Abstract

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

Highlights

  • The purpose of this work is to describe the geometry of a large critical percolation cluster in the Uniform Infinite Half-Planar Triangulation (UIHPT for short), which is the local limit of random triangulations with a boundary, upon letting first the volume and the perimeter tend to infinity

  • The study of local limits of large planar maps goes back to Angel & Schramm, who introduced in [5] the Uniform Infinite Planar Triangulation (UIPT), while the half-plane model was defined later on by Angel in [3]

  • Local limits of large planar maps equipped with a percolation model have been studied extensively

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Summary

Introduction

The purpose of this work is to describe the geometry of a large critical percolation cluster in the (type 2) Uniform Infinite Half-Planar Triangulation (UIHPT for short), which is the local limit of random triangulations with a boundary, upon letting first the volume and the perimeter tend to infinity. The probability measure PIIC is called (the law of) the Incipient Infinite Cluster of the UIHPT (IIC for short) and is supported on triangulations of the half-plane. In Theorem 2.1, we decompose the UIHPT into two infinite sub-maps distributed as the closed percolation hulls of the IIC, and glued along a uniform necklace The idea of such a decomposition goes back to [18]. The boundary of an infinite planar map is the embedding of edges and vertices of its root face. The probability measure P∞,k is called (the law of) the UIPT of the k-gon, while P∞,∞ is (the law of) the Uniform Infinite Half-Planar Triangulation (UIHPT). The hull H of C is the coloured map obtained by filling in the finite holes of C

Random trees and looptrees
Statement of the results
The contour function
Random walks
Decomposition of the UIHPT
Exploration process
Percolation hulls and necklace
Proof of the decomposition result
Distribution of the revealed map
The IIC probability measure
Proof of the IIC results
Scaling limits and perspectives
Full Text
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