2002/08/15 by Omer Angel, Angel, Omer
Mathematics · Physics and Astronomy · #05C30 #05C80 #81T40 #82B43 #Advanced Combinatorial Mathematics #Advanced Topology and Set Theory #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:05C30 #msc:05C80 #msc:81T40 #msc:82B43
paper · pdf · doi:10.48550/arxiv.math/0208123
arxiv created 2002/08/15 · openalex publication_date 2002/08/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 r4 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 r2 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 pc=1/2 and that at pc percolation does not occur.