Collision times of random walks and applications to the Brownian web

TL;DR

This paper introduces a new method using Lyapunov functions to verify a key convergence condition for the Brownian web, simplifying analysis for models with complex path interactions.

Contribution

It provides an alternative approach to verify the (B2) condition for convergence to the Brownian web using expected collision times and Lyapunov functions.

Findings

Expected collision times can be explicitly computed for independent random walks and Brownian motions.

The Lyapunov function method offers a simpler verification of the (B2) condition.

Applicable to models in the Brownian web's basin of attraction.

Abstract

Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length $t(>0)$, all starting within a segment of length $\varepsilon$, is of small order of $\varepsilon$. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first collision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([S67], [H71], [FLT04]).