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

In long-range percolation on Zᵈ, we connect each pair of distinct points x and y by an edge independently at random with probability 1-exp(-β\|x-y\|-d-α), where α>0 is fixed and β≥ 0 is a parameter. In a previous paper, we proved that if 0<α<d then the critical two-point function satisfies the spatially averaged upper bound \[ 1/r^d∑x∈ [-r,r]^d Pβ_c(0↔ x) r-d+α \] for every r≥ 1. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), and a matching lower bound for α<1 was proven by B\"aumler and Berger (AIHP 2022). In this paper, we prove pointwise upper and lower bounds of the same order under the same assumption that α<1. We also prove analogous two-sided pointwise estimates on the slightly subcritical two-point function under the same hypotheses, interpolating between \| x \|-d+α decay below the correlation length and \| x \|-d-α decay above the correlation length. In dimensions $d=1,2,3$, we deduce that the triangle condition holds under the minimal assumption that 0<α<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 α=d/3, implying that the triangle diagram diverges at most logarithmically and hence that mean-field critical behaviour holds to within polylogarithmic factors.