PreprintCC0arXiv
Quantum percolation conjecture on regular trees
Abstract
For Bernoulli bond percolation on every regular tree of degree at least three, we prove that infinite clusters appear strictly before absolutely continuous spectrum. Throughout an explicit supercritical interval, the adjacency operator of every open cluster almost surely has no absolutely continuous component. Together with Bordenave's theorem, this gives pc<pac<1.
Related
- Loop soup percolation and boundary dimension on metric trees
- Percolation of the contact process on the regular tree
- Weinstock Inequality on Regular Trees
- Note on near-spanning balanced antidirected trees missing from regular tournaments
- A finite-graph conjecture related to θ(pc)=0 for Bernoulli bond percolation on ℤd
BibTeX
@misc{becker2026quantum,
title = {{Quantum percolation conjecture on regular trees}},
author = {Simon Becker and Izak Oltman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38017},
eprint = {2609.38017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38017},
note = {CC0},
}
RIS
TY - UNPB
TI - Quantum percolation conjecture on regular trees
AU - Simon Becker
AU - Izak Oltman
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38017
UR - https://arxiv.org/abs/2609.38017
AB - For Bernoulli bond percolation on every regular tree of degree at least three, we prove that infinite clusters appear strictly before absolutely continuous spectrum. Throughout an explicit supercritical interval, the adjacency operator of every open cluster almost surely has no absolutely continuous component. Together with Bordenave's theorem, this gives p_c<p_ac<1.
ER -