Supercritical level-set percolation on finite graphs
Abstract
We continue the study of bond percolation on finite graphs induced by the level-sets of the zero-average Gaussian free field on the associated metric graph. In our previous work Goswami and Pal (2026), we analyzed the critical and near-critical behavior of this model on uniform expanders, focusing on the volume of the largest cluster. In the present paper, we investigate the size of the second largest cluster in the supercritical regime --- the phase where one expects a unique giant component. We prove that the second largest cluster is at most of logarithmic order with high probability. The estimates hold uniformly over deterministic graphs with a bounded maximum degree and a spectral gap bounded away from zero. In particular, we do not require any assumption regarding regularity, girth or local geometry. This picture is substantially different from Bernoulli percolation, where the second largest supercritical cluster can have polynomial volume under identical hypotheses. The proof relies on expansion properties of the underlying graph and the Dirichlet martingale framework developed in Goswami and Pal (2026).
Related
- Non-perturbative finite connectivities in supercritical Bernoulli percolation on ℤd with d≥ 3
- Percolation on Finite Graphs
- Distribution tail of the distance to infinity in supercritical bond percolation
- Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters
- Critical oriented percolation enhanced along random supercritical horizontal lines
BibTeX
@misc{goswami2026supercritical,
title = {{Supercritical level-set percolation on finite graphs}},
author = {Subhajit Goswami and Dipranjan Pal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05299},
eprint = {2611.05299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05299},
}
RIS
TY - UNPB
TI - Supercritical level-set percolation on finite graphs
AU - Subhajit Goswami
AU - Dipranjan Pal
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05299
UR - https://arxiv.org/abs/2611.05299
AB - We continue the study of bond percolation on finite graphs induced by the level-sets of the zero-average Gaussian free field on the associated metric graph. In our previous work Goswami and Pal (2026), we analyzed the critical and near-critical behavior of this model on uniform expanders, focusing on the volume of the largest cluster. In the present paper, we investigate the size of the second largest cluster in the supercritical regime --- the phase where one expects a unique giant component. We prove that the second largest cluster is at most of logarithmic order with high probability. The estimates hold uniformly over deterministic graphs with a bounded maximum degree and a spectral gap bounded away from zero. In particular, we do not require any assumption regarding regularity, girth or local geometry. This picture is substantially different from Bernoulli percolation, where the second largest supercritical cluster can have polynomial volume under identical hypotheses. The proof relies on expansion properties of the underlying graph and the Dirichlet martingale framework developed in Goswami and Pal (2026).
ER -