arXiv subjects

Andrew Wade

Publications and source records attributed to Andrew Wade.

18 recordsLinked to original sources

arXiv2026

Dynamics of the leftmost particle in heterogeneous semi-infinite exclusion systems

We study the behaviour of the leftmost particle in a semi-infinite particle system on $\mathbb{Z}$, where each particle performs a continuous-time nearest-neighbour random walk, with particle-specific jump rates, subject to the exclusion interaction (i.e., no more than one particle per site). We give conditions, in terms of the jump rates on the system, under which the leftmost particle is recurrent or transient, and develop tools to study its rate of escape in the transient case, including by comparison with an $M/G/\infty$ queue. In particular we show examples in which the leftmost particle can be null recurrent, positive recurrent, ballistically transient, or subdiffusively transient. Finally we indicate the role of the initial condition in determining the dynamics, and show, for example, that sub-ballistic transience can occur started from close-packed initial configurations but not from stationary initial conditions.

math.PR↗

arXiv2025

Fluctuations for diameter and perimeter of convex hulls of multiple random walks

We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctuations are governed by the geometry of the polygon formed by the drift vectors. We develop an $L^2$-approximation framework, based on Wald-type maximal central limit theorems, which reduces the asymptotic analysis of the hull to a finite collection of endpoint, maximal-projection, and Brownian support-function terms. For the diameter, we obtain general max-type limit theorems, Gaussian in the case of a unique extremal diametrical pair and typically non-Gaussian when several extremal pairs compete. For the perimeter, we prove a general distributional limit: non-zero extremal drifts contribute maxima of Gaussian projections, while zero-drift extremal walks contribute Brownian support-function terms. The results recover the previously known Gaussian regimes (the case of one or two walks) and identify the non-Gaussian limits in the degenerate and boundary cases left open (even for two walks). We also give $L^2$ approximations of the convex hull by simpler random sets, under Hausdorff and $\ell_1$ metrics on compact convex sets. Our proofs work under the optimal finite second moment assumption.

math.PR↗

arXiv2025

One-dimensional particle clouds with elastic collisions

We study an interacting particle system of a finite number of labelled particles on the integer lattice, in which particles have intrinsic masses and left/right jump rates. If a particle is the minimal-label particle at its site when it tries to jump left, the jump is executed. If not, `momentum' is transferred to increase the rate of jumping left of the minimal-label particle. Similarly for jumps to the right. The collision rule is `elastic' in the sense that the net rate of flow of mass is independent of the present configuration, in contrast to the exclusion process, for example. We show that the particle masses and jump rates determine explicitly, via a concave majorant of a simple `potential' function associated to the masses and jump rates, a unique partition of the system into maximal stable subsystems. The internal configuration of each stable subsystem remains tight, while the location of each stable subsystem obeys a strong law of large numbers with an explicit speed. We indicate connections to adjacent models, including diffusions with rank-based coefficients.

math.PR↗

arXiv2025

Rates of convergence for extremal spacings in Kakutani's random interval-splitting process

Kakutani's random interval-splitting process iteratively divides, via a uniformly random splitting point, the largest sub-interval in a partition of the unit interval. The length of the longest sub-interval after $n$ steps, suitably centred and scaled, is known to satisfy a central limit theorem as $n \to \infty$. We provide a quantitative (Berry-Esseen) upper bound for the finite-$n$ approximation in the central limit theorem, with conjecturally optimal rates in $n$. We also prove convergence to an exponential distribution for the length of the smallest sub-interval, with quantitative bounds. The Kakutani process can be embedded in certain branching and fragmentation processes, and we translate our results into that context also. Our proof uses conditioning on an intermediate time, a conditional independence structure for statistics involving small sub-intervals, an Hermite-Edgeworth expansion, and moments estimates with quantitative error bounds.

math.PR↗

arXiv2025

Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion, and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. We also obtain an exact expression for the expected perimeter length after an independent exponential random time.

math.PR↗

arXiv2024

Central limit theorem for superdiffusive reflected Brownian motion

We study the second-order asymptotics around the superdiffusive strong law~\cite{MMW} of a multidimensional driftless diffusion with oblique reflection from the boundary in a generalised parabolic domain. In the unbounded direction we prove the limit is Gaussian with the usual diffusive scaling, while in the appropriately scaled cross-sectional slice we establish convergence to the invariant law of a reflecting diffusion in a unit ball. Using the separation of time scales, we also show asymptotic independence between these two components. The parameters of the limit laws are explicit in the growth rate of the boundary and the asymptotic diffusion matrix and reflection vector field. A phase transition occurs when the domain becomes too narrow, in which case we prove that the central limit theorem for the unbounded component fails.

math.PR↗

arXiv2024

Long-range one-dimensional internal diffusion-limited aggregation

We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blach\`ere, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $\alpha$-stable law, $1 < \alpha <2$, we prove that the cluster contains a contiguous block of $\delta m +o(m)$ sites, where $0 < \delta < 1$, but, unlike the finite-variance case, one may not take $\delta=1$.

math.PR↗

arXiv2024

Experimental demonstration of the combined arm- and cavity-locking system for LISA

