Graph diameter in long-range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{\Delta+o(1)}$ where $\Delta^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius $\exp\{r^{1/\Delta+o(1)}\}$ in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph diameter, substantially revised and corrected, added a result on volume growth asymptoti

Transience and anchored isoperimetric dimension of supercritical percolation clusters

We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that if $G$ is a transient transitive graph then the infinite clusters of Bernoulli percolation on $G$ are transient for $p$ sufficiently close to $1$. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.Comment: 15 page

Scale-free percolation mixing time

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index τ −1 > 0. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent α > 0 in their distance. The resulting graph is called scalefree percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in α and τ . In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages

Simple random walk on long-range percolation clusters II: Scaling limits

Supported in part at the Technion by a Landau fellowship. Supported in part by an Alfred Sloan Fellowship in Mathematic