Persistence Exponent Research Articles

Persistence of some additive functionals of Sinai’s walk

Nous nous intéressons à la marche de Sinai $(S_{n})_{n\in\mathbb{N}}$. Nous prouvons que la probabilité annealed que $\sum_{k=0}^{n}f(S_{k})$ soit strictement positive pour tout $n\in[1,N]$ est égale à $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, pour une large classe de fonctions $f$, et en particulier pour $f(x)=x$. L’exposant de persistance $\frac{3-\sqrt{5}}{2}$ est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basée sur des techniques de localisation pour la marche de Sinai et utilise des résultats de Cheliotis sur les changements de signe des fonds de vallées d’un mouvement Brownien indexé par $\mathbb{R}$.

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents.

We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins 0≤x≤1. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for x≤0.5, persistence for the up (minority) spins shows the behavior P_{min}(t)∼t^{-γ}exp[-(t/τ)^{δ}] with time t, while for the down (majority) spins, P_{maj}(t) approaches a finite value. We find that the timescale τ diverges as (x_{c}-x)^{-λ}, where x_{c}=0.5 and λ≃2.31. The exponent γ varies as θ_{2d}+c_{0}(x_{c}-x)^{β} where θ_{2d}≃0.215 is very close to the persistence exponent in two dimensions; β≃1. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form E(x)=f[(x-x_{c}/x_{c})L^{1/ν}], with ν≈1.47. This result suggests that τ∼L^{z[over ̃]}, where z[over ̃]=λ/ν=1.57±0.11 is an exponent not explored earlier.

Spherical model of growing interfaces

Building on an analogy between the ageing behaviour of magnetic systems and growing interfaces, the Arcetri model, a new exactly solvable model for growing interfaces is introduced, which shares many properties with the kinetic spherical model. The long-time behaviour of the interface width and of the two-time correlators and responses is analysed. For all dimensions d ≠ 2, universal characteristics distinguish the Arcetri model from the Edwards–Wilkinson model, although for d > 2 all stationary and non-equilibrium exponents are the same. For d = 1 dimensions, the Arcetri model is equivalent to the p = 2 spherical spin glass. For 2 < d < 4 dimensions, its relaxation properties are related to the ones of a particle-reaction model, namely a bosonic variant of the diffusive pair-contact process. The global persistence exponent is also derived.

Persistence exponent for random processes in Brownian scenery

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian andnon-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup_{t\in[0,T]} \Delta_t \leq 1] $where $\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L_t(x); x\in\mathbb{R},t\geq 0}$ isthe local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.

No zero-crossings for random polynomials and the heat equation

Consider random polynomial $\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_{\alpha}+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_{\alpha}-2b_0+o(1)}$. Here, $b_{\alpha}=0$ when $\alpha\le-1$ and otherwise $b_{\alpha}\in(0,\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\phi_d({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\phi_d({\mathbf{x}},0)$, we confirm that the probability of $\phi_d({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_{\alpha}+o(1)}$, for $\alpha=d/2-1$.

Nonlinear fluctuation analysis for a set of 41 magnetic clouds measured by the Advanced Composition Explorer (ACE) spacecraft

Abstract. The statistical distribution of values in the signal and the autocorrelations (interpreted as the memory or persistence) between values are attributes of a time series. The autocorrelation function values are positive in a time series with persistence, while they are negative in a time series with anti-persistence. The persistence of values with respect to each other can be strong, weak, or nonexistent. A strong correlation implies a "memory" of previous values in the time series. The long-range persistence in time series could be studied using semivariograms, rescaled range, detrended fluctuation analysis and Fourier spectral analysis, respectively. In this work, persistence analysis is to study interplanetary magnetic field (IMF) time series. We use data from the IMF components with a time resolution of 16 s. Time intervals corresponding to distinct processes around 41 magnetic clouds (MCs) in the period between March 1998 and December 2003 were selected. In this exploratory study, the purpose of this selection is to deal with the cases presenting the three periods: plasma sheath, MC, and post-MC. We calculated one exponent of persistence (e.g., α, β, Hu, Ha) over the previous three time intervals. The persistence exponent values increased inside cloud regions, and it was possible to select the following threshold values: α(j) = 1.392, Ha(j) = 0.327, and Hu(j) = 0.875. These values are useful as another test to evaluate the quality of the identification. If the cloud is well structured, then the persistence exponent values exceed thresholds. In 80.5% of the cases studied, these tools were able to separate the region of the cloud from neighboring regions. The Hausdorff exponent (Ha) provides the best results.

