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Convergence of directed forests, spanning on random subsets of lattices or onpoint processes, towards the Brownian web has made the subject of an abundantliterature, 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 threedistinct paths for a time interval of length $t(>0)$, all starting within asegment of length $\varepsilon$, is of small order of $\varepsilon$. Thiscondition is often verified by applying an FKG type correlation inequalitytogether with a coalescing time tail estimate for two paths. For many modelswhere paths have complex interactions, it is hard to establish FKG typeinequalities. In this article, we show that for a non-crossing path model, withcertain assumptions, a suitable upper bound on expected first collision timeamong three paths can be obtained directly using Lyapunov functions. This, inturn, provides an alternate verification of Condition (B2). We further showthat in case of independent simple symmetric one dimensional random walks or incase of independent Brownian motions, the expected value can be computedexplicitly. We apply this alternate method of verification of (B2) to severalmodels in the basin of attraction of the Brownian web studied earlier in theliterature ([S67], [H71], [FLT04]).

Collision times of random walks and applications to the Brownian web