Growth and percolation on the uniform infinite planar triangulation

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.