Universality of the Brownian net

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite $(3+\epsilon )$-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing.