Appears in collection : 2015 - T1 - Disordered systems, random spatial processes and some applications

The Brownian web is the collection of one-dimensional coalescing Brownian motions starting from every point in space-time. Originally conceived by Arratia in the context of the one-dimensional voter model and its dual coalescing random walks, the Brownian web has since been shown to arise in the scaling limit of many one-dimensional interacting particle systems with coalescent interaction, including zero-temperature dynamics of Ising and Potts models, true self-avoiding random walks, drainage networks, Hastings-Levitov planar aggregation models, and super-critical oriented percolation. The Brownian net is an extension of the Brownian web, which also allows for branching of the Brownian motions. It has been shown to arise in the scaling limit of many one-dimensional interacting particle systems with branching-coalescing interactions, including the voter model with selection, dynamics of Ising and Potts models with boundary nucleation, and one-dimensional random walks in i. i. d. space-time random environments. The goal of the lecture series is to introduce the Brownian web and the Brownian net, discuss some of their properties, and study how they arise in the scaling limits of various models of interest.