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David A. Croydon

Publications and source records attributed to David A. Croydon.

At least 19 recordsLinked to original sources
arXiv2026

On the cover time of Brownian motion on the Brownian continuum random tree

Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion on the Brownian CRT (i.e.\ the time taken by the process in question to visit the entire state space) is equal to the infimum over the times at which the associated local times are strictly positive everywhere. The proof of this result depends on the recursive self-similarity of the Brownian CRT and a novel version of the first Ray-Knight theorem for trees, which is of independent interest. As a consequence, we obtain that the suitably-rescaled cover times of simple random walks on critical, finite variance Galton-Watson trees converge in distribution with respect to their annealed laws to the cover time of Brownian motion on the Brownian CRT. Other families of graphs that have the Brownian CRT as a scaling limit are also covered. Additionally, we partially confirm a 1991 conjecture of David Aldous regarding related cover-and-return times.

math.PR↗
arXiv2026

Collision properties of the four-dimensional random walk trace

We consider collisions of multiple random walks on the trace of a simple random walk on the four-dimensional integer lattice. For two independent walks (in continuous time), we apply a result of Noda to derive a scaling limit for the collision time process. For three independent walks (in discrete time), we demonstrate infinitely many triple collisions occur.

math.PR↗
arXiv2025

Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover

In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincaré inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal.

math.PR↗
arXiv2025

Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$

The $λ$-biased random walk on a binary tree of depth $n$ is the continuous-time Markov chain that has unit mean holding times and, when at a vertex other than the root or a leaf of the tree in question, has a probability of jumping to the parent vertex that is $λ$ times the probability of jumping to a particular child. (From the root, it chooses one of the two children with equal probability.) For this process, when $λ<1$, we derive an $n\rightarrow \infty$ scaling limit for the cover time, that is, the time taken to visit every vertex. The distributional limit is described in terms of a jump process on a Cantor set that can be seen as the asymptotic boundary of the tree. This conclusion complements previous results obtained when $λ\geq 1$.

math.PR↗
arXiv2024

Triple collisions on a comb graph

In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for $n\in \mathbb{Z}$, restricting the vertical segment of the graph located at horizontal coordinate $n$ to the interval $\{0,1,\dots,\log^α(|n|\vee 1)\}$. Specifically, we show the following phase transition: when $α\leq 1$, the three random walks collide infinitely many times almost-surely, whereas when $α>1$, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height $|n|^α$.

math.PR↗
arXiv2024

Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry-Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

math.PR↗
arXiv2024

Aging and sub-aging for one-dimensional random walks amongst random conductances

We consider random walks amongst random conductances in the cases where the conductances can be arbitrarily small, with a heavy-tailed distribution at 0, and where the conductances may or may not have a heavy-tailed distribution at infinity. We study the long time behaviour of these processes and prove aging statements. When the heavy tail is only at 0, we prove that aging can be observed for the maximum of the process, i.e. the same maximal value is attained repeatedly over long time-scales. When there are also heavy tails at infinity, we prove a classical aging result for the position of the walker, as well as a sub-aging result that occurs on a shorter time-scale.

math.PR↗
arXiv2023

Annealed transition density of simple random walk on a high-dimensional loop-erased random walk

We derive sub-Gaussian bounds for the annealed transition density of the simple random walk on a high-dimensional loop-erased random walk. The walk dimension that appears in these is the exponent governing the space-time scaling of the process with respect to the extrinsic Euclidean distance, which contrasts with the exponent given by the intrinsic graph distance that appears in the corresponding quenched bounds. We prove our main result using novel pointwise Gaussian estimates on the distribution of the high-dimensional loop-erased random walk.

math.PR↗
arXiv2022

Anomalous scaling regime for one-dimensional Mott variable-range hopping

We derive an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this 'blocking' mechanism with one of 'trapping'. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.

math.PR↗
arXiv2022

Extremal regime for one-dimensional Mott variable-range hopping

We study the asymptotic behaviour of a version of the one-dimensional Mott random walk in a regime that exhibits severe blocking. We establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, we give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it.

math.PR↗
arXiv2022

Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical 'Pitman's transformation' of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural 'carrier process' for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

nlin.SI↗
arXiv2022

Generalized hydrodynamic limit for the box-ball system

We deduce a generalized hydrodynamic limit for the box-ball system, which explains how the densities of solitons of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state-space analogue of the soliton decomposition of Ferrari, Nguyen, Rolla and Wang (cf. the original work of Takahashi and Satsuma), namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of 'effective distances', where the dynamics are linear. For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the 'effective speeds' of solitons locally.

math-ph↗
arXiv2022

On the stationary solutions of random polymer models and their zero-temperature limits

We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models (i.e. the inverse-gamma, gamma, inverse-beta and beta random polymers), highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an `independence preservation' property, these bijections characterise the associated stationary measures. We then derive similar descriptions for the corresponding zero-temperature maps, whereby each is written in terms of one of two bijections. One issue with this picture is that, unlike in the positive-temperature case, the change of variables required is degenerate in general, and so whilst the argument yields stationary solutions, it does not provide a complete characterisation of them. On the other hand, this degeneracy does allow us to explain the appearance of atoms in the stationary measures of certain zero-temperature models. We also derive from our viewpoint various links between random polymer models, some of which recover known results, some of which are novel, and some of which lead to further questions.

math.PR↗
arXiv2022

Biased random walk on supercritical percolation: Anomalous fluctuations in the ballistic regime

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. For this model, Fribergh and Hammond showed the existence of an exponent $γ$ such that: for $γ<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by $γ$; whereas for $γ>1$, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when $γ>2$. In this article, we explain how Fribergh and Hammond's percolation estimates further allow it to be established that for $γ\in(1,2)$ the fluctuations about the mean are of an anomalous polynomial order, with exponent given by $γ^{-1}$.

math.PR↗
arXiv2022

Spectral dimension of simple random walk on a long-range percolation cluster

Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e.\ for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters $s\in (d,2d)$. We further note that our approach is applicable to short-range models as well.

math.PR↗
arXiv2021

Detailed balance and invariant measures for discrete KdV- and Toda-type systems

In order to study the invariant measures of discrete KdV- and Toda-type systems, this article focusses on models, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. A condition for establishing ergodicity of the dynamics is also given. These results are applied to various examples of discrete integrable systems, namely the ultra-discrete and discrete KdV equations, for which it is shown that the relevant invariant measures are of exponential/geometric and generalized inverse Gaussian form, respectively, as well as the ultra-discrete and discrete Toda lattice equations, for which the relevant invariant measures are found to be of exponential/geometric and gamma form. Ergodicity is demonstrated in the case of the KdV-type models. Links between the invariant measures of the different systems are presented, as are connections with stochastic integrable models and iterated random functions. Furthermore, a number of conjectures concerning the characterization of standard distributions are posed.

math.PR↗
arXiv2021

Scaling limit for random walk on the range of random walk in four dimensions

We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk's invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.

math.PR↗