Convergence of trees with a given degree sequence and of their associated laminations

TL;DR

This paper investigates the convergence of uniform rooted plane trees with specified degree sequences to the Inhomogeneous Continuum Random Tree, and explores the related lamination processes, providing a unified framework for their asymptotic behavior.

Contribution

It establishes the convergence of trees with given degree sequences to the Inhomogeneous Continuum Random Tree and links this to lamination-valued processes, unifying discrete and continuous models.

Findings

Trees with given degree sequences converge to the Inhomogeneous Continuum Random Tree.

The lamination-valued processes derived from trees converge and are equivalent to Gromov-weak convergence.

A unified approach to the limit behavior of fragmentation processes on trees.

Abstract

In this paper, we study uniform rooted plane trees with given degree sequence. We show, under some natural hypotheses on the degree sequence, that these trees converge toward the so-called Inhomogeneous Continuum Random Tree after renormalisation. Our proof relies on the convergence of a modification of the well-known Lukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting-down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations that are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.