Explicit upper bounds on the threshold of one-dimensional long-range percolation
Abstract
We study bond percolation on the one-dimensional lattice in which two sites at distance r are connected with probability pr=C/r1+σ, where 0 < σ< 1. An infinite cluster exists if C exceeds a critical value Cc(σ), which is not known exactly: the available results are the lower bound of Schulman, Cc≥1/[2ζ(1+σ)], and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives Cc≤σ+O(σ2) as σ→0 and is below $1$ for σ< 0.667. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: Cc→ζ(1+σ)→1 as σ→0. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound Cc < 1 for every σ< 1. We compare the bounds with the known numerical values of Cc.
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BibTeX
@misc{amit2026explicit,
title = {{Explicit upper bounds on the threshold of one-dimensional long-range percolation}},
author = {Guy Amit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11872},
eprint = {2610.11872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11872},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit upper bounds on the threshold of one-dimensional long-range percolation
AU - Guy Amit
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11872
UR - https://arxiv.org/abs/2610.11872
AB - We study bond percolation on the one-dimensional lattice in which two sites at distance r are connected with probability p_r=C/r^1+σ, where 0 < σ< 1. An infinite cluster exists if C exceeds a critical value C_c(σ), which is not known exactly: the available results are the lower bound of Schulman, C_c≥1/[2ζ(1+σ)], and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives C_c≤σ+O(σ^2) as σ→0 and is below $1$ for σ< 0.667. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: C_c^→ζ(1+σ)→1 as σ→0. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound C_c < 1 for every σ< 1. We compare the bounds with the known numerical values of C_c.
ER -