A Drainage Network with Dependence and the Brownian Web

We study a system of coalescing random walks on the integer lattice $$ {\mathbb {Z}}^{d} in which walk is oriented d-th direction and follows certain specified rules. first geometry paths show that, almost surely, form graph consisting just one tree for dimensions d=2,3 infinitely many disjoint trees d\ge 4 . Also, there no bi-infinite path surely 2 Subsequently, we prove that d=2 diffusive scaling this converges distribution to Brownian web.