  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]).
