# Scaling limit of random forests with prescribed degree sequences

## TL;DR

This paper investigates the scaling limits of random plane forests with fixed degree sequences, demonstrating convergence to a continuum random tree in the Gromov-Hausdorff-Prokhorov topology, extending Aldous's framework.

## Contribution

It establishes the Gromov-Hausdorff-Prokhorov convergence of large random forests with prescribed degrees to a continuum limit, using excursions of first passage bridges.

## Key findings

- Convergence of random forests to Brownian Continuum Random Tree
- Identification of the limit as a sequence of real trees encoded by excursions
- Utilization of Lukasiewicz walks to study scaling limits

## Abstract

In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov-Hausdorff-Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov-Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree, pioneered by the work of Aldous. In fact we identify the limiting random object as a sequence of random real trees encoded by excursions of some first passage bridges reflected at minimum. We establish such convergence by studying the associated Lukasiewicz walk of the degree sequences. In particular, our work is closely related to and uses the results from the recent work of Broutin and Marckert on scaling limit of random trees with prescribed degree sequences, and the work of Addario-Berry on tail bounds of the height of a random tree with prescribed degree sequence.