Spatial persistence of fluctuating interfaces

Geometrical optics of first-passage functionals of random acceleration

Random acceleration is a fundamental stochastic process encountered in many applications. In the one-dimensional version of the process a particle is randomly accelerated according to the Langevin equation x[over ̈](t)=sqrt[2D]ξ(t), where x(t) is the particle's coordinate, ξ(t) is Gaussian white noise with zero mean, and D is the particle velocity diffusion constant. Here, we evaluate the A→0 tail of the distribution P_{n}(A|L) of the functional I[x(t)]=∫_{0}^{T}x^{n}(t)dt=A, where T is the first-passage time of the particle from a specified point x=L to the origin, and n≥0. We employ the optimal fluctuation method akin to geometrical optics. Its crucial element is determination of the optimal path-the most probable realization of the random acceleration process x(t), conditioned on specified A, n, and L. The optimal path dominates the A→0 tail of P_{n}(A|L). We show that this tail has a universal essential singularity, P_{n}(A→0|L)∼exp(-α_{n}L^{3n+2}/DA^{3}), where α_{n} is an n-dependent number which we calculate analytically for n=0, 1, and 2 and numerically for other n. For n=0 our result agrees with the asymptotic of the previously found first-passage time distribution.

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Persistence, defined as the probability that a signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. Often, persistence decays algebraically with time with non trivial exponents. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non-perturbative determination of persistence exponents of Gaussian non-Markovian processes with non stationary dynamics relaxing to a steady state after an initial perturbation. Two situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.

Position distribution in a generalized run-and-tumble process

We study a class of stochastic processes of the type d^{n}x/dt^{n}=v_{0}σ(t) where n>0 is a positive integer and σ(t)=±1 represents an active telegraphic noise that flips from one state to the other with a constant rate γ. For n=1, it reduces to the standard run-and-tumble process for active particles in one dimension. This process can be analytically continued to any n>0, including noninteger values. We compute exactly the mean-squared displacement at time t for all n>0 and show that at late times while it grows as ∼t^{2n-1} for n>1/2, it approaches a constant for n<1/2. In the marginal case n=1/2, it grows very slowly with time as ∼lnt. Thus, the process undergoes a localization transition at n=1/2. We also show that the position distribution p_{n}(x,t) remains time-dependent even at late times for n≥1/2, but approaches a stationary time-independent form for n<1/2. The tails of the position distribution at late times exhibit a large deviation form, p_{n}(x,t)∼exp[-γtΦ_{n}(x/x^{*}(t))], where x^{*}(t)=v_{0}t^{n}/Γ(n+1). We compute the rate function Φ_{n}(z) analytically for all n>0 and also numerically using importance sampling methods, finding excellent agreement between them. For three special values n=1, n=2, and n=1/2 we compute the exact cumulant-generating function of the position distribution at all times t.

Lacunarity of the zero crossings of Gaussian processes

A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.

Statistics of zero crossings in rough interfaces with fractional elasticity

We study numerically the distribution of zero crossings in one-dimensional elastic interfaces described by an overdamped Langevin dynamics with periodic boundary conditions. We model the elastic forces with a Riesz-Feller fractional Laplacian of order z=1+2ζ, such that the interfaces spontaneously relax, with a dynamical exponent z, to a self-affine geometry with roughness exponent ζ. By continuously increasing from ζ=-1/2 (macroscopically flat interface described by independent Ornstein-Uhlenbeck processes [Phys. Rev. 36, 823 (1930)PHRVAO0031-899X10.1103/PhysRev.36.823]) to ζ=3/2 (super-rough Mullins-Herring interface), three different regimes are identified: (I) -1/2<ζ<0, (II) 0<ζ<1, and (III) 1<ζ<3/2. Starting from a flat initial condition, the mean number of zeros of the discretized interface (I) decays exponentially in time and reaches an extensive value in the system size, or decays as a power-law towards (II) a subextensive or (III) an intensive value. In the steady state, the distribution of intervals between zeros changes from an exponential decay in (I) to a power-law decay P(ℓ)∼ℓ^{-γ} in (II) and (III). While in (II) γ=1-θ with θ=1-ζ the steady-state persistence exponent, in (III) we obtain γ=3-2ζ, different from the exponent γ=1 expected from the prediction θ=0 for infinite super-rough interfaces with ζ>1. The effect on P(ℓ) of short-scale smoothening is also analyzed numerically and analytically. A tight relation between the mean interval, the mean width of the interface, and the density of zeros is also reported. The results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints, along with discretization effects, are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis.

Records in fractal stochastic processes

The record statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions, the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes; otherwise, it has no impact on the record statistics.