Persistence of integrated stable processes

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Shi (Lower tails of some integrated processes. In: Small deviations and related topics (problem panel 2003). Along the way, we investigate the law of the stable process $$L$$ evaluated at the first time its integral $$X$$ hits zero, when the bivariate process $$(X,L)$$ starts from a coordinate axis. This extends classical formulæ by McKean (J Math Kyoto Univ 2:227–235, 1963) and Gor’kov (Soviet. Math. Dokl. 16:904–908, 1975) for integrated Brownian motion.

Persistence of random walk records

We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk ‘superior’ if the record is always above average, and conversely, the walk is said to be ‘inferior’ if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ∼ t−β, in the limit t → ∞, and that the persistence exponent is nontrivial, β = 0.382 258…. The fraction of inferior walks, I, also decays as a power law, I ∼ t−α, but the persistence exponent is smaller, α = 0.241 608…. Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with a given record and position, while for inferior walks it suffices to study the density as a function of position.

Effects of initial height on the steady-state persistence probability of linear growth models

The effects of the initial height on the temporal persistence probability of steady-state height fluctuations in up-down symmetric linear models of surface growth are investigated. We study the (1+1)-dimensional Family model and the (1+1)- and (2+1)-dimensional larger curvature (LC) model. Both the Family and LC models have up-down symmetry, so the positive and negative persistence probabilities in the steady state, averaged over all values of the initial height h(0), are equal to each other. However, these two probabilities are not equal if one considers a fixed nonzero value of h(0). Plots of the positive persistence probability for negative initial height versus time exhibit power-law behavior if the magnitude of the initial height is larger than the interface width at saturation. By symmetry, the negative persistence probability for positive initial height also exhibits the same behavior. The persistence exponent that describes this power-law decay decreases as the magnitude of the initial height is increased. The dependence of the persistence probability on the initial height, the system size, and the discrete sampling time is found to exhibit scaling behavior.

Nature versus nurture: Predictability in low-temperature Ising dynamics

Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state ("nature") versus the realization of the stochastic dynamics ("nurture") in predicting the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from T=∞ to T=0. We performed Monte Carlo studies on the overlap between "identical twins" raised in independent dynamical environments, up to size L=500. Our results suggest an overlap decaying with time as t(-θ)(h) with θ(h)=0.22 ± 0.02; the same exponent holds for a quench to low but nonzero temperature. This "heritability exponent" may equal the persistence exponent for the two-dimensional Ising ferromagnet, but the two differ more generally.

Traces of self-organisation and long-range memory in variations of environmental radon in soil: comparative results from monitoring in Lesvos Island and Ileia (Greece)

