Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on $\mathbb{Z}^d$

Abstract: Consider long-range Bernoulli percolation on $\mathbb{Z}^d$ 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 $\beta\geq 0$ is a parameter. We prove that if $0<\alpha<d$ then the critical two-point function satisfies [ \frac{1}{|\Lambda_r|}\sum_{x\in \Lambda_r} \mathbf{P}_{\beta_c}(0\leftrightarrow x) \preceq r^{-d+\alpha} ] for every $r\geq 1$, where $\Lambda_r=[-r,r]^d \cap \mathbb{Z}^d$. In other words, the critical two-point function on $\mathbb{Z}^d$ 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 $\mathbb{Z}^d$ and the hierarchical lattice are believed to be equal.