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Mixing time and isoperimetry of long-range percolation in the diffusive regime
Abstract
We consider long-range percolation (LRP) on ℤd, d ≥ 1 with connection probabilities p(x, y) ≈ β/(|x−y|s) for β>0 and s > min(2d, d+2). Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length n is of order n2. To establish the upper bound, we show that a d-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.
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BibTeX
@misc{mitsche2026mixing,
title = {{Mixing time and isoperimetry of long-range percolation in the diffusive regime}},
author = {Dieter Mitsche and Carlo Scali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11808},
eprint = {2610.11808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11808},
}
RIS
TY - UNPB
TI - Mixing time and isoperimetry of long-range percolation in the diffusive regime
AU - Dieter Mitsche
AU - Carlo Scali
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11808
UR - https://arxiv.org/abs/2610.11808
AB - We consider long-range percolation (LRP) on ℤ^d, d ≥ 1 with connection probabilities p(x, y) ≈ β/(|x−y|^s) for β>0 and s > min(2d, d+2). Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length n is of order n^2. To establish the upper bound, we show that a d-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.
ER -