4 research outputs found
Critical percolation of free product of groups
In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability of a free product of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups . For finite groups these
equations are polynomial and can be explicitly written down. The expected
subcritical cluster size of the free product is also found in terms of the
subcritical cluster sizes of the factors. In particular, we prove that
for the Cayley graph of the modular group (with the
standard generators) is , the unique root of the polynomial
in the interval .
In the case when groups can be "well approximated" by a sequence of
quotient groups, we show that the critical probabilities of the free product of
these approximations converge to the critical probability of
and the speed of convergence is exponential. Thus for residually finite groups,
for example, one can restrict oneself to the case when each free factor is
finite.
We show that the critical point, introduced by Schonmann,
of the free product is just the minimum of for the factors
Critical percolation of virtually free groups and other tree-like graphs
This article presents a method for finding the critical probability 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 ) which has finite expected population size if and only if the
expected percolation cluster size is finite. This provides sufficient
information about . In particular, if the pairwise intersections of pieces
are finite, then is the smallest positive such that , where is the first-moment matrix of the branching process.
If the pieces of the tree-like structure are finite, then is an algebraic
number and we give an algorithm computing 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 for the Cayley graph of a free group
with respect to any finite generating set.Comment: Published in at http://dx.doi.org/10.1214/09-AOP458 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org