Nic Freeman (University of Sheffield) – The Augmented Brownian Web

We consider coalescing random walks in 1+1 dimensional space-time, with a jump kernel that has finite moments up to order alpha. We view this system as a random set of coalescing paths, with one path starting from each point of space-time. When alpha>3, the diffusive scaling limit is known to be the Brownian web. We study the regime in which alpha is between 2 and 3. In this regime tightness fails (in the sense of continuous paths) due to erratic behaviour near the start times of some of the paths. We show that a surprising transition in behaviour occurs at alpha=9/4; when alpha>9/4 a diffusive scaling limit exists in which paths are essentially Brownian but some paths possess jumps at their initial times, whilst when alpha<9/4 tightness "truly" fails.