Critical percolation of virtually free groups and other tree-like graphs

Mathematics – Probability

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

Published in at http://dx.doi.org/10.1214/09-AOP458 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP458

This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end. The idea of the method is to find a multi-type Galton--Watson branching process (with a parameter $p$) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides sufficient information about $p_c$. In particular, if the pairwise intersections of pieces are finite, then $p_c$ is the smallest positive $p$ such that $\operatorname {det}(M-1)=0$, where $M$ is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic number and we give an algorithm computing $p_c$ as a root of some algebraic function. We show that any Cayley graph of a virtually free group (i.e., a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular, we show how to compute $p_c$ for the Cayley graph of a free group with respect to any finite generating set.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Critical percolation of virtually free groups and other tree-like graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Critical percolation of virtually free groups and other tree-like graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Critical percolation of virtually free groups and other tree-like graphs will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-156341

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.