Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd

Consider long-range Bernoulli percolation on Zd in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖−d−α), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical two-point function satisfies 1|Λr|∑x∈ΛrPβc(0↔x)⪯r−d+α for every r ⩾ 1, where Λr=[−r,r]d∩Zd. In other words, the critical two-point function on Zd is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), where the values of several critical exponents for long-range percolation on Zd and the hierarchical lattice are believed to be equal.