Persistence in Random Walk in Composite Media

Quantum Persistence: A Random Walk Scenario

In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum wa…

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^(-beta), in the limit t --> infty, and that the …

Phase transitions in persistent and run-and-tumble walks

We calculate the large deviation function of the end-to-end distance and the corresponding extension-versus-force relation for (isotropic) random walks, on and off-lattice, with and without persistence, and in any spatial dimension. For off-lattice random walks with persistence, the large deviation function undergoes a first order phase transition in dimension $d> 5$. In the corresponding force-versus-extension relation, the extension becomes independent of the force beyond a critical value. The transition is anticipated in dimensions $d=4$ and $d=5$, where full extension is reached at a finit…

Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation

The persistence probability, $P_C(t)$, of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^γ$. In the mean-field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For $γ\ge 0$ the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For $γ< 0$ the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory.…

Survival probability of stochastic processes beyond persistence exponents

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^θ$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $θ$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first…

Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model

The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t) -> const + t^{-θ}$, where $θ$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $ρ>> 1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $θ\appr…

自相互作用随机游走的持续指数

Persistence exponents of self-interacting random walks

本文精确计算了自相互作用随机游走的持续指数及劈裂概率，为具有长程记忆的探索过程提供了定量表征，应用于觅食、细胞生物学和机器学习等领域。

具有竞争动力学的相序系统的持久性性质

Persistence Properties of a Phase-ordering System with Competing Dynamics

通过零温度蒙特卡洛模拟，研究二维Ising模型在竞争非守恒自旋翻转和守恒自旋交换动力学下的持久性，发现总持久性和自旋翻转持久性具有相同的幂律衰减，而复合持久性表现出不同的标度行为。

Correlated Gaussian random walk models of animal dispersal

A correlated Gaussian random walk(CGRW) model is proposed as a simple model of animal dispersal. The general features of CGRW is described. We will discuss how from this single model a number of different kinds of correlated random walk can be studied. The CGRWs in one dimension is studied in detail and the special limiting cases are discussed where the probability densities are found analytically. Numerical simulations are performed and the results are found to be in good agreement with the theoretical predictions. Directional persistence in CGRWs is discussed for the one dimensional cases. W…

Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model

The Langevin equation for a particle (`random walker') moving in d-dimensional space under an attractive central force, and driven by a Gaussian white noise, is considered for the case of a power-law force, F(r) = - Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not visited the origin up to time t, is calculated. For sigma > 1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P_0(t) are those of a free random walker. For sigma < 1, the noise is (dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a wea…

Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes

We consider the problem of `discrete-time persistence', which deals with the zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T = n(ΔT). For a Gaussian Stationary Process the persistence (no crossing) probability decays as exp(-θ_D T) = [ρ(a)]^n for large n, where a = \exp[-(ΔT)/2], and the discrete persistence exponent, θ_D, is given by θ_D = \ln(ρ)/2\ln(a). Using the `Independent Interval Approximation', we show how θ_D varies with (ΔT) for small (ΔT) and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are le…

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

Persistence, defined as the probability that a fluctuating signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. It quantifies the kinetics of processes as varied as phase ordering, reaction diffusion or interface relaxation dynamics. The fact that persistence can decay algebraically with time with non trivial exponents has triggered a number of experimental and theoretical studies. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing…

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In 1-dimension, the system models the zero temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d < 2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) ~ m^(z/d) t^(…

Persistence effects in deterministic diffusion

In systems which exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient can often be understood in terms of random walk models. Provided the decay of correlations is fast enough, one can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments. By successively including the effects of one and two steps of memory on this approximation, we examine the effects of ``persistence'' on the diffusion coefficients of extended two-dimensional billiard tables and show how to properly account for these effects, using walks in …

Random walks and branching processes in correlated Gaussian environment

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.

Scaling laws in the diffusion limited aggregation of persistent random walkers

We investigate the diffusion limited aggregation of particles executing persistent random walks. The scaling properties of both random walks and large aggregates are presented. The aggregates exhibit a crossover between ballistic and diffusion limited aggregation models. A non-trivial scaling relation $ξ\sim\ell^{1.25}$ between the characteristic size $ξ$, in which the cluster undergoes a morphological transition, and the persistence length $\ell$, between ballistic and diffusive regimes of the random walk, is observed.

Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length ηdrawn from a continuous distribution f(η) which is symmetric around a constant drift c. We focus in particular on the case were f(η) is a symmetric stable law with a Lévy index 0 < μ\leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit o…

Coarsening and persistence in a class of stochastic processes interpolating between the Ising and voter models

We study the dynamics of a class of two dimensional stochastic processes, depending on two parameters, which may be interpreted as two different temperatures, respectively associated to interfacial and to bulk noise. Special lines in the plane of parameters correspond to the Ising model, voter model and majority vote model. The dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing, annihilating, and branching random walkers. We use the freedom allowed by the space of parameters to measure, by numerical simulations, the persi…

Non-monotonic displacement distribution of active random walkers

We consider a simple model for active random walk with general temporal correlations, and investigate the shape of the probability distribution function of the displacement during a short time interval. We find that under certain conditions the distribution is non-monotonic and we show analytically and numerically that the existence of the non-monotonicity is governed by the walker's tendency to move forward, while the correlations between the timing of its active motion control the magnitude and shape of the non-monotonicity. In particular, we find that in a homogeneous system such non-monoto…

Holey Random Walks: Optics of Heterogeneous Turbid Composites

We present a probabilistic theory of random walks in turbid media with non-scattering regions. It is shown that important characteristics such as diffusion constants, average step lengths, crossing statistics and void spacings can be analytically predicted. The theory is validated using Monte Carlo simulations of light transport in heterogeneous systems in the form of random sphere packings, and good agreement is found. The role of step correlations is discussed, and differences between unbounded and bounded systems are investigated. Our results are relevant to the optics of heterogeneous syst…

Convex hulls of several multidimensional Gaussian random walks

We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the $d$-dimensional Gaussian polytope with or without the origin.