This paper addresses issues of self-affinity, long-memory and self-organisation in variations of radon in soil recorded in Lesvos Island, Greece. Several techniques were employed, namely (a) power-law wavelet spectral fractal analysis, (b) estimation of Hurst exponents through (b1) rescaled-range, (b2) roughness-length, (b3) variogram and (a), (c) detrended fluctuation analysis, (d) investigation of fractal dimensions and (e) analysis of five block entropies: (e1) Shannon entropy, (e2) Shannon entropy per letter, (e3) conditional entropy, (e4) Tsallis entropy, and (e5) normalised Tsallis entropy. Long-lasting antipersistency was identified during a period of anomalous radon variations following fractional Brownian modelling. Remaining variations did not exhibit analogous behaviour and followed fractional Gaussian modelling. Antipersistent power-law-beta-exponent-values between 1.5 and 2.0 were detected during anomalies. Persistent values were also found. Hurst exponents were mainly within 0 < H < 0.5. Some persistent exponents (0.5 < H < 1) were also observed. Fractal dimensions were within 1.5 < D < 2. Radon anomalies presented lower fractal dimensions. Shannon entropy ranged between 0.77 ≤ H(n) ≤ 2.38, Shannon entropy per letter, between 0.19 ≤ h(n) ≤ 0.59, conditional entropy, between 0.01 ≤ h(n) ≤ 0.58, Tsallis entropy, between 0.55 ≤ Sq ≤ 1.01 and normalised Tsallis entropy between, 0.98 ≤ \(\hat{S}\) ≤ 5.42 (block-size n = 4). Entropies were lower during anomalies, indicating strong self-organisation. Persistency–antipersistency switching was observed, consistent with long-memory dynamics. Potential geological sources were discussed. The asperity-model was proposed. Findings were compared to results obtained under analogous methodologies in Ileia, Greece.

Universal persistence exponent in transition to antiferromagnetic order in coupled logistic maps

We study coupled logistic maps on a one-dimensional lattice. We coarse grain the system by labeling the local state + if the value of the variable is above the fixed point and - if the value is below the (nonzero) fixed point, x(*)=1-1/μ. We find that the system exhibits a transition from a global state, in which moving defects occur, to a global state, which is frozen in time (modulo-2). All such states display nonzero value of persistence. Besides, we also observe a state which displays long-range antiferromagnetic order. Onset of such a state is accompanied with a power-law behavior of persistence P(t) as a function of time at the critical line. Usually, persistence transitions show nonuniversal exponent. However, here persistence exponent for synchronous dynamics is 3/8 over the entire lower critical curve. This exponent is the same as one observed for asynchronous Glauber dynamics simulation of zero-temperature Ising model in one dimension. The expected value for synchronous dynamics is 3/4. We present a plausible argument for this scaling.

Time-dependent Monte Carlo simulations of critical and Lifshitz points of the axial-next-nearest-neighbor Ising model

In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: <M>(m(0)=1)~t(-β/νz), which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent z, evaluated from the behavior of the ratio F(2)(t)=<M(2)>(m(0)=0)/<M>(2)(m(0)=1)~t(3/z), along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent θ that characterizes the initial slip of magnetization and the global persistence exponent θ(g) associated with the probability P(t) that magnetization keeps its signal up to time t. Our estimates for the dynamic critical exponents at the Lifshitz point are z=2.34(2) and θ(g)=0.336(4), values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the z axis, i.e., J(2)=0), i.e., z≈2.07 and θ(g)≈0.38. We also present estimates for the static critical exponents β and ν, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.

Long-range memory patterns in variations of environmental radon in soil

This paper addresses issues of long-range memory and self-organisation in variations of radon in soil in Greece. The methods of rescaled-range, roughness-length, variogram, fractal dimension and block entropy were employed through lumping. Sliding was utilised with the wavelet spectral fractal technique. Antipersistent Hurst exponents in the range 0 < H < 0.5 were mainly identified. Persistent exponents (0.5 < H < 1) were also detected. Switching between persistency and antipersistency was observed and considered consistent with an underlying geo-environmental long-memory self-organisation. Fractal dimensions were in the range 1.2 < D < 2. The anomalous parts of the 2008 radon signal presented significantly lower fractal dimensions. Value ranges of Shannon, Shannon-per-letter, conditional, Tsallis and normalised Tsallis block entropies were 0.67 ≤ H(n) ≤ 2.73, 0.2 ≤ h(n) ≤ 0.7, 0.2 ≤ h(n) ≤ 0.6, 0.36 ≤ Sq ≤ 1.11, 0.50 ≤ Ŝq ≤ 9.55 respectively. The entropy values were affected by the block-size n. The entropic index values of the radon anomalies were significantly lower indicating long-memory underlying patterns. Underlying sources are discussed. The asperity-model is proposed.

