Bi-infinite incipient cluster in high dimensions

TL;DR

This paper introduces the bi-infinite incipient percolation cluster in high dimensions, a new measure describing two disjoint infinite paths emerging at criticality, constructed via a double lace expansion.

Contribution

It constructs and characterizes a novel bi-infinite incipient percolation measure in high dimensions, distinct from previously known incipient infinite clusters.

Findings

Defines the bi-infinite incipient percolation cluster measure.

Shows mutual singularity with existing incipient infinite clusters.

Constructs the measure using a double lace expansion.

Abstract

We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity. This limiting procedure gives rise to a new percolation measure that locally resembles critical percolation but is concentrated on configurations with two disjoint infinite occupied paths. We coin this the bi-infinite incipient percolation cluster. It is mutually singular with respect to incipient infinite clusters that have been constructed in the literature. We achieve the construction through a double lace expansion of the cluster.