The scaling limit of the root component in the Wired Minimal Spanning Forest of the Poisson Weighted Infinite Tree

TL;DR

This paper establishes a scaling limit for the root component of the Wired Minimal Spanning Forest on the Poisson-Weighted Infinite Tree, revealing how it converges to a structure formed by Brownian trees through specific gluing procedures.

Contribution

It introduces a novel scaling limit for the root component of the WMSF on the PWIT, connecting it to Brownian tree aggregation and chain constructions.

Findings

The root component converges to a limit described by Brownian trees.

The limit involves two types of gluing procedures: Brownian tree aggregation and chain construction.

Provides a new perspective on the structure of minimal spanning forests in infinite graphs.

Abstract

In this paper we prove a scaling limit result for the component of the root in the Wired Minimal Spanning Forest (WMSF) of the Poisson-Weighted Infinite Tree (PWIT), where the latter tree arises as the local weak limit of the Minimal Spanning Tree (MST) on the complete graph endowed with i.i.d. weights on its edges. The limiting object can be obtained by aggregating independent Brownian trees using two types of gluing procedures: one that we call the Brownian tree aggregation process and resembles the so-called stick-breaking construction of the Brownian tree; and another one that we call the chain construction, which simply corresponds to gluing a sequence of metric spaces along a line.