Probability Theory and Related Fields

, Volume 134, Issue 1, pp 81–126

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

Article

## 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→∞ 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→∞ 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.

Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields.

### Key words or phrases

Continuum random tree Brownian excursion Real tree Gromov-Hausdorff metric Hausdorff metric Aldous-Broder algorithm Piecewise-deterministic Markov process

### Mathematics Subject Classification (2000)

60B05 60J27 Secondary: 60J80 60B99

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