Convergence of Coalescing Nonsimple Random Walks to the Brownian Web

TL;DR

This paper establishes new convergence criteria for coalescing random walks with crossing paths to the Brownian Web, extending previous results to more complex models and analyzing their scaling limits.

Contribution

It introduces the first convergence proof for models with crossing paths to the Brownian Web, broadening the scope of models that can be analyzed.

Findings

Verified convergence for non-simple coalescing random walks with crossing paths

Developed new criteria applicable to models with crossing paths

Analyzed the scaling limit of voter model interfaces

Abstract

The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently, Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and general convergence criteria allowing either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. In this thesis, we formulate new convergence criteria for crossing paths, and verify them for non-simple coalescing random walks (both discrete and continuous time) satisfying a finite fifth moment condition. This is the first time convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.