Convergence of Discrete Percolation Models to the Brownian Web Distance

Abstract

The Brownian web is a collection of coalescing Brownian motions started from every space-time point in R2. The Brownian web can be constructed as a scaling limit of coalescing one-dimensional simple random walks started at every point in a two-dimensional space-time lattice. Veto and Virag (2023) introduced a family of discrete random distance functions defined on these sequences of rescaled lattices. It was shown that, given the appropriate notion of convergence, these discrete distance functions converge to a function known as the Brownian web distance. We introduce a new method of argument that allows us to show that a broad class of discrete first passage percolation models also converge to the Brownian web distance. Unlike the arguments used in Veto and Virag (2023), our methods do not depend on the use of planar dual graphs. This allows our methods to be applied to models that allow random walks to cross one another before coalescing.