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 \(\frac{1}{r^d}\sum _{x\in [-r,r]^d} \mathbb {P}_{\beta _c}(0\leftrightarrow x) \preceq r^{-d+\alpha }\) 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.
