The Incipient Infinite Cluster of the Uniform Infinite Half-Planar Triangulation

TL;DR

This paper introduces the Incipient Infinite Cluster (IIC) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation, providing an explicit construction and analysis of its properties.

Contribution

It defines the IIC in the UIHPT and shows how to explicitly construct it by adding an infinite triangulation with a boundary.

Findings

The IIC can be obtained by conditioning the percolation cluster to be infinite.

An explicit distribution for the added infinite triangulation is derived.

The construction links percolation theory with random triangulation models.

Abstract

We introduce the Incipient Infinite Cluster (IIC) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation (UIHPT), which is the local limit of large random triangulations with a boundary. The IIC is defined from the UIHPT by conditioning the open percolation cluster of the origin to be infinite. We prove that the IIC can be obtained by adding within the UIHPT an infinite triangulation with a boundary whose distribution is explicit.