PreprintCC BY 4.0arXiv
Anchored isoperimetry in percolation
Abstract
We show that, for any infinite transitive graph G, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of G. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, ℤd, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.
Related
BibTeX
@misc{diskin2026anchored,
title = {{Anchored isoperimetry in percolation}},
author = {Sahar Diskin and Lyuben Lichev and Ritvik Ramanan Radhakrishnan and Benny Sudakov and Laury Verhoeven},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38941},
eprint = {2610.38941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38941},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Anchored isoperimetry in percolation
AU - Sahar Diskin
AU - Lyuben Lichev
AU - Ritvik Ramanan Radhakrishnan
AU - Benny Sudakov
AU - Laury Verhoeven
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38941
UR - https://arxiv.org/abs/2610.38941
AB - We show that, for any infinite transitive graph G, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of G. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, ℤ^d, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.
ER -