Volume growth and heat kernel estimates for the continuum random tree

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 almostsurely 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.

Scaling limits for simple random walks on random ordered graph trees

“…By comparison with results appearing in [10] for random walks on infinite variance Galton-Watson trees conditioned to survive, we expect that the polynomial term in the following lemma is the best possible. For an analogous estimate in the Brownian case, see [8].…”

“…Moreover, that X T ,µ admits jointly measurable local times (L t (σ )) σ ∈T , t≥0 can be checked as in the proof of [8,Lemma 8.2]. In the arguments of subsequent sections we will require further that (L t (σ )) σ ∈T , t≥0 is jointly continuous in t and σ , and we will demonstrate that this is the case whenever µ satisfies a polynomial lower bound of the form of (1.1).…”

“…Conditional on the event where the jump chain J n,k hits σ before σ occurs, we have, by a simple calculation, 8) where N i is the number of visits by J n,k to σ between the ith and (i + 1)th visits to σ , and m≥1 , where F m is the σ -algebra generated by J n,k up to the (m + 1)th hitting time of σ , and…”

Heat kernel fluctuations for a resistance form with non-uniform volume growth

“…Possessing a natural shortest path metric, it fits naturally into the resistance form framework, and so the problem of establishing good heat kernel bounds reduces to that of finding good measure bounds for the set. This is the aim of [9], in which logarithmic global measure fluctuations about a leading order r 2 term are demonstrated. Recall, for a dendrite the resistance metric is actual identical to the original one if this is a shortest path metric, and so the conclusions drawn there may be taken to be for resistance balls.…”

“…Although the assumptions that make a space a dendrite are restrictive, there are many important examples, including the continuum random tree of Aldous, see [1]. This is a random dendrite that arises naturally as the scaling limit of various families of random graph trees, and demonstrates measure fluctuations of the kind considered here, [9]. For further discussion, see Section 9.…”