Coarsening and persistence in a one-dimensional system of orienting arrowheads: Domain-wall kinetics with A+B→0

We demonstrate the large-scale effects of the interplay between shape and hard-core interactions in a system with left- and right-pointing arrowheads <> on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain walls, >< (A) and <> (B). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In this high-density limit, the A domain walls diffuse, while the B walls are static. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain-wall annihilation A+B→0, quite different from the A+A→0 kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially with time, in contrast to the power-law decay known for A+A→0. In the thermodynamic limit with a finite density of walls, coarsening as a function of time t is studied by simulation. While the number of walls falls as t^{-1/2}, the fraction of persistent arrowheads decays as t^{-θ} where θ is close to 1/4, quite different from the Ising value. The global persistence too has θ=1/4, as follows from a heuristic argument. In a generalization where the B walls diffuse slowly, θ varies continuously, increasing with increasing diffusion constant.

Distribution of zeros in the rough geometry of fluctuating interfaces

We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce nontrivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium nonstationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.

Numerical approach to unbiased and driven generalized elastic model

From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent β characterizing the growth of the mean squared displacement 〈(δh)(2)〉 of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments 〈∣δh∣(q)〉 with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe.

Order statistics of 1/fα signals

Order statistics of periodic, Gaussian noise with 1/f(α) power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d(k) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), known for independent, identically distributed variables, remains valid for 0 ≤ α < 1. Nontrivial, α-dependent scaling exponents, d(k) k((α-3)/2), emerge for 1 < α < 5, and, finally, α-independent scaling, d(k) ~ k, is obtained for α > 5. The spectra of average ordered values ε(k) =(x(1) - x(k))~ k(β) is also examined. The exponent β is derived from the gap scaling as well as by relating ε(k) to the density of near-extreme states. Known results for the density of near-extreme states combined with scaling suggest that β(α = 2) = 1/2, β(4) = 3/2, and β(∞) = 2 are exact values. We also show that parallels can be drawn between ε(k) and the quantum mechanical spectra of a particle in power-law potentials.

Maximum relative height of elastic interfaces in random media

The distribution of the maximal relative height (MRH) of self-affine one-dimensional elastic interfaces in a random potential is studied. We analyze the ground-state configuration at zero driving force, and the critical configuration exactly at the depinning threshold, both for the random-manifold and random-periodic universality classes. These configurations are sampled by exact numerical methods, and their MRH distributions are compared with those with the same roughness exponent and boundary conditions, but produced by independent Fourier modes with normally distributed amplitudes. Using Pickands' theorem we derive an exact analytical description for the right tail of the latter. After properly rescaling the MRH distributions we find that corrections from the Gaussian independent modes approximation are, in general, small, as previously found for the average width distribution of depinning configurations. In the large size limit all corrections are finite except for the ground state in the random-periodic class whose MRH distribution becomes, for periodic boundary conditions, indistinguishable from the Airy distribution. We find that the MRH distributions are, in general, sensitive to changes of boundary conditions.

Perturbation theory for fractional Brownian motion in presence of absorbing boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations (x(t(1))x(t(2)))=D(t(1)(2H)+t(2)(2H)-|t(1)-t(2)|(2H)), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(+)(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(+)(x,t)~t(-H)R(+)(x/t(H)). Our objective is to compute the scaling function R(+)(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H=1/2+ε, to calculate the scaling function R(+)(y) to first order in ε. We find that R(+)(y) behaves as R(+)(y)~y(ϕ) as y→0 (near the absorbing boundary), while R(+)(y)~y(γ)exp(-y(2)/2) as y→∞, with ϕ=1-4ε+O(ε(2)) and γ=1-2ε+O(ε(2)). Our ε-expansion result confirms the scaling relation ϕ=(1-H)/H proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for H=2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

Correlations in a generalized elastic model: fractional Langevin equation approach

The generalized elastic model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, and growing interfaces. On the other hand a probe (tracer) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a fractional Langevin equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree with the corresponding quantities calculated from the GEM. Furthermore we show that the Fox H -function formalism appears to be very convenient to describe the correlation properties within the FLE approach.

Generalized elastic model yields a fractional Langevin equation description

Starting from a generalized elastic model which accounts for the stochastic motion of several physical systems such as membranes, (semi)flexible polymers, and fluctuating interfaces among others, we derive the fractional Langevin equation (FLE) for a probe particle in such systems, in the case of thermal initial conditions. We show that this FLE is the only one fulfilling the fluctuation-dissipation relation within a new family of fractional Brownian motion equations. The FLE for the time-dependent fluctuations of the donor-acceptor distance in a protein is shown to be recovered. When the system starts from nonthermal conditions, the corresponding FLE, which does not fulfill the fluctuation-dissipation relation, is derived.

