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AB percolation on plane triangulations is unimodal

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  14 July 2016

Martin J. B. Appel
Martin J. B. Appel*
Affiliation:
University of Iowa
*
∗ Postal address∗. Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA.
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Abstract

Let ℱ be a countable plane triangulation embedded in ℝ2 in such a way that no bounded region contains more than finitely many vertices, and let Pp be Bernoulli (p) product measure on the vertex set of ℱ. Let E be the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that the AB percolation probability function θΑΒ (p) = Pp(E) is non-decreasing in p for 0 ≦ p ≦ ½. By symmetry, θ AΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.

Keywords

AB PERCOLATION PROBABILITY FUNCTION

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 193 - 204
Copyright
Copyright © Applied Probability Trust 1994 

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