In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability pc of a free product G1∗G2∗...∗Gn of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups G1,G2,...,Gn. 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 pc
for the Cayley graph of the modular group PSL2(Z) (with the
standard generators) is .5199..., the unique root of the polynomial
2p5−6p4+2p3+4p2−1 in the interval (0,1).
In the case when groups Gi 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 G1∗G2∗...∗Gn
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, pexp
of the free product is just the minimum of pexp for the factors