Persistence in reactive-wetting interfaces

In this paper, we report on persistence results of reactive-wetting advancing interfaces performed with mercury on silver at room temperature. Earlier kinetic roughening studies of reactive-wetting systems at room temperature as well as at high temperatures revealed some limited information on the spatiotemporal behavior of these systems. However, by calculating the persistence exponent, we were able to identify two distinct kinetic time regimes in this process. In the first one, while the interface is moving but its width is not yet growing, the persistence exponent is θ=0.55±0.05, which is typical for a random, noisy behavior. In the second regime, there is an effective growth of the interface width with a growth exponent β=0.67±0.06 followed by saturation, according to the Family-Vicsek description of interface growth. The persistence exponent in this regime is θ=0.37±0.05, which indicates that the relation θ=1-β seems to hold even for this nonlinear experimental system.

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback

We investigate coupled circle maps in the presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence acts as an excellent quantifier to describe this transition. Taking the location of the fixed point of circle map (which does not change with feedback) as a reference point, we compute a number of sites which have been greater than (less than) the fixed point until time t. Though local dynamics is high dimensional in this case, this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional scaling as seen in second order phase transitions. This indicates that persistence could work as a good order parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in eigenvalue spectrum of the Jacobian of localized state.

Local and global persistence exponents of two quenched continuous-lattice spin models

Local and global persistence exponents associated with zero-temperature quenched dynamics of two-dimensional XY model and three-dimensional Heisenberg model have been estimated using numerical simulations. The method of block persistence has been used to find the global and local exponents simultaneously (in a single simulation). Temperature universality of both the exponents for three-dimensional Heisenberg model has been confirmed by simulating the stochastic (with noise) version of the equation of motion. The noise amplitudes added were small enough to retain the dynamics below criticality. In the second part of our work we have studied scaling associated with correlated persistence sites in the three-dimensional Heisenberg model in the later stages of the dynamics. The relevant length scale associated with correlated persistent sites was found to behave in a manner similar to the dynamic length scale associated with the phase ordering dynamics.

The persistence probability and the price–price correlation functions in the Korean stock market

We consider the one minute time series of the Korean stock market index (KOSPI). We defined the persisting time as the time interval when the index remains above (or below) an initial index. We observed that the average persistence probability P ( t ) followed a power-law behavior like P ( t ) ∼ t − θ with the persistence exponent θ = 0.477 ( 2 ) . The persistence exponent in the Korean stock market deviates slightly from a random process. The Korean stock market has shown a weak anti-persistence. We measured the persistence properties of the stock market by the generalized price–price correlation function F q ( t ) . The price–price correlation function followed a power law, F q ( t ) ∼ t h q , where h q is called the generalized Hurst exponent. The generalized Hurst exponent depends on the order q which means that there are multiscaling properties in the time series of the stock index. We observed the relationship θ + h 2 = 1 between the error bars where h 2 is the fractal dimension of the time series.

Dynamic phase transition in the prisoner’s dilemma on a lattice with stochasticmodifications

We present a detailed study of the prisoner’s dilemma game with stochastic modificationson a two-dimensional lattice, in the presence of evolutionary dynamics. By very nature ofthe rules, the cooperators have incentives to cheat and fear being cheated. They may cheateven when this is not dictated by the evolutionary dynamics. We consider two variants here. Ineach case, the agents mimic the action (cooperation or defection) in the previous time stepof the most successful agent in the neighborhood. But over and above this, the fractionp of cooperators spontaneously change their strategy to pure defector atevery time step in the first variant. In the second variant, there areno pure cooperators. All cooperators keep defecting with probabilityp at every time step. In both cases, the system switches from acoexistence state to an all-defector state for higher values ofp. We showthat the transition between these states unambiguously belongs to the directed percolation universalityclass in 2 + 1 dimensions. We also study the local persistence. The persistence exponents obtained arehigher than the ones obtained in previous studies, underlining their dependence on detailsof the dynamics.

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.