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Rayleigh processes, real trees, and root growth with re-grafting
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  • Published: 15 March 2005

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

  • Steven N. Evans1,
  • Jim Pitman1 &
  • Anita Winter2 

Probability Theory and Related Fields volume 134, pages 81–126 (2006)Cite this article

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  • 81 Citations

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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.

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Author information

Authors and Affiliations

  1. Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA, 94720-3860, USA

    Steven N. Evans & Jim Pitman

  2. Mathematisches Institut, Universität Erlangen–Nürnberg, Bismarckstraße 11/2, 91054, Erlangen, Germany

    Anita Winter

Authors
  1. Steven N. Evans
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  2. Jim Pitman
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  3. Anita Winter
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Corresponding author

Correspondence to Steven N. Evans.

Additional information

SNE supported in part by NSF grants DMS-0071468 and DMS-0405778, and a Miller Institute for Basic Research in Science research professorship

JP supported in part by NSF grants DMS-0071448 and DMS-0405779

AW supported by a DFG Forchungsstipendium

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Evans, S., Pitman, J. & Winter, A. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Relat. Fields 134, 81–126 (2006). https://doi.org/10.1007/s00440-004-0411-6

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  • Received: 02 February 2004

  • Revised: 28 October 2004

  • Published: 15 March 2005

  • Issue Date: January 2006

  • DOI: https://doi.org/10.1007/s00440-004-0411-6

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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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