Markov properties for the vertical edge profile in random labelled trees

TL;DR

This paper demonstrates that the vertical edge profile in a broad class of random labelled trees forms a Markov chain, providing explicit transition kernels and connecting to super-Brownian motion theory.

Contribution

It introduces the vertical edge profile as a Markovian structure in labelled trees and derives explicit transition kernels for incomplete binary trees.

Findings

Vertical edge profile forms a time-homogeneous Markov chain.

Enrichment with total mass yields Markovian structure conditioned on tree size.

Explicit transition kernels are provided for labelled incomplete binary trees.

Abstract

We study a broad class of random labelled trees in which integer-valued labels evolve along the edges according to increments in $\{-1, 0, 1\}$. These models include e.g. branching random walks, embedded complete and incomplete binary trees, random Cayley and plane trees with uniform displacements along edges. Motivated by recent work suggesting a Markovian structure in the vertical profile of such trees, we introduce the vertical edge profile, which counts both oriented edges connecting label $k-1$ to label $k$ and oriented edges connecting label $k$ to label $k-1$. We show that the vertical edge profile forms a time-homogeneous Markov chain for a wide class of models, and this remains true (provided we enrich this process by the total mass of the tree below each label) if we condition on the total size of the tree. We give explicit transition kernels in the case of labelled incomplete binary trees, which are closely related to known enumeration formulas. To establish these results, we study a decomposition of labelled trees into excursions above and below fixed label levels, yielding a forest structure with tractable probabilistic laws. We further explain briefly how these findings connect to the theory of super-Brownian motion and the Integrated Super-Brownian Excursion (ISE). In a companion paper, we show that the vertical edge profile converges, after rescaling,to the local time and its derivative of Brownian motion indexed by the Brownian tree.