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- (-\|x-y\|^-d-), where >0 is fixed and 0 is a parameter. In a previous paper, we proved that if 0<<d then the critical two-point function satisfies the spatially averaged upper bound \ 1rᵈₗ [-ₑ, ₑ㵧 P₂ (0 x) r^-d+ \] for every r 1. This upper bound is believed to be sharp for values of strictly below the crossover value c (d), and a matching lower bound for <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 <1. We also prove analogous two-sided pointwise estimates on the slightly subcritical two-point function under the same hypotheses, interpolating between \| x \|^-d+ decay below the correlation length and \| x \|^-d- decay above the correlation length. In dimensions d=1, 2, 3, we deduce that the triangle condition holds under the minimal assumption that 0<<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 =d/3, implying that the triangle diagram diverges at most logarithmically and hence that mean-field critical behaviour holds to within polylogarithmic factors.