Volume growth and heat kernel estimates for the continuum random tree |
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Authors: | David A Croydon |
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Institution: | (1) Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK |
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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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Keywords: | Continuum random tree Brownian excursion Heat kernel estimates Volume fluctuations |
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