High-dimensional incipient infinite clusters revisited

TL;DR

This paper constructs and compares multiple methods for defining the incipient infinite cluster (IIC) measure in high-dimensional percolation, demonstrating its robustness and analyzing its structural properties.

Contribution

It introduces three new constructions of the IIC measure in high dimensions, showing their equivalence and extending applicability to spread-out models.

Findings

All constructions yield the same IIC measure.

The methods apply to both finite-range and long-range percolation models.

Structural estimates include volume bounds of IIC intersections with Euclidean balls.

Abstract

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second author and Jarai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also obtain estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.