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Volume growth and heat kernel estimates for the continuum random tree
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  • Published: 03 March 2007

Volume growth and heat kernel estimates for the continuum random tree

  • David A. Croydon1 

Probability Theory and Related Fields volume 140, pages 207–238 (2008)Cite this article

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Abstract

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.

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Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    David A. Croydon

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  1. David A. Croydon
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Correspondence to David A. Croydon.

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Croydon, D.A. Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Relat. Fields 140, 207–238 (2008). https://doi.org/10.1007/s00440-007-0063-4

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  • Received: 26 June 2006

  • Revised: 09 February 2007

  • Published: 03 March 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0063-4

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Keywords

  • Continuum random tree
  • Brownian excursion
  • Heat kernel estimates
  • Volume fluctuations

Mathematics Subject Classification (2000)

  • Primary: 60D05
  • Secondary: 60G57
  • Secondary: 60H25
  • Secondary: 60J35
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