Pointwise two-point function estimates and a non-pertubative proof of mean-field critical behaviour for long-range percolation

Abstract

In long-range percolation on ${\mathbb{Z}}^d$, we connect each pair of distinct points $x$ and $y$ by an edge independently at random with probability $1-\exp (-\beta \Vert x-y{\Vert }^{-d-\alpha })$, where $\alpha >0$ is fixed and $\beta \ge 0$ is a parameter. In a previous paper, we proved that if $0<\alpha <d$ then the critical two-point function satisfies the spatially averaged upper bound \[ \frac{1}{r^d}\sum_{x\in [-r,r]^d} \mathbb{P}_{β_c}(0\leftrightarrow x) \preceq r^{-d+α} \] for every $r\ge 1$. This upper bound is believed to be sharp for values of $\alpha$ strictly below the crossover value ${\alpha }_c(d)$, and a matching lower bound for $\alpha <1$ was proven by Bäumler and Berger (AIHP 2022). In this paper, we prove pointwise upper and lower bounds of the same order under the same assumption that $\alpha <1$. We also prove analogous two-sided pointwise estimates on the slightly subcritical two-point function under the same hypotheses, interpolating between $\Vert x{\Vert }^{-d+\alpha }$ decay below the correlation length and $\Vert x{\Vert }^{-d-\alpha }$ decay above the correlation length. In dimensions $d=1,2,3$, we deduce that the triangle condition holds under the minimal assumption that $0<\alpha <d/3$. While this result had previously been established under additional perturbative assumptions using the lace expansion, our proof is completely non-perturbative and does not rely on the lace expansion in any way. In dimensions $1$ and $2$ our results also treat the marginal case $\alpha =d/3$, implying that the triangle diagram diverges at most logarithmically and hence that mean-field critical behaviour holds to within polylogarithmic factors.

17 pages