Markov properties for the vertical edge profile in random labelled trees

TL;DR

The paper develops a general Markov-structure framework for the vertical edge profile of random labelled trees with edge increments in $\{-1,0,1\}$, using an excursion-forest decomposition to reveal a time-homogeneous Markov chain for $(X_m^+,X_m^-)_{m\ge1}$ and its size-conditioned extension to $(X_m^+,X_m^-,M_m^-)_{m\ge1}$. It provides explicit kernels for the labelled binary-tree model and connects these probabilistic objects to combinatorial enumeration via the excursion decomposition, with broader implications for random maps. The work also situates these discrete results in the context of super-Brownian motion and the Integrated Super-Brownian Excursion (ISE), outlining scaling limits and a companion paper that identifies convergence to continuous local-time processes. By bridging probabilistic decomposition with combinatorial structure, the paper offers new tools for analysing vertical profiles in labelled trees and related random-geometric models, with potential to inform SDE descriptions and universality across scaling regimes.

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.

Figures (4)

• Figure 1: Example of the construction of the excursion forest for $m=2$. Consider a labelled plane tree rooted at $0$ (left), edges between labels $1$ and $2$ are in red in the figure. Disconnect each of these edges to get a forest of labelled rooted plane (sub) trees. The associated excursion forest at level $2$ (right) has one vertex per component of this forest that does not contain the root.
• Figure 2: Consider a positive binary tree excursion $\tau$ and highlight the vertices with label $1$ (left). Connect these vertices through the geodesics in the tree $\tau$, keeping the planar structure in $\tau$ (middle). The tree $\mathfrak T_\tau$ is the tree obtained this way (right). The vertices with a right child (resp. left child) in $\tau$ are marked with a white square (resp. a black circle) and are the elements of $\mathcal{R}(\tau)$ (resp. $\mathcal{L}(\tau)$).
• Figure 3: An element $\mathbf q$ of $\mathcal{Q}_{50}^\bullet$, the distinguished vertex is in blue and the root edge is marked by an arrow. The numbers in the circles correspond to the distance to the distinguished point. The ball of radius $3$ is obtained by only keeping edges between vertices of label $3$ or less. In the example, this creates $5$ external faces of respective degrees $2, 2, 4, 4$ and $22$. We have in this case $C_3(\mathbf q)=5$ and $P_3(\mathbf q)=34$.
• Figure 4: An example of a possible external face $\mathtt f$ (left) of perimeter $2p=14$ in the ball of radius $3$ (note that it may have pendant edges). Again, the numbers in the circles indicate the distance to the distinguished point $\mathbf x_\star$. The map $\hat{\mathbf{q}}_{\mathtt f}$ (middle) is obtained by only keeping edges connecting vertices at distance at least $3$ from $\mathbf x_\star$ that originally lie inside $\mathtt f$, it again has perimeter $14$. The number of internal faces is in this case $F=5$. The set $V_{\mathtt f}$ of vertices with label strictly greater than $k-d_\star$ is in green. The tree induced by the Schaeffer bijection (right, in green) connects all elements of $V_{\mathtt f}$.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2: Excursion Forest
• Definition 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• proof