Random fractal dendrites - Croydon, D; Croydon, David (2006)
﻿ Dendrites are tree-like topological spaces, and in this thesis, the physical characteristics of various random fractal versions of this type of set are investigated. This work will contribute to the development of analysis on fractals, an area which has grown considerably over the last twenty years. First, a collection of random self-similar dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have often relied on the scaling factors being bounded uniformly away from zero. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that this condition is not necessary; a simple condition on the tail of the distribution of the scaling factors at zero is all that is assumed. The scaling factors of these recursively defined structures form what is known as a multiplicative cascade, and results about the height of this random object are also obtained. With important physical and probabilistic applications, the heat equation has justifiably received a substantial amount of attention in a variety of settings. For certain types of fractals, it has become clear that a key factor in estimating the heat kernel is the volume growth with respect to the resistance metric on the space. In particular, uniform polynomial volume growth, which occurs for many deterministic self-similar fractals, immediately implies uniform (on-diagonal) heat kernel behaviour. However, in the random fractal setting, this is frequently not the case, and volume fluctuations are often observed. Motivated by this, an analysis of how volume fluctuations lead to corresponding heat kernel fluctuations for measure-metric spaces equipped with a resistance form is conducted here. These results apply to the aforementioned random self-similar dendrites, amongst other examples. The continuum random tree (CRT) of Aldous is an important random example of a measure-metric space, and fits naturally into the framework of the previous paragraph. In this thesis, quenched (almost-sure) volume growth asymptotics for the CRT are deduced, which show that the behaviour in almost-every realisation is not uniform. Applying the results introduced above, these yield heat kernel bounds for the CRT, demonstrating that heat kernel fluctuations occur almost-surely. Finally, a new representation of the CRT as a random self-similar dendrite is presented.