Persistence Exponent Research Articles

Analyzing asymmetry in the Wolf-Villain model via the study of the persistence of various initial height fluctuations

The Wolf-Villain (WV) model, which, in some literature, has shown trait of up-down asymmetry, is investigated through the study of the persistence probability of height fluctuations in simulated film surfaces. The persistence probability is the probability that the height fluctuation of each site does not cross its initial value (h 0) along a time interval. When averaged over all possible values of h 0, the probability is known to exhibit a power law decay behavior with the exponent called persistence exponent. In this work, instead of averaging over all h 0, we consider a fixed value of h 0 and find that the persistence probability of the WV model still decreases with time as a power law if the fixed value meets a necessary condition, i.e. h 0 is roughly equal to or larger than the saturated roughness of the film. The persistence exponent for each h 0 is measured and found to decrease as the value of h 0 increases. Scaling form of the persistence probability at a specific h 0 is also studied. Notably, all results obtained here are in agreement with those of the up-down symmetric models appeared in literatures despite the fact that the WV model is an asymmetric model. These results support the idea from other studies that the asymmetry in the WV is very weak.

Approach to zigzag and checkerboard patterns in spatially extended systems

Zigzag patterns in one dimension or checkerboard patterns in two dimensions occur in a variety of pattern-forming systems. We introduce an order parameter ‘phase defect’ to identify this transition and help to recognize the associated universality class on a discrete lattice. In one dimension, if xi(t) is a variable value at site i at time t. We assign spin si(t)=1 for xi(t)>xi−1(t), si(t)=−1 if xi(t)<xi−1(t), and si(t)=0 if xi(t)=xi−1(t). The phase defect D(t) is defined as D(t)=∑i=1N|si(t)+si−1(t)|2N for a lattice of N sites with periodic boundary conditions. It is zero for a zigzag pattern. In two dimensions, D(t) is the sum of row-wise as well as column-wise phase defects and is zero for the checkerboard pattern. The persistence P(t) is the fraction of sites whose spin value did not change even once at all even times till time t. We find that D(t)∼t−δ and P(t)∼t−θ for the parameter range over which the zigzag or checkerboard pattern is realized with δ=0.5 and θ=3/8 for 1-d coupled logistic or Gauss maps, and δ=0.45 and θ=0.22 for 2-d logistic or Gauss maps. The exponent θ matches with the persistence exponent at zero temperature for the Ising model and δ matches with the exponent for the Ising model at the critical temperature. This power-law decay is observed over a range of parameter values and not just the critical point.

Persistence probabilities of weighted sums of stationary Gaussian sequences

With {ξi}i≥0 being a centered stationary Gaussian sequence with non-negative correlation function ρ(i)≔E[ξ0ξi] and {σ(i)}i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑i=1ℓσ(i)ξi, ℓ≥1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non-summable ρ, even for polynomial weight functions σ(i)∼ip the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(H,p) and study its properties as a function of (p,H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations – e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero – that might be of independent interest.

On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations

For a fractionally integrated Brownian motion (FIBM) of order α∈(0,1],Xα(t), we investigate the decaying rate of P(τSα&gt;t) as t→+∞, where τSα=inf{t&gt;0:Xα(t)≥S} is the first-passage time (FPT) of Xα(t) through the barrier S&gt;0. Precisely, we study the so-called persistent exponent θ=θ(α) of the FPT tail, such that P(τSα&gt;t)=t−θ+o(1), as t→+∞, and by means of numerical simulation of long enough trajectories of the process Xα(t), we are able to estimate θ(α) and to show that it is a non-increasing function of α∈(0,1], with 1/4≤θ(α)≤1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function θ=θ(α), for α∈(0,1]. Such a numerical validation is carried out in two ways: in the first one, we estimate θ(α), by using the simulated FPT density, obtained for any α∈(0,1]; in the second one, we estimate the persistent exponent by directly calculating Pmax0≤s≤tXα(s)&lt;1. Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of Xα(t) and we find the upper bound of its covariance function.

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.

Persistence probabilities of mixed FBM and other mixed processes

We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under the assumption of non-negative correlations and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in (Cheridito 2001 Bernoulli 7 913–34) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent 1 − max(1/2, H), where H is the Hurst parameter of the underlying fractional Brownian motion.

Asymptotics of the Persistence Exponent of Integrated Fractional Brownian Motion and Fractionally Integrated Brownian Motion

We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.

