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 …

The main theme of these lecture notes is to analyze heat conduction on disordered media
such as fractals and percolation clusters by means of both probabilistic and analytic …

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when
edges are assigned independent random weights. Endow this tree with the graph distance …

Coupling is a powerful method in probability theory through which random variables can be
compared with each other. Coupling has been applied in a broad variety of contexts, eg to …

We consider simple random walk on the incipient infinite cluster for the spread-out model of
oriented percolation on Z^ d * Z _+. In dimensions d> 6, we obtain bounds on exit times …

The nature of fluid flow in hydrocarbon reservoirs is very complex because of the
complicated geology formed as a result of the intricate sedimentary processes, which …

We introduce a general model of trapping for random walks on graphs. We give the possible
scaling limits of these Randomly Trapped Random Walks on Z. These scaling limits include …

We study invasion percolation in two dimensions. We compare connectivity properties of the
origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the …

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of
the triangular lattice in the complex plane have unique scaling limits, which are invariant …

We study invasion percolation in two dimensions, focusing on properties of the outlets of the
invasion and their relation to critical percolation and to incipient infinite clusters (IICs). First …