Abstract

The Brownian web was introduced by Arratia (1979) and Toth—Werner (1997), and corresponds informally to starting coalescing Brownian motions from every space-time point in 1+1 dimensions. Inspired by the work of Schramm and Smirnov on the scaling limit of critical percolation, we provide a new state space and topology for this process. In particular, this allows us to derive an invariance principle for coalescing random walks under an optimal second moment condition on the increments of the underlying random walk. This stands in contrast with a series of previous papers working under a different topology, where it was shown that existence of third moments was both necessary and sufficient for this convergence to hold. Our approach is sufficiently simple and general that we can handle substantially more complicated coalescing flows with little extra work. For instance similar results are obtained in the case of coalescing Brownian motions on recurrent fractals such as the Sierpinsky gasket. This is the first such result where the limiting paths do not enjoy the non-crossing property.

Joint work with Christophe Garban (ENS Lyon) and Arnab Sen (Minnesota).