arXiv subjects

David A. Croydon

Publications and source records attributed to David A. Croydon.

40 records · Page 3Linked to original sources

arXiv2012

The Hausdorff dimension of a class of random self-similar fractal trees

In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical properties of an underlying metric space or the scaling factors being bounded uniformly away from 0. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that conditions such as these are not necessary. The scaling factors of the recursively defined structures in consideration form what is known as a multiplicative cascade, and results about the height of this random object are also obtained.

math.PR↗

arXiv2012

Scaling limit for the random walk on the largest connected component of the critical random graph

A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.

math.PR↗

arXiv2012

Biased random walk on critical Galton-Watson trees conditioned to survive

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.

math.PR↗

arXiv2009

Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

A Brownian spatial tree is defined to be a pair $(\mathcal{T},ϕ)$, where $\mathcal{T}$ is the rooted real tree naturally associated with a Brownian excursion and $ϕ$ is a random continuous function from $\mathcal{T}$ into $\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $ϕ$ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in $\mathbb{R}^d$ run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric $d_{\mathcal{S}}$ on the set $\mathcal{S}:=ϕ(\mathcal{T})$. Applications of this result include the recovery of the spatial tree $(\mathcal{T},ϕ)$ from the set $\mathcal{S}$ alone, which implies in turn that a Dawson--Watanabe super-process can be recovered from its range. Furthermore, $d_{\mathcal{S}}$ can be used to construct a Brownian motion on $\mathcal{S}$, which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained.

math.PR↗