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

Abstract

Consider long-range Bernoulli percolation on [Formula: see text] 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 [Formula: see text] for every r ⩾ 1, where [Formula: see text]. In other words, the critical two-point function on [Formula: see text] 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 [Formula: see text] and the hierarchical lattice are believed to be equal.