17,552 research outputs found
Sharpness for Inhomogeneous Percolation on Quasi-Transitive Graphs
In this note we study the phase transition for percolation on
quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention
probabilities. A quasi-transitive graph is an infinite graph with finitely many
different "types" of edges and vertices. We prove that the transition is sharp
almost everywhere, i.e., that in the subcritical regime the expected cluster
size is finite, and that in the subcritical regime the probability of the
one-arm event decays exponentially. Our proof extends the proof of sharpness of
the phase transition for homogeneous percolation on vertex-transitive graphs by
Duminil-Copin and Tassion [Comm. Math. Phys., 2016], and the result generalizes
previous results of Antunovi\'c and Veseli\'c [J. Stat. Phys., 2008] and
Menshikov [Dokl. Akad. Nauk 1986].Comment: 9 page
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Indistinguishability of Percolation Clusters
We show that when percolation produces infinitely many infinite clusters on a
Cayley graph, one cannot distinguish the clusters from each other by any
invariantly defined property. This implies that uniqueness of the infinite
cluster is equivalent to non-decay of connectivity (a.k.a. long-range order).
We then derive applications concerning uniqueness in Kazhdan groups and in
wreath products, and inequalities for .Comment: To appear in Ann. Proba
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot have
infinitely many infinite clusters at critical Bernoulli percolation. In the
case of heavy clusters, this result has already been established, but it also
follows from one of our results. We give a general necessary condition for
nonunimodular graphs to have a phase with infinitely many heavy clusters. We
present an invariant spanning tree with on some nonunimodular graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a new way of
constructing nonunimodular graphs that have properties more peculiar than the
ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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