The Uncover Process for Random Labeled Trees

TL;DR

This paper analyzes the uncovering process of random labeled trees, revealing phase transitions and convergence behaviors of the number of edges and connected components as vertices are uncovered sequentially.

Contribution

It introduces a detailed probabilistic analysis of the uncovering process, including convergence to stochastic processes and phase transition phenomena in connected components.

Findings

Number of edges converges to a Brownian bridge-like process.

Identified phases and limiting distributions for the component of a fixed vertex.

Observed phase transition in the size of the largest connected component.

Abstract

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this work: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition.