Small trees in supercritical random forests

TL;DR

This paper investigates the asymptotic behavior of small trees within supercritical random forests, revealing their limiting structure through connections to cyclic forests and lattice paths.

Contribution

It provides a novel description of the limit of small trees in supercritical random forests using combinatorial and probabilistic techniques.

Findings

Characterizes the limit of small trees in supercritical forests

Relates plane forests to cyclic forests and lattice paths

Provides a framework for understanding forest decomposition

Abstract

We study the scaling limit of random forest with prescribed degree sequence in the regime that the largest tree consists of all but a vanishing fraction of nodes. We give a description of the limit of the forest consisting of the small trees, by relating plane forest to marked cyclic forest and its corresponding lattice path.