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 $1-\exp(-\beta\|x-y\|^{ -d-\alpha})$, where $\alpha>0$ is fixed and $\beta\geq 0$ 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 \[ \frac{ 1}{ r^d}\sum_{ x\in [-r,r]^d} \mathbb{ P}_{ \beta_c}(0\leftrightarrow x) \preceq r^{ -d+\alpha} \] for every $r\geq 1$. 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 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 $\alpha<1$. We also prove analogous two-sided pointwise estimates on the slightly subcritical two-point function under the same hypotheses, interpolating between $\| x \|^{ -d+\alpha}$ decay below the correlation length and $\| x \|^{ -d-\alpha}$ 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.
Categories: math.PR math-ph math.MP