Growth and Percolation on the Uniform Infinite Planar Triangulation

TL;DR

This paper introduces an efficient growth process for sampling the uniform infinite planar triangulation (UIPT), analyzes its growth rate, and studies percolation properties, confirming physics heuristics and revealing new probabilistic behaviors.

Contribution

It provides a novel algorithmic growth process for UIPT, establishes its growth rate, and analyzes percolation thresholds and scaling limits, advancing understanding of random planar maps.

Findings

UIPT has growth rate r^4 up to polylogarithmic factors

Boundary component growth rate is r^2 up to polylogarithmic factors

Critical percolation probability p_c=1/2 with no percolation at p_c

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.