Isoperimetric lower bounds for critical exponents for long-range percolation

Abstract

We study independent long-range percolation on ${\mathbb{Z}}^d$ where the vertices $x$ and $y$ are connected with probability $1-e^{-\beta \Vert x-y{\Vert }^{-d-\alpha }}$ for $\alpha >0$. Provided the critical exponents $\delta$ and $2-\eta$ defined by $\delta ={\lim }_{n\to \infty }\frac{-\log (n)}{\log \left({\mathbb{P}}_{{\beta }_c}\left(|K_0|\ge n\right)\right)}$ and $2-\eta ={\lim }_{x\to \infty }\frac{\log \left({\mathbb{P}}_{{\beta }_c}\left(0\leftrightarrow x\right)\right)}{\log (\Vert x\Vert )}+d$ exist, where $K_0$ is the cluster containing the origin, we show that \begin{equation*} δ\geq \frac{d+(α\wedge 1)}{d-(α\wedge 1)} \ \text{ and } \ 2-η\geq α\wedge 1 \text. \end{equation*} The lower bound on $\delta$ is believed to be sharp for $d=1,\alpha \in \left[\frac{1}{3},1\right)$ and for $d=2,\alpha \in \left[\frac{2}{3},1\right]$, whereas the lower bound on $2-\eta$ is sharp for $d=1,\alpha \in (0,1)$, and for $\alpha \in \left(0,1\right]$ for $d>1$, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside ${\mathbb{Z}}^d$.

13 pages, 2 figures

References in corpus (4)

• On the scaling of the chemical distance in long-range percolation models
• Critical random graphs: Diameter and mixing time
• 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 ${\mathbb{Z}}^d$

Cited by in corpus (1)

• Behavior of the distance exponent for $\frac{1}{|x-y{|}^{2d}}$ long-range percolation