Growth and Percolation on the Uniform Infinite Planar Triangulation

Abstract

A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, 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, confirming 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 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.