On the convergence of the Drainage Network Model with branching

Abstract

Systems of coalescing random walks have been object of extensive studies, with applications in many areas. The Drainage Network is a system of coalescing random walks that have dependence before coalescence, introduced by Gangopadhyay, Roy and Sarkar in 2004. Few years later, in 2009, Coletti, Fontes and Dias have proved the convergence of the Drainage Network under diffusive scaling to the Brownian Web. Besides that, Rongfeng Sun and Swart have introduced the Brownian Net in 2008, which arises as the limit under diffusive scaling of systems of coalescing random walks with branching. In this work we introduce the Drainage Network with branching, which is a system of coalescing random walks with paths that can branch and that exhibit some dependence before coalescence. The study of this model contributes to the understanding of the universality class related to the Brownian Web and Net. The main objective of this work is to study the convergence of the Drainage Network with branching, under diffusive scaling, to the Brownian Web or Net, according to specific conditions for the branching probability. We show that based on the specification of the branching probability, we can have convergence to the Brownian Web or we can have a tight family such that any weak limit point contains a Brownian Net. In the latter case, we conjecture that the limit is indeed the Brownian Net.