Volume growth and heat kernel estimates for the continuum random tree
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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.
KeywordsContinuum 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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