Self-Interacting Random Walks: Aging, Exploration, and First-Passage Times

Self-interacting random walks are endowed with long-range memory effects that emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t′<t. This class of non-Markovian random walks has applications in a broad range of examples, from insects to living cells, where a random walker locally modifies its environment—leaving behind footprints along its path and, in turn, responding to its own footprints. Because of their inherent non-Markovian nature, the exploration properties of self-interacting random walks have remained elusive. Here, we show that long-range memory effects can have deep consequences on the dynamics of generic self-interacting random walks; they can induce aging and nontrivial persistence and transience exponents, which we determine quantitatively, in both infinite and confined geometries. Based on this analysis, we quantify the search kinetics of self-interacting random walkers and show that the distribution of the first-passage time to a target site in a confined domain takes universal scaling forms in the large-domain size limit, which we characterize quantitatively. We argue that memory abilities induced by attractive self-interactions provide a decisive advantage for local space exploration, while repulsive self-interactions can significantly accelerate the global exploration of large domains.Received 13 July 2021Revised 8 November 2021Accepted 20 December 2021DOI:https://doi.org/10.1103/PhysRevX.12.011052Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDiffusionFluctuations & noiseRandom walksStochastic processesStatistical PhysicsInterdisciplinary Physics

The Persistence Exponents of Gaussian Random Fields Connected by the Lamperti Transform

The (fractional) Brownian sheet is the simplest example of a Gaussian random field $$X$$ whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self- similar Gaussian processes. We consider the homogeneous Gaussian field $$Y$$ obtained by applying to X the exponential change of time (more precisely, the Lamperti transform). Under some assumptions, we prove the existence of the persistence exponents for both fields, $$X$$ and $$Y$$ , and find the relation between them. The exponent for any random function $$Z$$ is $$\psi (T)$$ if lim $$\ln P(Z({\mathbf{t}}) \le ,{\mathbf{t}} \in TG)/\psi (T) = - 1$$ , $$T > > {1}$$ , where $$G$$ is a d-dimensional region containing 0, and $$T$$ is a similarity coefficient. The considered problem was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals.

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Transition to Fully or Partially Arrested State in Coupled Logistic Maps on a Ladder

Layered structures are an object of interest for theoretical and experimental reasons. In this work, we study coupled map lattice on a ladder. The ladder consists of two one-dimensional chains coupled at every point. We study linearly and nonlinearly coupled logistic maps in this system and study transition to nonzero persistence, in particular. We coarse-grain the variable value by assigning spin [Formula: see text] ([Formula: see text]) to sites that have value greater (less) than the fixed point and compute the number of sites that have not changed their spin values at all even times till the given time [Formula: see text]. The fraction of such sites at a given time [Formula: see text] is known as persistence. In our system, we observe a power-law of persistence at the critical value of coupling. This transition is also accompanied by long-range antiferromagnetic ordering for nonlinear coupling and long-range ferromagnetic ordering for linear coupling. The number of domain walls decay as [Formula: see text] at the critical point in both cases. The persistence exponent is 0.375 for a nonlinear case with two layers which is an exponent for the voter model on the ladder as well as for the Ising model at zero temperature or voter model in 1D. For linear coupling, we obtain a smaller persistence exponent.

Finding self-similar behavior in quantum many-body dynamics via persistent homology

Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider data from a classical-statistical simulation of a two-dimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium self-similar phenomena. A possible explanation of the underlying processes is provided in terms of mixing strong wave turbulence and anomalous vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques.

Stochastic Processes under Constraints

The analysis of random processes under various constraints and conditions has been a central theme in the theory of stochastic processes, which links together several mathematical subdisciplines. The connection between potential theory and a certain type of conditioning of Markov processes via Doob’s h-transform can be seen as a classical highlight. The last decades have seen further exciting and highly interesting developments which are related to the title of the workshop such as the analysis of persistence exponents for various classes of processes and various types of penalization problems. Many of these problems are rooted in questions from statistical mechanics. The workshop aims to investigate the topic stochastic processes under constraints from all these different perspectives.

Active Brownian motion with directional reversals

Active Brownian motion with intermittent direction reversals is common in bacteria like Myxococcus xanthus and Pseudomonas putida. We show that, for such a motion in two dimensions, the presence of the two timescales set by the rotational diffusion constant D_{R} and the reversal rate γ gives rise to four distinct dynamical regimes: (I) t≪min(γ^{-1},D_{R}^{-1}), (II) γ^{-1}≪t≪D_{R}^{-1}, (III) D_{R}^{-1}≪t≪γ^{-1}, and (IV) t≫max(γ^{-1},D_{R}^{-1}), showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly nondiffusive and anisotropic behavior at short times to a diffusive isotropic behavior via an intermediate regime, II or III. In regime II, we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable z∝x_{⊥}/t with a nontrivial scaling function, f(z)=(2π^{3})^{-1/2}Γ(1/4+iz)Γ(1/4-iz). Furthermore, by computing the exact first-passage time distribution, we show that a persistence exponent α=1 emerges due to the direction reversal in this regime.

Fractional Brownian motion of worms in worm algorithms for frustrated Ising magnets.

