Bi-infinite incipient cluster in high dimensions

Abstract

We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x^{\prime }$, and subsequently take the limit as $|x|$, $|x^{\prime }|$ as well as $|x-x^{\prime }|$ 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.

70 pages, 4 figures