Growth and percolation on the uniform infinite planar triangulation

Abstract

A construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.

By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r ⁴ up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r ² up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.

By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p _c = 1/2 and that at p _c percolation does not occur.