2006/11/30 by Iva Kozakova · 1 citation
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.GR #math.PR #msc:20P05 #msc:60K35 #msc:82B43
paper · pdf · doi:10.1142/s0218196708004524
published as International Journal of Algebra and Computation, Volume No.18, Issue No.4, June 2008, Page: 683 - 704
openalex publication_date 2008/06/01 · arxiv created 2008/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we study percolation on the Cayley graph of a free product of groups. The critical probability p c of a free product G 1 * G 2 * ⋯ * G n of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G 1 , G 2 , …, G n . For finite groups this equation is 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 p c for the Cayley graph of the modular group PSL 2 (ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p 5 - 6p 4 + 2p 3 + 4p 2 - 1 in the interval (0, 1). In the case when groups G i 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 G 1 * G 2 * ⋯ * G n 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, p exp of the free product is just the minimum of p exp for the factors.