consider this question for first-passage
percolation on a wide class of subgraphs of : those whose vertex
set is infinite and connected with an infinite connected complement. This
includes, for instance, slit planes, half-planes and sectors. Writing for
the sequence of boundary vertices, we show that the sequence of geodesics from
any point to has an almost sure limit assuming only existence of finite
geodesics. For all passage-time configurations, we show existence of a limiting
Busemann function. Specializing to the case of the half-plane, we prove that
the limiting geodesic graph has one topological end; that is, all its infinite
geodesics coalesce, and there are no backward infinite paths. To do this, we
prove in the Appendix existence of geodesics for all product measures in our
domains and remove the moment assumption of the Wehr-Woo theorem on absence of
bigeodesics in the half-plane.Comment: Published in at http://dx.doi.org/10.1214/13-AAP999 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
50 years of first passage percolation
We celebrate the 50th anniversary of one the most classical models in
probability theory. In this survey, we describe the main results of first
passage percolation, paying special attention to the recent burst of advances
of the past 5 years. The purpose of these notes is twofold. In the first
chapters, we give self-contained proofs of seminal results obtained in the '80s
and '90s on limit shapes and geodesics, while covering the state of the art of
these questions. Second, aside from these classical results, we discuss recent
perspectives and directions including (1) the connection between Busemann
functions and geodesics, (2) the proof of sublinear variance under 2+log
moments of passage times and (3) the role of growth and competition models. We
also provide a collection of (old and new) open questions, hoping to solve them
before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with
expanded and additional material. Small typos corrected throughou