No infinite cluster at criticality for long-range percolation with fast-decaying interactions
Abstract
We consider independent long-range bond percolation on ℤd in which x and y are joined with probability 1−e−βJ(x−y). Let d≥2 and let J be invariant under coordinate permutations and sign changes, positive on nearest-neighbour pairs, and satisfy liminfn→∞ nd∑|x|>nJ(x)=0. We prove that there is no infinite cluster at the critical parameter. Together with Berger's theorem for heavy tails, this gives continuity of the phase transition for J(x)asymp|x|−d−α for every α>0 except α=d, in every dimension d≥2, and for every exponentially decaying kernel satisfying these hypotheses. Previously the fast-decaying case was known for nearest-neighbour and uniform finite-range kernels, in d=2 for finite-range and exponential kernels, and in high dimension for sufficiently spread-out kernels. The proof adapts a recent renormalization argument for nearest-neighbour percolation on ℤ3 that is driven by a gluing inequality conjectured by Kozma and Nitzan. Long edges enter in two places: they can jump over the surfaces that a nearest-neighbour path must cross, and they let a cluster outside a box enter it from far away. We control the first with thick outer faces and the tail condition, and the second by measuring entrances through the total edge weight between a cluster and a box rather than through the number of boundary vertices.
Related
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BibTeX
@misc{pei2026infinite,
title = {{No infinite cluster at criticality for long-range percolation with fast-decaying interactions}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01490},
eprint = {2611.01490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01490},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - No infinite cluster at criticality for long-range percolation with fast-decaying interactions
AU - Yan Ru Pei
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01490
UR - https://arxiv.org/abs/2611.01490
AB - We consider independent long-range bond percolation on ℤ^d in which x and y are joined with probability 1−e^−βJ(x−y). Let d≥2 and let J be invariant under coordinate permutations and sign changes, positive on nearest-neighbour pairs, and satisfy liminf_n→∞ n^d∑_|x|>nJ(x)=0. We prove that there is no infinite cluster at the critical parameter. Together with Berger's theorem for heavy tails, this gives continuity of the phase transition for J(x)asymp|x|^−d−α for every α>0 except α=d, in every dimension d≥2, and for every exponentially decaying kernel satisfying these hypotheses. Previously the fast-decaying case was known for nearest-neighbour and uniform finite-range kernels, in d=2 for finite-range and exponential kernels, and in high dimension for sufficiently spread-out kernels. The proof adapts a recent renormalization argument for nearest-neighbour percolation on ℤ^3 that is driven by a gluing inequality conjectured by Kozma and Nitzan. Long edges enter in two places: they can jump over the surfaces that a nearest-neighbour path must cross, and they let a cluster outside a box enter it from far away. We control the first with thick outer faces and the tail condition, and the second by measuring entrances through the total edge weight between a cluster and a box rather than through the number of boundary vertices.
ER -