Hitting probability for anomalous diffusion processes

We present the universal features of the hitting probability Q(x,L), the probability that a generic stochastic process starting at x and evolving in a box [0, L] hits the upper boundary L before hitting the lower boundary at 0. For a generic self-affine process, we show that Q(x,L)=Q(z=x/L) has a scaling Q(z) approximately z;{phi} as z-->0, where phi=theta/H, H, and theta being the Hurst and persistence exponent of the process, respectively. This result is verified in several exact calculations, including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical support for our analytical results.

Crossing intervals of non-Markovian Gaussian processes

We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.

Extreme statistics for time series: distribution of the maximum relative to the initial value

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Fractional Laplacian in bounded domains

The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.

Maximal height statistics for 1/f(alpha) signals

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.

Spatial first-passage statistics of Al/Si(111)-(square root 3 x square root 3) step fluctuations

Spatial step edge fluctuations on a multicomponent surface of Al/Si(111)-(square root 3 x square root 3) were measured via scanning tunneling microscopy over a temperature range of 720-1070 K, for step lengths of L=65-160 nm. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature range, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be theta=0.498+/-0.062 in agreement with the theoretical prediction theta=1/2. The survival probability is found to scale directly with y/L, where y is the distance along the step edge. The form of the survival probabilities agrees quantitatively with the theoretical prediction, which yields exponential decay in the limit of small y/L. The decay constant is found experimentally to be y(s)/L=0.076+/-0.033 for y/L<or=0.2.

Probability distribution of the maximum of a smooth temporal signal

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated with the probability that the process remains below a nonzero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=X(0)X(t).

Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models

We study the probability density function P(h(m), L) of the maximum relative height h(m) in a wide class of one-dimensional solid-on-solid models of finite size L. For all these lattice models, in the large-L limit, a central limit argument shows that, for periodic boundary conditions, P(h(m), L) takes a universal scaling form P(h(m), L) approximately radical(12w(L))(-1) f(h(m)radical(12w(L))(-1), with w(L) the width of the fluctuating interface f(x) and the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using the transfer matrix technique, valid for any L > 0. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a nonuniversal amplitude, and simply given by the derivative of the Airy distribution function f'(x).

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.

Persistence of randomly coupled fluctuating interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2) , respectively, each growing over a d -dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2) , however, is coupled to h(1) via a quenched random velocity field. In the limit d-->0 , our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0) -->infinity , the stochastic process h(2) , at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H2 =1- beta(1) /2 , where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(2)(s) =1- H2 = beta(1) /2 . These analytical results are verified by numerical simulations.

Mapping spatial persistent large deviations of nonequilibrium surface growth processes onto the temporal persistent large deviations of stochastic random walk processes

Spatial persistent large deviations probability of surface growth processes governed by the Edwards-Wilkinson dynamics, Px(x,s), with -1< or =s< or =1 is mapped isomorphically onto the temporal persistent large deviations probability Pt(t,s) associated with the stochastic Markovian random walk problem. We show using numerical simulations that the infinite family of spatial persistent large deviations exponents thetax(s) characterizing the power-law decay of Px(x,s ) agrees, as predicted on theoretical grounds by Phys. Rev. Lett. 86, 3700 (2001)], with the numerical measurements of thetat(s), the continuous family of exponents characterizing the long-time power law behavior of Pt(t,s). We also discuss the simulations of the spatial persistence probability corresponding to a discrete model in the Mullins-Herring universality class, where our discrete simulations do not agree well with the theoretical predictions perhaps because of the severe finite-size corrections which are known to strongly inhibit the manifestation of the asymptotic continuum behavior in discrete models involving large values of the dynamical exponent and the associated extremely slow convergence to the asymptotic regime.

Exact maximal height distribution of fluctuating interfaces

We present an exact solution for the distribution P(h(m),L) of the maximal height h(m) (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h(m),L)=L(-1/2)f(h(m)L(-1/2)) for all L>0, where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds, but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.

Spatial persistence and survival probabilities for fluctuating interfaces

We report the results of numerical investigations of the steady-state (SS) and finite-initial-conditions (FIC) spatial persistence and survival probabilities for (1+1) -dimensional interfaces with dynamics governed by the nonlinear Kardar-Parisi-Zhang equation and the linear Edwards-Wilkinson (EW) equation with both white (uncorrelated) and colored (spatially correlated) noise. We study the effects of a finite sampling distance on the measured spatial persistence probability and show that both SS and FIC persistence probabilities exhibit simple scaling behavior as a function of the system size and the sampling distance. Analytical expressions for the exponents associated with the power-law decay of SS and FIC spatial persistence probabilities of the EW equation with power-law correlated noise are established and numerically verified.

