The critical two-point function for long-range percolation on the hierarchical lattice

TL;DR

This paper establishes bounds on the critical two-point function for long-range percolation on hierarchical lattices, revealing when the model exhibits mean-field critical behavior based on the decay parameter.

Contribution

It provides the first rigorous bounds on the two-point function for critical long-range percolation on hierarchical lattices, clarifying the conditions for mean-field behavior.

Findings

Critical two-point function scales as rac{ ext{(distance)}}

Model exhibits mean-field behavior when rac{ ext{(parameter)}}

Non-mean-field behavior occurs beyond a threshold rac{ ext{(parameter)}}

Abstract

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points $x$ and $y$ by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $0<\alpha<d$ is fixed and $\beta\geq 0$ is a parameter, then the critical two-point function satisfies \[ \mathbb{P}_{\beta_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+\alpha} \] for every pair of distinct points $x$ and $y$. We deduce in particular that the model has mean-field critical behaviour when $\alpha<d/3$ and does not have mean-field critical behaviour when $\alpha>d/3$.