Power-law bounds for critical long-range percolation below the upper-critical dimension
Authors
Publication Date
2021-11Journal Title
Probability Theory and Related Fields
ISSN
0178-8051
Publisher
Springer Science and Business Media LLC
Volume
181
Issue
1-3
Pages
533-570
Language
en
Type
Article
This Version
VoR
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Hutchcroft, T. (2021). Power-law bounds for critical long-range percolation below the upper-critical dimension. Probability Theory and Related Fields, 181 (1-3), 533-570. https://doi.org/10.1007/s00440-021-01043-7
Abstract
We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two
vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta
\|x-y\|^{-d-\alpha})$. It is a theorem of Noam Berger (CMP, 2002) that if
$0<\alpha<d$ then there is no infinite cluster at the critical parameter
$\beta_c$. We give a new, quantitative proof of this theorem establishing the
power-law upper bound \[ \mathbf{P}_{\beta_c}\bigl(|K|\geq n\bigr) \leq C
n^{-(d-\alpha)/(2d+\alpha)} \] for every $n\geq 1$, where $K$ is the cluster of
the origin. We believe that this is the first rigorous power-law upper bound
for a Bernoulli percolation model that is neither planar nor expected to
exhibit mean-field critical behaviour.
As part of the proof, we establish a universal inequality implying that the
maximum size of a cluster in percolation on any finite graph is of the same
order as its mean with high probability. We apply this inequality to derive a
new rigorous hyperscaling inequality $(2-\eta)(\delta+1)\leq d(\delta-1)$
relating the cluster-volume exponent $\delta$ and two-point function exponent
$\eta$.
Keywords
Article, Primary 60K35, Secondary 82B27, 82B34
Identifiers
s00440-021-01043-7, 1043
External DOI: https://doi.org/10.1007/s00440-021-01043-7
This record's URL: https://www.repository.cam.ac.uk/handle/1810/330857
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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