Distances in critical long range percolation

TL;DR

This paper investigates the behavior of graph distances in a long-range percolation model on the integers, revealing a power-law growth in typical distances at the critical exponent s=2, confirming a conjecture about the model's scaling.

Contribution

We prove that at the critical exponent s=2, the typical distance scales as a power law with an exponent depending on the parameter β, confirming a conjecture about the model's self-similar structure.

Findings

Distances grow as a power law at s=2

Confirmed the conjecture of Benjamini and Berger

Identified the exponent θ(β) for the growth rate

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 $n$.