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 $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, 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 $\alpha>d/3$.