In 2008, Tóth and Vető defined the self-repelling random walk with directed edges as a non-Markovian random walk on $\unicode{x2124}$: in this model, the probability that the walk moves from a point of $\unicode{x2124}$ to a given neighbor depends on the number of previous crossings of the directed edge from the initial point to the target, called the local time of the edge. Tóth and Vető found that this model exhibited very peculiar behavior, as the process formed by the local times of all the edges, evaluated at a stopping time of a certain type and suitably renormalized, converges to a deterministic process, instead of a random one as in similar models. In this work, we study the fluctuations of the local times process around its deterministic limit, about which nothing was previously known. We prove that these fluctuations converge in the Skorokhod $M_1$ topology, as well as in the uniform topology away from the discontinuities of the limit, but not in the most classical Skorokhod topology. We also prove the convergence of the fluctuations of the aforementioned stopping times.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

Networked dynamical systems, i.e., systems of dynamical units coupled via nontrivial interaction topologies, constitute models of broad classes of complex systems, ranging from gene regulatory and metabolic circuits in our cells to pandemics spreading across continents. Most of such systems are driven by irregular and distributed fluctuating input signals from the environment. Yet how networked dynamical systems collectively respond to such fluctuations depends on the location and type of driving signal, the interaction topology and several other factors and remains largely unknown to date. As a key example, modern electric power grids are undergoing a rapid and systematic transformation towards more sustainable systems, signified by high penetrations of renewable energy sources. These in turn introduce significant fluctuations in power input and thereby pose immediate challenges to the stable operation of power grid systems. How power grid systems dynamically respond to fluctuating power feed-in as well as other temporal changes is critical for ensuring a reliable operation of power grids yet not well understood. In this work, we systematically introduce a linear response theory (LRT) for fluctuation-driven networked dynamical systems. The derivations presented not only provide approximate analytical descriptions of the dynamical responses of networks, but more importantly, also allow to extract key qualitative features about spatio-temporally distributed response patterns. Specifically, we provide a general formulation of a LRT for perturbed networked dynamical systems, explicate how dynamic network response patterns arise from the solution of the linearised response dynamics, and emphasise the role of LRT in predicting and comprehending power grid responses on different temporal and spatial scales and to various types of disturbances. Understanding such patterns from a general, mathematical perspective enables to estimate network responses quickly and intuitively, and to develop guiding principles for, e.g., power grid operation, control and design.

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

We analyze the fluctuations in the X-ray flux of 20 AGN (mainly Seyfert 1 galaxies) monitored by RXTE and XMM-Newton with a sampling frequency ranging from hours to years, using structure function (SF) analysis. We derive SFs over four orders of magnitude in the time domain (0.03-300 days). Most objects show a characteristic time scale, where the SF flattens or changes slope. For 10 objects with published power-spectral density (PSD) the break time scales in the SF and PSD are similar and show a good correlation. We also find a significant correlation between the SF timescale and the mass of the central black hole, determined for most objects by reverberation mapping.

We analyze the Lattice Boltzmann method for the simulation of fluctuating hydrodynamics by Adhikari et al. [Europhys. Lett., 71 (2005), 473-479] and find that it shows excellent agreement with theory even for small wavelengths as long as a stationary system is considered. This is in contrast to other finite difference and older lattice Boltzmann implementations that show convergence only in the limit of large wavelengths. In particular cross correlators vanish to less than 0.5%. For larger mean velocities, however, Galilean invariance violations manifest themselves.

We perform a numerical study of the fluctuations of the rescaled hydrodynamic transversevelocity field during the cooling state of a homogeneous granular gas. We are interestedin the role of Molecular Chaos for the amplitude of the hydrodynamic noise and itsrelaxation in time. For this purpose we compare the results of Molecular Dynamics (MD,deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, randomprocess), where Molecular Chaos can be directly controlled. It is seen that the large timedecay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficientpredicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC).On the other side, the noise amplitude in Molecular Dynamics, which is known toviolate the equilibrium Fluctuation-Dissipation relation, is not alwaysaccurately reproduced in a DSMC scheme. The agreement between the two models improves ifthe probability of recollision (controlling Molecular Chaos) is reduced by increasing thenumber of virtual particles per cells in the DSMC. This result suggests that DSMC is notnecessarily more efficient than MD, if the real number of particles is small(~103 ± 104) and if one is interested in accurately reproducefluctuations. An open question remains about the small-times behavior of theautocorrelation function in the DSMC, which in MD and in kinetic theory predictions is nota straight exponential.

Consider a random environment in ${\mathbb Z}^d$ given by i.i.d. conductances.In this work, we obtain tail estimates for the fluctuations about themean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environmentsand the spectral gap of the random walk in a finite cube.

The charge on plasma ions may fluctuate due to random ionization and recombination processes. The resulting motion has a time dependent Hamiltonian and in simple cases, the ions steadily gain energy. Quantitative rates are estimated for some cases involving high Z plasmas.

This article is an edited transcript based on the David Turnbull Lecture given by Ellen D. Williams of the University of Maryland on December 2, 2003, at the Materials Research Society Fall Meeting in Boston.Williams received the award for “groundbreaking research on the atomic-scale science of surfaces and for leadership, writing, teaching, and outreach that convey her deep understanding of and enthusiasm for materials research.” This article focuses on the special properties of small structures that provide much of the exciting potential of nanotechnology.One aspect of small structures—their susceptibility to thermal fluctuations—may create or necessitate new ways of exploiting nanostructures.The advent of scanned probe imaging techniques created new opportunities for observing and understanding such structural fluctuations and the related evolution of nanostructure.Direct observations show that it is relatively easy for large numbers of atoms—the kinds of numbers that are present in nanoscale structures—to pick up and move about on the surface cooperatively with substantial impact on nano-to micron-scale structures.Such labile evolution of structure can be predicted quantitatively by using length-scale bridging techniques of statistical mechanics coupled with scanned probe observations of structural and temporal distributions.The same measurements also provide direct information about the stochastic paths of structural fluctuations that can be used outside of the traditional thermodynamic framework.Future work involves moving beyond the classical thermodynamic picture to assess the impact that the stochastic behavior has on the physical properties of individual nanostructures.

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.