A lower bound for the chemical distance in sparse long-range percolation models

TL;DR

This paper establishes a linear lower bound on the chemical distance in sparse long-range percolation models for dimensions where the connection probability decays sufficiently fast.

Contribution

It provides a new lower bound for the chemical distance in long-range percolation models when the decay parameter exceeds twice the dimension.

Findings

Chemical distance grows at least linearly with Euclidean distance.

Results apply to models with decay parameter s > 2d.

Supports understanding of geometric properties in long-range percolation.

Abstract

We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.