ABpercolation on plane triangulations is unimodal

Abstract

Let ℱ be a countable plane triangulation embedded in ℝ2in such a way that no bounded region contains more than finitely many vertices, and letPpbe Bernoulli (p) product measure on the vertex set of ℱ. LetEbe the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that theAB percolation probability function θΑΒ(p) =Pp(E) is non-decreasing inpfor 0 ≦p≦ ½. By symmetry,θAΒ(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.