Distances in critical long range percolation

Abstract

We study the long range percolation model on $\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $\beta |i-j{|}^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{\theta (\beta )}$ up to a multiplicative constant for some exponent $0<\theta (\beta )<1$ as does the diameter of the graph on a box of length $n$.

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References in corpus (2)

• On the scaling of the chemical distance in long-range percolation models
• The local weak limit of the minimum spanning tree of the complete graph

Cited by in corpus (6)

• Arithmetic oscillations of the chemical distance in long-range percolation on ${\mathbb{Z}}^d$
• Long-range contact process and percolation on a random lattice
• Graph distances in scale-free percolation: the logarithmic case
• Graph distances of continuum long-range percolation
• Chemical distance in geometric random graphs with long edges and scale-free degree distribution
• Inhomogeneous Long-Range Percolation for Real-Life Network Modeling