Multi-scale Webs of Coalescing Random Walks

Consider coalescing random walks on the one-dimensional integer lattice, started from every point in space and time, whose jump kernel has finite moments up to order alpha. Thanks to the work of Belhaouari, Mountford, Newman, Ravishankar, Sun, and Valle, we know that if alpha is greater than three, then the collection of all random walk paths converges in law with respect to a rather strong topology to an object called the Brownian web, which informally consists of coalescing Brownian motions, started from every point in space and time. I will report about work in progress with Nic Freeman where we aim to understand what happens in the regime where alpha lies between two and three. In this regime, a single random walk path converges to Brownian motion, but the collection of all random walk paths fails to converge to the Brownian web. Our main tool is a new multi-scale decomposition of discrete webs.