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

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 $1-exp(-\mathit{\beta }\Vert \mathit{x}-\mathit{y}{\Vert }^{-\mathit{d}-\mathit{\alpha }})$, where $0<\mathit{\alpha }<\mathit{d}$ is fixed and $\mathit{\beta }\ge 0$ is a parameter, then the critical two-point function satisfies

$${\mathbb{P}}_{{\mathit{\beta }}_{\mathit{c}}}(\mathit{x}\leftrightarrow \mathit{y})\asymp \Vert \mathit{x}-\mathit{y}{\Vert }^{-\mathit{d}\mathbf{+}\mathit{\alpha }}$$

for every pair of distinct points x and y. We deduce in particular that the model has mean-field critical behaviour when $\mathit{\alpha }<\mathit{d}/3$ and does not have mean-field critical behaviour when $\mathit{\alpha }>\mathit{d}/3$.