Probability Theory and Related Fields

, Volume 140, Issue 1, pp 207–238

Volume growth and heat kernel estimates for the continuum random tree


DOI: 10.1007/s00440-007-0063-4

Cite this article as:
Croydon, D.A. Probab. Theory Relat. Fields (2008) 140: 207. doi:10.1007/s00440-007-0063-4


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.


Continuum random treeBrownian excursionHeat kernel estimatesVolume fluctuations

Mathematics Subject Classification (2000)

Primary: 60D05Secondary: 60G57Secondary: 60H25Secondary: 60J35

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WarwickCoventryUK