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arXiv · 2506.06559

Bi-infinite incipient cluster in high dimensions

Manuel Cabezas·Alexander Fribergh·Markus Heydenreich·Antal A. Járai
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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.

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Manuel Cabezas, Alexander Fribergh, Markus Heydenreich, Antal A. Járai. 2025-06-06. Bi-infinite incipient cluster in high dimensions. https://arxiv.org/abs/2506.06559

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