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

Consider long-range Bernoulli percolation on Z(d) in which we connect each pair of distinct points x and y by an edge with probability 1 - exp(-beta||x - y||(-d-alpha)), where alpha > 0 is fixed and beta >= 0 is a parameter. We prove that if 0 < alpha < d, then the critical two-point function satisfies 1|Lambda r| Sigma(x is an element of Lambda) P-beta c (0 <-> x) less than or similar to r(-d+alpha) for every r >= 1, where Lambda(r)=[-r,r](d) boolean AND Z(d). In other words, the critical two-point function on Z(d) 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 alpha strictly below the crossover value alpha(c)(d), where the values of several critical exponents for long-range percolation on Z(d) and the hierarchical lattice are believed to be equal. Published under an exclusive license by AIP Publishing.

Automated summary

This passage describes the properties of long-range Bernoulli percolation on Z(d) and proves that the critical two-point function on Z(d) is always bounded above by the critical two-point function on the hierarchical lattice.