Laser frequency noise suppression is a critical requirement for the Laser Interferometer Space Antenna (LISA) mission to detect gravitational waves. The baseline laser stabilization is achieved using cavity pre-stabilization and a post-processing technique called Time-Delay-Interferometry (TDI). To enhance the margins for TDI, alternate laser locking schemes should be investigated. A novel stabilisation blending the excellent stability of the arm with the existing cavity reference has been shown theoretically to meet the first-generation TDI margins. This locking system was designed to be implemented as a firmware change and have minimal or no changes to the LISA hardware. This paper experimentally verifies the hybrid laser locking technique by utilizing two references - an optical cavity, and an interferometer with delay imparted using 10 km of optical fiber. The results indicate the viability of the combination of arm-cavity locking system for LISA. They show the key benefits envisioned by this technique; suppression of the cavity fluctuations by the arm sensor (by 21 dB in this demonstration) and reduction of Doppler pulling of the laser frequency, a key technical challenge for arm locking.

gr-qc↗

arXiv2024

Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates

We study semi-infinite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. Under appropriate hypotheses on the jump rates (uniformly bounded rates is sufficient) and started from an initial condition that is a finite perturbation of the close-packed configuration, we give conditions under which the particles evolve as a single, semi-infinite "stable cloud". More precisely, we show that inter-particle separations converge to a product-geometric stationary distribution, and that the location of every particle obeys a strong law of large numbers with the same characteristic speed.

math.PR↗

arXiv2024

Superdiffusive limits for stochastic kinetics driven by self-similar drifts

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous self-similar noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove strong and weak convergence results on the corresponding scale. Our framework admits self-similar noise that is either a Bessel process, or, more generally, a self-similar continuous-state branching process with immigration, as well as more general processes satisfying certain asymptotic conditions.

math.PR↗

arXiv2024

Superdiffusive planar random walks with polynomial space-time drifts

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent $3/4$. The self-interacting process originated in discussions with Francis Comets.

math.PR↗

arXiv2023

Iterated-logarithm laws for convex hulls of random walks with drift

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.

math.PR↗

arXiv2023

Passage-times for partially-homogeneous reflected random walks on the quadrant

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peign\'e and Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.

math.PR↗

arXiv2023

Energy-constrained random walk with boundary replenishment

We study an energy-constrained random walker on a length-$N$ interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of~$M$ on each boundary visit. We establish large $N, M$ distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. Three phases are exhibited. When $M \ll N^2$ (energy is scarce), we show that there is an $M$-scale limit distribution related to a Darling-Mandelbrot law, while when $M \gg N^2$ (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where $M / N^2 \to \rho \in (0,\infty)$, we show that there is an $M$-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.

math.PR↗

arXiv2023

Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability

We quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field. Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applicable to general continuous semimartingales. Narrowing domains fall outside of the standard literature, in part because boundary local time can accumulate arbitrarily rapidly. Establishing Feller continuity (essential for characterizing stability) thus requires an extension of the usual approach. Our recurrence/transience classification extends previous work on strictly normal reflections, and expands the range of phenomena observed across all dimensions. For all recurrent cases, we provide quantitative information through upper and lower bounds on tails of return times to compact sets (see https://www.youtube.com/watch?v=oDDDWdbPx74 for a short YouTube video describing the results).

math.PR↗

arXiv2022

Supercell low-level mesocyclones: Origins of inflow and vorticity

The development of low-level mesocyclones in supercell thunderstorms has often been explained via the development of storm-generated streamwise vorticity along a baroclinic gradient in the forward flank of supercells. However, the ambient streamwise vorticity of the environment (often quantified via storm-relative helicity), especially near the ground, is particularly skillful at discriminating between nontornadic and tornadic supercells. This study investigates whether the origins of the inflow air into supercell low-level mesocyclones, both horizontally and vertically, can help explain the dynamical role of environmental versus storm-generated vorticity in the development of low-level mesocyclone rotation. Simulations of supercells, initialized with wind profiles common to supercell environments observed in nature, show that the air bound for the low-level mesocyclone primarily originates from the undisturbed, ambient environment, rather than from along the forward flank, and from very close to the ground, often in the lowest 200 - 400 m of the atmosphere. Given that the near-ground environmental air comprises the bulk of the inflow into low-level mesocyclones, this likely explains the forecast skill of environmental streamwise vorticity in the lowest few hundred meters of the atmosphere. The low-level mesocyclone does not appear to require much augmentation from the development of additional horizontal vorticity in the forward flank. Instead, the dominant contributor to vertical vorticity within the low-level mesocyclone is from the environmental horizontal vorticity. This study hopefully clarifies the development of low-level mesocyclones in supercells.

physics.ao-ph↗

arXiv2022

Dynamics of finite inhomogeneous particle systems with exclusion interaction

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm using only elementary arithmetic. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and the location of each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we provide a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on partially-stable Jackson queueing networks.

math.PR↗

arXiv2020

A Cryogenic Silicon Interferometer for Gravitational-wave Detection

The detection of gravitational waves from compact binary mergers by LIGO has opened the era of gravitational wave astronomy, revealing a previously hidden side of the cosmos. To maximize the reach of the existing LIGO observatory facilities, we have designed a new instrument that will have 5 times the range of Advanced LIGO, or greater than 100 times the event rate. Observations with this new instrument will make possible dramatic steps toward understanding the physics of the nearby universe, as well as observing the universe out to cosmological distances by the detection of binary black hole coalescences. This article presents the instrument design and a quantitative analysis of the anticipated noise floor.

astro-ph.IM↗