Consider long-range Bernoulli percolation on 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 for every r ⩾ 1, where . In other words, the critical two-point function on 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 and the hierarchical lattice are believed to be equal.
The fact that 2 − η = α ∧ 2 is the mean-field value of this exponent is related to the fact that the inverse of the fractional Laplacian (−Δ)α/2 decays as ‖x‖−d+α for α < 2 and as ‖x‖−d+2 for α > 2 (Ref. 32, Secs. 2 and 3).
Here, we are avoiding making the stronger statement that the models belong to the same universality class since this claim would arguably be too strong. Indeed, while the models may share exponents, they should not have a common scaling limit because in the Euclidean case, the relevant continuum limit should be defined on , while in the hierarchical case, it should be defined on the xmltex \beginMathtotextLxmltex \endMathtotext-adic numbers. See Ref. 48 for detailed discussions of related phenomena. It is also unclear at present whether one should expect the exponents describing off-critical behavior to coincide.
See Remark 4.2.