# A version of the random directed forest and its convergence to the Brownian web

## TL;DR

This paper proves that a generalized non-Markovian random path system with crossing paths converges to the Brownian web, extending previous results on non-crossing systems.

## Contribution

It introduces a generalized non-Markovian Random Directed Forest allowing crossings and demonstrates its convergence to the Brownian web, broadening applicability of existing techniques.

## Key findings

- Convergence to the Brownian web established for crossing paths system.
- Extension of techniques from non-crossing to crossing path systems.
- Provides a new example of non-Markovian system converging to the Brownian web.

## Abstract

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