Percolation probability and critical exponents for site percolation on the UIPT

TL;DR

This paper derives critical exponents for Bernoulli site percolation on the UIPT, providing explicit formulas and decay rates for cluster size and perimeter probabilities at criticality.

Contribution

It introduces new explicit formulas for critical exponents and cluster laws in site percolation on the UIPT, extending previous knowledge.

Findings

Probability of an infinite root cluster computed explicitly.

Off-critical exponent for site percolation is 1/2.

At criticality, cluster size decays as n^{-1/7} and perimeter as n^{-4/3}.

Abstract

We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least $n$ vertices decays like $n^{-1/7}$. Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to $n$ decays like $n^{-4/3}$. Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, as well as analytic combinatorics.