Critical percolation of free product of groups

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dc.creatorKozakova, Iva
dc.date2006-11-22
dc.date2008-09-26
dc.date.accessioned2026-07-07T10:05:23Z
dc.date.available2026-07-07T10:05:23Z
dc.descriptionIn 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 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 $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, 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, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors.
dc.identifierhttps://arxiv.org/abs/math/0611668
dc.identifierhttp://arxiv.org/abs/math/0611668
dc.identifierInternational Journal of Algebra and Computation, Volume No.18, Issue No.4, June 2008, Page: 683 - 704
dc.identifierdoi:10.1142/S0218196708004524
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/169994
dc.subjectProbability
dc.subjectGroup Theory
dc.subject60K35, 82B43, 20P05
dc.titleCritical percolation of free product of groups
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