Convergence of Discrete Percolation Models to the Brownian Web Distance

TL;DR

This work establishes that a broad class of discrete directed first passage percolation models on nonplanar random walk webs converge to the Brownian web distance $D_{BR}$ under appropriate scaling, without relying on planar dual graphs. The authors introduce journeys and itineraries to approximate Brownian-web geodesics and develop a robust epigraph-convergence framework for $D_{RW}^{n}$ to $D_{BR}$, along with stronger convergence of journeys in the path space $(\Pi,d)$. The approach handles crossing paths before coalescence, extending prior results that relied on planar duality, and relies on a coupling between the discrete web and the Brownian web. The results provide a general, constructive method for proving scaling limits of discrete percolation on webs and highlight the role of forward-time boundary structures in understanding geodesic behavior. Overall, the paper advances the understanding of scaling limits for nonplanar percolation models and broadens the applicability of the Brownian web framework to more complex path interactions.

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.

Figures (10)

• Figure 1: The discrete paths on the left start five vertices apart from one another. In a discrete model which allows a particle to jump only to directly adjacent vertices, a particle cannot travel from the red curve to the blue curve using a single jump. However, since $5 \ll \sqrt{n}$, the distance between the start points of red and blue paths can disappear in the scaling limit. As such, in the scaling limit, the red and blue paths may start from the same space-time point, hence a particle can jump between the paths in the model $D_{BR}$.
• Figure 2: A discrete web defined on the even integer lattice shown in black, and the associated dual web, defined on the odd integer lattice, shown in blue. Figure taken from ESS
• Figure 3: A spatial interval $I$ is depicted in black. The backward-time distance zero boundary curves from I are depicted in blue, while the forward-time distance zero boundary curves are depicted in red. Any point $u$ between the blue curves always satisfies $D_{BR}(u, v) = 0$ for some $v \in I$. This is because at any such point $u$, there originates a Brownian motion path and that path cannot cross the blue boundary curves, lest it coalesce with them. However, for many (in fact, almost all) points $v$ between the red curves, there are no points $u \in I$ such that $D_{BR}(u, v) = 0$. This is because the paths originating along $I$ rapidly coalesce with one another, so they cover very little space. The same idea holds for higher distance boundaries.
• Figure 4:
• Figure 5: A depiction of the percolation model on two random walks. $\mathcal{K}$ is chosen to be the five vertices adjacent to and 'above' $(0,0)$: $\mathcal{K} = \{ (-1, 0), (1,0), (-1,1), (0,1), (1,1) \}$. Since $(1, 3) \in \mathcal{K}_{(0,3)}$, a jump can be made from $R(t)$ to $B(t)$ at the time $t=3$ for a cost of one. As a result, the first passage percolation distance from $(0,0) \to (9,6)$ is one. The first passage percolation distance from $(0,0) \to (0,6)$ is zero, as a walker can freely follow the upward path of $R(t)$. Note that we can only follow the upward path freely. As a result, the model is asymmetric in time; the first passage percolation distance from $(0,6) \to (0,0)$ would instead be infinity.

Theorems & Definitions (30)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof