We discuss several connections between discrete and continuous random trees. In the
discrete setting, we focus on Galton-Watson trees under various conditionings. In particular …

A measure-valued process describes the evolution of a population that evolves according to
the law of chance. In this chapter we provide some basic characterizations and constructions …

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 consider spread-out models of self-avoiding walk, bond percolation, lattice trees and
bond lattice animals on ${\mathbb {Z}^ d} $, having long finite-range connections, above …

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 …

In a previous paper we defined a version of the percolation triangle condition that is suitable
for the analysis of bond percolation on a finite connected transitive graph, and showed that …

We survey recent developments about random real trees, whose prototype is the Continuum
Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real …

The class of random-cluster models is a unification of a variety of stochastic processes of
significance for probability and statistical physics, including percolation, Ising, and Potts …