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 $1-\exp (-\beta \Vert x-y{\Vert }^{-d-\alpha })$, where $\alpha >0$ is fixed and $\beta \ge 0$ is a parameter. We prove that if $0<\alpha <d$ then the critical two-point function satisfies \[ \frac{1}{|Λ_r|}\sum_{x\in Λ_r} \mathbf{P}_{β_c}(0\leftrightarrow x) \preceq r^{-d+α} \] for every $r\ge 1$, 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.

26 pages, 4 figures. V2: Various minor corrections

References in corpus (4)

• The crossover region between long-range and short-range interactions for the critical exponents
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$
• Arithmetic oscillations of the chemical distance in long-range percolation on ${\mathbb{Z}}^d$

Cited by in corpus (4)

• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$
• Distances in $\frac{1}{|x-y{|}^{2d}}$ percolation models for all dimensions
• Isoperimetric lower bounds for critical exponents for long-range percolation
• Crossing exponent in the Brownian loop soup