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David A. Croydon

Publications and source records attributed to David A. Croydon.

At least 37 records · Page 2Linked to original sources
arXiv2021

Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

math.PR↗
arXiv2021

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

math.PR↗
arXiv2020

Discrete integrable systems and Pitman's transformation

We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice equations. It is explained how this connection enables the dynamics of the integrable systems to be initiated from infinite configurations, which is important in the study of invariant measures. In the special case of spatially independent and identically distributed configurations, progress on the latter topic is also reported.

math.PR↗
arXiv2020

An introduction to the trapping experienced by biased random walk on the trace of biased random walk

We introduce and summarise results from the recent paper 'Biased random walk on the trace of biased random walk on the trace of ...', which was written jointly with M. P. Holmes (University of Melbourne). We also present additional discussion on some of the conjectures made in that work. The content of this article is loosely based on the presentation given by the author at the Probability Symposium held at the Research Institute for Mathematical Sciences, Kyoto University in December 2018

math.PR↗
arXiv2020

The random conductance model with heavy tails on nested fractal graphs

Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.

math.PR↗
arXiv2020

The number of spanning clusters of the uniform spanning tree in three dimensions

Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).

math.PR↗
arXiv2019

Dynamics of the ultra-discrete Toda lattice via Pitman's transformation

By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between $-1$ and $1$, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in a parallel work. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than $\pm1$.

math-ph↗
arXiv2019

Duality between box-ball systems of finite box and/or carrier capacity

We construct the dynamics of the box-ball system with box capacity $J$ and carrier capacity $K$, which we abbreviate to BBS($J$,$K$), in the case of infinite initial configurations, and show that this system is dual to the analogous BBS($K$,$J$) model. Towards this end, we build on previous work for the original box-ball system, that is BBS($1$,$\infty$), to show that when the box capacity $J$ and carrier capacity $K$ satisfy $J<K$ the dynamics can be represented by a Pitman-type transformation. These ideas are applied in the case of random initial configurations to show that the distributional properties of spatial stationarity and invariance under the BBS dynamics are dual. Moreover, for independent and identically distributed configurations, we derive a characterisation of invariant measures in terms of a detailed balance equation, which captures the duality of the system locally; this is used to find all invariant measures in this class. Finally, we deduce the speed of a tagged particle, and show that this also satisfies a natural duality relation.

math.PR↗
arXiv2019

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the BBS based on two-sided stationary Markov chains. In this article, we survey the invariant measures that were presented in the latter work, and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures. Moreover, we show that the former examples can be obtained as infinite volume limits of the latter. Another aspect of the previous work was to describe scaling limits for the box-ball system; here, we review these results, and also present scaling limits other than those that were covered there. One, the zigzag process has previously been observed in the context of queuing; another, a periodic version of the zigzag process, is apparently novel. Furthermore, we demonstrate that certain Palm measures associated with the stationary and periodic versions of the zigzag process yield natural invariant measures for the dynamics of corresponding versions of the ultra-discrete Toda lattice.

math.PR↗
arXiv2018

Dynamics of the box-ball system with random initial conditions via Pitman's transformation

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al.\ recently showed the invariance in distribution of Bernoulli product measures with density strictly less than $\frac{1}{2}$, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by `reflection in the past maximum', which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible for all times. We give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter $p\uparrow\frac12$ (which may be considered the `high density' regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.

math.PR↗
arXiv2016

Central limit theorems for the spectra of classes of random fractals

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known.

math.PR↗
arXiv2015

Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

math.PR↗
arXiv2015

Moduli of continuity of local times of random walks on graphs in terms of the resistance metric

In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.

math.PR↗
arXiv2012

Scaling limits for simple random walks on random ordered graph trees

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as $n\rightarrow\infty$, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be rescaled to converge to the Brownian motion on a related $α$-stable tree.

math.PR↗
arXiv2012

Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive

We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index $α\in(1,2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $2α/(2α-1)$. Furthermore, we demonstrate that when $α\in(1,2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $α=2$. In the course of our arguments, we obtain tail bounds for the distribution of the $n$th generation size of a Galton-Watson branching process with offspring distribution $Z$ conditioned to survive, as well as tail bounds for the distribution of the total number of individuals born up to the $n$th generation, that are uniform in $n$.

math.PR↗
arXiv2012

Random walk on the range of random walk

We study the random walk $X$ on the range of a simple random walk on $\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched and annealed scaling limits for the process $X$, which show that the intersections of the original simple random walk path are essentially unimportant. For $d=4$ our results are less precise, but we are able to show that any scaling limit for $X$ will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when $d=4$ similar logarithmic corrections are necessary in describing the asymptotic behaviour of the return probability of $X$ to the origin.

math.PR↗
arXiv2012

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

In this article, we consider the problem of estimating the heat kernel on measure-metric spaces equipped with a resistance form. Such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set. In this situation, it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on-diagonal part of the heat kernel is completely determined by this rate of volume growth. However, recent results have shown that for certain random fractal sets, there are global and local (point-wise) fluctuations in the volume as $r\rightarrow 0$ and so these uniform results do not apply. Motivated by these examples, we present global and local on-diagonal heat kernel estimates when the volume growth is not uniform, and demonstrate that when the volume fluctuations are non-trivial, there will be nontrivial fluctuations of the same order (up to exponents) in the short-time heat kernel asymptotics. We also provide bounds for the off-diagonal part of the heat kernel. These results apply to deterministic and random self-similar fractals, and metric space dendrites (the topological analogues of graph trees).

math.PR↗