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Published November 2022 | public
Journal Article

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

Abstract

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/|Λ_r|)∑_(xϵΛ_(r))P_(β_(c))(0 ↔ x) ≤ r^(−d+a) for every r ⩾ 1, where Λ_r = [−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 α_(c)(d), where the values of several critical exponents for long-range percolation on ℤᵈ and the hierarchical lattice are believed to be equal.

Additional Information

We thank Philip Easo, Emmanuel Michta, Gordon Slade, and the anonymous referee for helpful comments on earlier versions of the manuscript.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023