PreprintCC BY-NC-ND 4.0arXiv
Applications of the cluster graphing
Abstract
In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at pu in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.
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BibTeX
@misc{chu2026applications,
title = {{Applications of the cluster graphing}},
author = {Tasmin Chu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06644},
eprint = {2608.06644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06644},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Applications of the cluster graphing
AU - Tasmin Chu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.06644
UR - https://arxiv.org/abs/2608.06644
AB - In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at p_u in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.
ER -