We study the distribution of lengths and other statistical properties of worms constructed by Monte Carlo worm algorithms in the power-law three-sublattice ordered phase of frustrated triangular and kagome lattice Ising antiferromagnets. Viewing each step of the worm construction as a position increment (step) of a random walker, we demonstrate that the persistence exponent θ and the dynamical exponent z of this random walk depend only on the universal power-law exponents of the underlying critical phase and not on the details of the worm algorithm or the microscopic Hamiltonian. Further, we argue that the detailed balance condition obeyed by such worm algorithms and the power-law correlations of the underlying equilibrium system together give rise to two related properties of this random walk: First, the steps of the walk are expected to be power-law correlated in time. Second, the position distribution of the walker relative to its starting point is given by the equilibrium position distribution of a particle in an attractive logarithmic central potential of strength η_{m}, where η_{m} is the universal power-law exponent of the equilibrium defect-antidefect correlation function of the underlying spin system. We derive a scaling relation, z=(2-η_{m})/(1-θ), that allows us to express the dynamical exponent z(η_{m}) of this process in terms of its persistence exponent θ(η_{m}). Our measurements of z(η_{m}) and θ(η_{m}) are consistent with this relation over a range of values of the universal equilibrium exponent η_{m} and yield subdiffusive (z>2) values of z in the entire range. Thus, we demonstrate that the worms represent a discrete-time realization of a fractional Brownian motion characterized by these properties.

Zero-temperature coarsening in the two-dimensional long-range Ising model.

We investigate the nonequilibrium dynamics following a quench to zero temperature of the nonconserved Ising model with power-law decaying long-range interactions ∝1/r^{d+σ} in d=2 spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent α, the persistence exponent θ, and the fractal dimension d_{f}. It is found that the growth exponent α≈3/4 is independent of σ and different from α=1/2, as expected for nearest-neighbor models. In the large σ regime of the tunable interactions only the fractal dimension d_{f} of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponents θ this is a direct consequence of the different growth exponents α as can be understood from the relation d-d_{f}=θ/α; they just differ by the ratio of the growth exponents ≈3/2. This relation has been proposed for annihilation processes and later numerically tested for the d=2 nearest-neighbor Ising model. We confirm this relation for all σ studied, reinforcing its general validity.

Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

AbstractWe show that $$\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}$$ P ( ℓ X ( 0 , T ] ≤ 1 ) = ( c X + o ( 1 ) ) T - ( 1 - H ) , where $$\ell _X$$ ℓ X is the local time measure at 0 of any recurrent H-self-similar real-valued process X with stationary increments that admits a sufficiently regular local time and $$c_X$$ c X is some constant depending only on X. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $$1-H$$ 1 - H on the decay exponent of $$\mathbb {P}( \ell _X(0,T] \le 1)$$ P ( ℓ X ( 0 , T ] ≤ 1 ) . Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case.

Effect of a Mode of Update on Universality Class for Coupled Logistic Maps: Directed Ising to Ising Class

Ising model at zero temperature leads to a ferromagnetic state asymptotically. There are two such possible states linked by symmetry, and Glauber–Ising dynamics are employed to reach them. In some stochastic or deterministic dynamical systems, the same absorbing state with [Formula: see text] symmetry is reached. This transition often belongs to the directed Ising (DI) class where dynamic exponents and persistence exponent are different. In asymmetrically coupled sequentially updated logistic maps, the transition belongs to the DI class. We study changes in the nature of transition with an update scheme. Even with the synchronous update, the transition still belongs to the DI class. We also study a synchronous probabilistic update scheme in which each site is updated with the probability [Formula: see text]. The order parameter decays with an exponent [Formula: see text] in this scheme. Nevertheless, the dynamic exponent [Formula: see text] is less than [Formula: see text] even for small values of [Formula: see text] indicating a very slow crossover to the Ising class. However, with a random asynchronous update, we recover [Formula: see text]. In the presence of feedback, synchronous update leads to a transition in the DI universality class which changes to Ising class for synchronous probabilistic update.

Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for Any Integer Dimension

The persistence exponent for the simple diffusion equation , with random Gaussian initial condition, has been calculated exactly using a method known as selective averaging. The probability that the value of the field at a specified spatial coordinate remains positive throughout for a certain time t behaves as for asymptotically large time t. The value of , calculated here for any integer dimension d, is for and 1 otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values for respectively.

Anomalous persistence exponents for normal yet aging diffusion.

The persistence exponent θ, which characterizes the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. So far, anomalous values of the persistence exponent (θ≠1/2) were obtained, but only for anomalous processes (i.e., with Hurst exponent H≠1/2). Here we exhibit examples of ageing processes which, even if they display asymptotically a normal diffusive scaling (H=1/2), are characterized by anomalous persistent exponents that we determine analytically. Based on this analysis, we propose the following general criterion: The persistence exponent of asymptotically diffusive processes is anomalous if the increments display ageing and depend on the observation time T at all timescales.