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Rayleigh processes, real trees, and root growth with re-grafting

Steven N. Evans, Jim Pitman, Anita Winter

TL;DR

This work extends the CRT/real-tree framework by introducing a root-growth with re-grafting Markov process on rooted compact ${\mathbb R}$-trees, with the Brownian CRT as its unique stationary distribution. The construction uses a projective structure and the ${\bf T}^{\mathrm{root}}$-valued state space equipped with the rooted Gromov–Hausdorff distance, enabling infinite-edge-length and infinite-branching limits via trimming. It connects to Aldous's CRT through a scaling limit of the Aldous–Broder algorithm and proves key properties: recurrence to stationarity, Feller semigroup, and convergence of finite-tree dynamics to the CRT. Additionally, the height process of a fixed point follows a Rayleigh process with explicit stationary behavior, enriching the probabilistic dynamics on real trees and their scaling limits.

Abstract

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.

Rayleigh processes, real trees, and root growth with re-grafting

TL;DR

This work extends the CRT/real-tree framework by introducing a root-growth with re-grafting Markov process on rooted compact -trees, with the Brownian CRT as its unique stationary distribution. The construction uses a projective structure and the -valued state space equipped with the rooted Gromov–Hausdorff distance, enabling infinite-edge-length and infinite-branching limits via trimming. It connects to Aldous's CRT through a scaling limit of the Aldous–Broder algorithm and proves key properties: recurrence to stationarity, Feller semigroup, and convergence of finite-tree dynamics to the CRT. Additionally, the height process of a fixed point follows a Rayleigh process with explicit stationary behavior, enriching the probabilistic dynamics on real trees and their scaling limits.

Abstract

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.

Paper Structure

This paper contains 15 sections, 11 theorems, 133 equations.

Table of Contents

  1. Motivation and background
  2. $\mathbb{R}$-trees
  3. Unrooted trees
  4. Unrooted trees with $4$ leaves
  5. Rooted $\mathbb{R}$-trees
  6. Length measure
  7. Rooted subtrees and trimming
  8. Root growth with re-grafting
  9. Beginning the construction
  10. Finishing the construction
  11. Connection with Aldous's construction of the CRT
  12. Recurrence and convergence to stationarity
  13. Feller property
  14. Asymptotics of the Aldous-Broder algorithm
  15. Rayleigh process

Key Result

Theorem 1

The metric space $({\bf T},d_{{\mathrm{GH}}})$ is complete and separable.

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • proof
  • proof
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • Theorem 3
  • ...and 20 more