Convergence of drainage networks to the Brownian web

Fontes et. al. (2004), obtained criteria to study weak convergence to the Brownian web. Among them condition $(B_2)$ is often verified by applying FKG type positive correlation inequality, together with a bound on the distribution of the coalescence time between two paths. However, FKG is a strong property that is difficult to verify for most models. Here we propose an alternate methodology to verify condition $(B_2)$ which depends mainly on the Markovian structure and does not use any correlation inequality. We also show that this new method can be applied for three discrete drainage network models in the basin of attraction of Brownian web to verify $(B_2)$.