Temporal and spatial persistence of combustion fronts in paper

The spatial and temporal persistence, or first-return distributions are measured for slow-combustion fronts in paper. The stationary temporal and (perhaps less convincingly) spatial persistence exponents agree with the predictions based on the front dynamics, which asymptotically belongs to the Kardar-Parisi-Zhang universality class. The stationary short-range and the transient behavior of the fronts are non-Markovian, and the observed persistence properties thus do not agree with the predictions based on Markovian theory. This deviation is a consequence of additional time and length scales, related to the crossovers to the asymptotic coarse-grained behavior.

Experimental persistence probability for fluctuating steps

The persistence behavior for fluctuating steps on the Si(111)-(sqrt[3]xsqrt[3])R30 degrees -Al surface was determined by analyzing time-dependent STM images for temperatures between 770 and 970 K. Using the standard persistence definition, the measured persistence probability displays power-law decay with an exponent of theta=0.77+/-0.03. This is consistent with the value of theta=3/4 predicted for attachment-detachment limited step kinetics. If the persistence analysis is carried out in terms of return to a fixed-reference position, the measured probability decays exponentially. Numerical studies of the Langevin equation used to model step motion corroborate the experimental observations.

Persistence in cluster-cluster aggregation

Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For gamma>0 the universal persistences decay algebraically with the exponent 2/(2-gamma). For the empty site case the exponent remains the same for gamma<0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.

Persistence in a stationary time series

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times. We then construct a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. We show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice.

Fluctuation-dominated phase ordering driven by stochastically evolving surfaces: depth models and sliding particles

We study an unconventional phase ordering phenomenon in coarse-grained depth models of the hill-valley profile of fluctuating surfaces with zero overall tilt, and for hard-core particles sliding on such surfaces under gravity. We find that several such systems approach an ordered state with large scale fluctuations which make them qualitatively different from conventional phase ordered states. We consider surfaces in the Edwards-Wilkinson (EW), Kardar-Parisi-Zhang (KPZ) and noisy surface-diffusion (NSD) universality classes. For EW and KPZ surfaces, coarse-grained depth models of the surface profile exhibit coarsening to an ordered steady state in which the order parameter has a broad distribution even in the thermodynamic limit, the distribution of particle cluster sizes decays as a power-law (with an exponent straight theta), and the scaled two-point spatial correlation function has a cusp (with an exponent alpha=1/2) at small values of the argument. The latter feature indicates a deviation from the Porod law which holds customarily, in coarsening with scalar order parameters. We present several numerical and exact analytical results for the coarsening process and the steady state. For linear surface models with a dynamical exponent z, we show that alpha=(z-1)/2 for z<3 and alpha=1 for z>3, and there are logarithmic corrections for z=3, implying alpha=1/2 for the EW surface and 1 for the NSD surface. Within the independent interval approximation we show that alpha+straight theta=2. We also study the dynamics of hard-core particles sliding locally downward on these fluctuating one-dimensional surfaces, and find that the surface fluctuations lead to large-scale clustering of the particles. We find a surface-fluctuation driven coarsening of initially randomly arranged particles; the coarsening length scale grows as approximately t(1/z). The scaled density-density correlation function of the sliding particles shows a cusp with exponents alpha approximately 0.5 and 0.25 for the EW and KPZ surfaces. The particles on the NSD surface show conventional coarsening (Porod) behavior with alpha approximately 1.

Exact tagged particle correlations in the random average process

We study analytically the correlations between the positions of tagged particles in the random average process, an interacting particle system in one dimension. We show that in the steady state, the mean-squared autofluctuation of a tracer particle grows subdiffusively sigma(2)0(t) approximately t(1/2) for large time t in the absence of external bias but grows diffusively sigma(2)0(t) approximately t in the presence of a nonzero bias. The prefactors of the subdiffusive and diffusive growths, as well as the universal scaling function describing the crossover between them, are computed exactly. We also compute sigma(2)(r)(t), the mean-squared fluctuation in the position difference of two tagged particles separated by a fixed tag shift r in the steady state and show that the external bias has a dramatic effect on the time dependence of sigma(2)(r)(t). For fixed r,sigma(2)(r)(t) increases monotonically with t in the absence of bias, but has a nonmonotonic dependence on t in the presence of bias. Similarities and differences with the simple exclusion process are also discussed.

First-passage-time exponent for higher-order random walks: using Lévy flights

We present a heuristic derivation of the first-passage-time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. 90, 219 (1992)]. Building on this derivation, we construct an estimation scheme to understand the first-passage-time exponent for the integral of the integral of a random walk, which is numerically observed to be 0.220+/-0.001. We discuss the implications of this estimation scheme for the nth integral of a random walk. For completeness, we also address the n=infinity case. Finally, we explore an application of these processes to an extended, elastic object being pulled through a random potential by a uniform applied force. In so doing, we demonstrate a time reparametrization freedom in the Langevin equation that maps nonlinear stochastic processes into linear ones.