Robust construction of the incipient infinite cluster in high dimensional critical percolation

TL;DR

This paper introduces a new, minimal-assumption method for constructing the incipient infinite cluster in high-dimensional percolation, avoiding traditional diagrammatic techniques and broadening applicability.

Contribution

A novel construction of the IIC that relies on conditioning in supercritical regimes and at criticality, applicable in all dimensions with known two-point function asymptotics.

Findings

Constructs IIC by conditioning on infinite clusters in supercritical percolation.

Generalizes IIC construction to arbitrary distant sets at criticality.

Applicable in all dimensions with known two-point function asymptotics.

Abstract

We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we do not directly use the machinery of the lace expansion or similar diagrammatic expansions. We show that the IIC may be constructed by conditioning on the cluster of a vertex being infinite in the supercritical regime $p > p_c$ and then taking $p \searrow p_c$. Furthermore, at criticality, we show that the IIC may be constructed by conditioning on a connection to an arbitrary distant set $V$, generalizing previous constructions where one conditions on a connection to a single distant vertex or the boundary of a large box. The input to our proof are the asymptotics for the two-point function obtained by Hara, van der Hofstad, and Slade. Our construction thus applies in all dimensions for which those asymptotics are known, rather than an unspecified high dimension considered in previous works. The results in this paper will be instrumental in upcoming work related to structural properties and scaling limits of various objects involving high-dimensional percolation clusters at and near criticality.