Counterexamples to the site-percolation analogues of the Easo-Severo-Tassion cutset theorems
Abstract
Easo, Severo and Tassion [EST25] proved two theorems about minimal edge cutsets in infinite graphs: that non-triviality of the uniform critical percolation parameter is equivalent to exponential growth of the number of minimal edge cutsets of size n separating a vertex from infinity, and that uniform transience is an alternative sufficient assumption. We show that the naive extensions of (the original contribution to) these theorems to site percolation and vertex cutsets are false; our counterexample involves modifying a rooted binary tree and replacing edges with gadgets that allow many choices for where to block (in constructing a cutset) one of two local routes, while keeping every vertex uniformly close to a supercritical, uniformly transient shortcut skeleton.
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BibTeX
@misc{bassil2026counterexamples,
title = {{Counterexamples to the site-percolation analogues of the Easo-Severo-Tassion cutset theorems}},
author = {Joel Bassil},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29940},
eprint = {2608.29940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to the site-percolation analogues of the Easo-Severo-Tassion cutset theorems
AU - Joel Bassil
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29940
UR - https://arxiv.org/abs/2608.29940
AB - Easo, Severo and Tassion [EST25] proved two theorems about minimal edge cutsets in infinite graphs: that non-triviality of the uniform critical percolation parameter is equivalent to exponential growth of the number of minimal edge cutsets of size n separating a vertex from infinity, and that uniform transience is an alternative sufficient assumption. We show that the naive extensions of (the original contribution to) these theorems to site percolation and vertex cutsets are false; our counterexample involves modifying a rooted binary tree and replacing edges with gadgets that allow many choices for where to block (in constructing a cutset) one of two local routes, while keeping every vertex uniformly close to a supercritical, uniformly transient shortcut skeleton.
ER -