Critical random forests

TL;DR

This paper investigates the scaling limits of the largest components in critical random forests, extending known results from Erdős-Rényi graphs to acyclic forests with new diffusion-based models.

Contribution

It introduces a novel scaling limit for critical random forests using a reflected diffusion process, generalizing Aldous's Brownian motion approach for Erdős-Rényi graphs.

Findings

Derived scaling limits for largest components in critical forests

Connected the limits to a reflected diffusion with space and time-dependent drift

Extended Aldous's Brownian motion framework to forest structures

Abstract

Let $F(N,m)$ denote a random forest on a set of $N$ vertices, chosen uniformly from all forests with $m$ edges. Let $F(N,p)$ denote the forest obtained by conditioning the Erdos-Renyi graph $G(N,p)$ to be acyclic. We describe scaling limits for the largest components of $F(N,p)$ and $F(N,m)$, in the critical window $p=N^{-1}+O(N^{-4/3})$ or $m=N/2+O(N^{2/3})$. Aldous described a scaling limit for the largest components of $G(N,p)$ within the critical window in terms of the excursion lengths of a reflected Brownian motion with time-dependent drift. Our scaling limit for critical random forests is of a similar nature, but now based on a reflected diffusion whose drift depends on space as well as on time.