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