A Version of the Random Directed Forest and its Convergence to the Brownian Web

Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of systems Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, is proved for system coalescing paths—the directed forest—which are not Markovian. Paths forest do cross each other before coalescence. Here, we study variation where paths can prove web. This provides an example how techniques allowing crossings be applied non-Markovian systems.