Power-law bounds for critical long-range percolation below the upper-critical dimension

Abstract

We study long-range Bernoulli percolation on Zd$${\mathbb {Z}}^d$$ in which each two vertices x and y are connected by an edge with probability 1-exp(-β‖x-y‖-d-α)$$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$. It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if 0<α