Collision Times of Random Walks and Applications to the Brownian Web

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), roughly states that the probability that the first collision time of three paths, crossing a small segment of length $\varepsilon$, bigger than $t (>0)$ is of small order of $\varepsilon$, i.e., $o(\varepsilon)$. Condition (B2) 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 paper, we show that for a non-crossing path model, with some homogeneity and Markovian properties, a suitable upper bound on expected first collision time for three paths can be obtained directly using Lyapunov functions and that 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.