We use the lace expansion to study the standard self-avoiding walk in the d-dimensional
hypercubic lattice, for d≧ 5. We prove that the number cn of n-step self-avoiding walks …

The self-avoiding walk is a classical model in statistical mechanics, probability theory and
mathematical physics. It is also a simple model of polymer entropy which is useful in …

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 …

A mathematical model for distribution of mass in d-dimensional space, based upon
randomly embedding random trees into space, is introduced and studied. The model is a …

These lectures describe the lace expansion and its role in proving mean-field critical
behaviour for self-avoiding walks, lattice trees and animals, and percolation, above their …

We prove that above eight dimensions the scaling limit of sufficiently spread-out lattice trees
is the variant of super-Brownian motion known as integrated super-Brownian excursion …

We establish an exact relation between self-avoiding branched polymers in D+ 2 continuum
dimensions and the hard-core continuum gas at negative activity in D dimensions. We …

We consider general classes of lattice clusters, including various kinds of animals and trees
on different lattices. We prove that if a given local configuration (“pattern”) of sites and bonds …

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 …