This book focuses on percolation on high-dimensional lattices. We give a general
introduction to percolation, stating the main results and defining the central objects. We …

We consider the (unoriented) long-range percolation on ℤ d in dimensions d≥ 1, where
distinct sites x, y∈ ℤ d get connected with probability p xy∈[0, 1]. Assuming p xy=| x− y|− s+ …

We consider independent Bernoulli bond percolation on the integer lattice Zd, with edge
(bond) set consisting of pairs {x, y} of vertices of Zd with y− x∈ Ω, where Ω defines either the …

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-
field behavior has been established, namely when the dimension d is large enough or when …

We study random subgraphs of an arbitrary finite connected transitive graph 𝔾 obtained by
independently deleting edges with probability 1− p. Let V be the number of vertices in 𝔾, and …

We study the probability that the origin is connected to the sphere of radius $ r $(an arm
event) in critical percolation in high dimensions, namely when the dimension $ d $ is large …

We consider long-range Bernoulli bond percolation on the $ d $-dimensional hierarchical
lattice in which each pair of points $ x $ and $ y $ are connected by an edge with probability …

For independent nearest-neighbor bond percolation on Z d with d≫ 6, we prove that the
incipient infinite cluster's two-point function and three-point function converge to those of …

Van der Hofstad, Remco; Slade, Gordon. Convergence of critical oriented percolation to
super-brownian motion above $4+ 1$ dimensions. Annales de l'IHP Probabilités et …

We consider bond percolation on Z^ d at the critical occupation density pc for d> 6 in two
different models. The first is the nearest-neighbor model in dimension d≫ 6. The second …