Joint statistics of space and time exploration of $1d$ random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on …

极持久的随机游走揭示景观拓扑的转变

Very persistent random walkers reveal transitions in landscape topology

极持久的随机游走揭示了景观拓扑转变与非ergodic性转变之间的对应关系，为确定拓扑转变能量提供了新方法。

环形搜索域中持久随机游走的平均首通时间

Mean First Passage Time for Persistent Random Walks in Annular Search Domains

研究二维环形域中随机游走者到达中心小吸收目标的平均首通时间，通过速度跳跃模型分析方向持久性和趋化偏置对搜索效率的影响，结合解析与数值方法。

Inhomogeneous ensembles of correlated random walkers

Discrete time random walks, in which a step of random sign but constant length $δx$ is performed after each time interval $δt$, are widely used models for stochastic processes. In the case of a correlated random walk, the next step has the same sign as the previous one with a probability $q \neq 1/2$. We extend this model to an inhomogeneous ensemble of random walkers with a given distribution of persistence probabilites $p(q)$ and show that remarkable statistical properties can result from this inhomogenity: Depending on the distribution $p(q)$, we find that the probability density $p(Δx, Δt)…

Anomalous Diffusion in Aperiodic Environments

We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings we obtain many new analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations…

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase

We study analytically the distribution of fluctuations of the quantities whose average yield the usual two-point correlation and linear response functions in three unfrustrated models: the random walk, the $d$ dimensional scalar field and the 2d XY model. In particular we consider the time dependence of ratios between composite operators formed with these fluctuating quantities which generalize the largely studied fluctuation-dissipation ratio, allowing us to discuss the relevance of the effective temperature notion beyond linear order. The behavior of fluctuations in the aforementioned solvab…

Persistence and First-Passage Properties in Non-equilibrium Systems

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and ap…

扩散到 ballistic 转变在持久随机游走中

Diffusive-to-Ballistic transition in a Persistent Random Walk

研究了具有时间依赖速度反转概率的持久随机游走，确定了非平衡动力学转变的准则。以幂律形式的反转概率p(t)~t^{-α}为例，证明了在α=1时系统发生转变，将α<1的超扩散区域与α≥1的ballistic区域分开。通过速度相关性和持久性统计的结果，结合有限时间尺度的Binder cumulant和位移波动，详细表征了转变及其性质。进一步论证该转变不仅限于幂律形式，其他满足相同准则的时间依赖反转概率也能引发转变。在任意空间维度中，只要保持速度空间的各向同性，转变持续存在。

Continuous Time Random Walks for the Evolution of Lagrangian Velocities

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop…

Comment on "Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-Dimensional Random Environments"

In a recent letter [PRL 80 (1998) 3539] Fisher, Le Doussal and Monthus report new predictions for the persistence properties of Sinai's model, which they obtain by using an approximate real space renormalization group scheme. In this comment we show that these predictions can be derived in a rigorous and more transparent way by reducing the quantities of interest to particular statistical properties of a simple random walk in a homogeneous environment. We also point out that they are valid in a much more general context: our results are neither restricted to the random force model but are vali…

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, $P_x(x,s)$, with $-1 \leq s \leq 1$ is mapped isomorphically onto the temporal persistent large deviations probability $P_t(t,s)$ associated with the stochastic Markovian random walk problem. We show using numerical simulations that the infinite family of spatial persistent large deviations exponents $θ_x(s)$ characterizing the power law decay of $P_x(x,s)$ agrees, as predicted on theoretical grounds by Majumdar and Bray [Phys. Rev. Lett. {\bf 86}, 3700 (2001)] with the numer…

Reaction fronts in persistent random walks with demographic stochasticity

Standard Reaction-Diffusion (RD) systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that when the number of individuals is very large, the individual-based model is well described by the continuous Reactive Cattaneo Equation (RCE), but for smaller values of the carrying capacity important finite-population effects aris…

作为相关随机游走的气体扩散

Gaseous Diffusion as a Correlated Random Walk

本文把气体中分子的扩散建模为相关随机游走，通过级数展开表明扩散系数可由均方自由程和平均持续率两个量确定，并用 Boltzmann 方程模拟验证了改进扩散系数公式的准确性。

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, ranging from insects to living cells, where a random walker modifies locally its environment -- leaving behind footprints along its path, and in turn responds to its own footprints. Because of their inherent non Markovian nature, the exploration properties of self-interacting random walks have remained elusi…

A dimensional acceleration of gradient descent-like methods, using persistent random walkers

Finding a local minimum or maximum of a function is often achieved through the gradient-descent optimization method. For a function in dimension d, the gradient requires to compute at each step d partial derivatives. This method is for instance used in machine-learning, to fit the models parameters so as to minimize the error rate on a given data set, or in theoretical chemistry, to obtain molecular conformation. Since each step requires to obtain d partial derivatives, it can quickly become time-consuming when d grows and when each computation of the function is complex. If the computation ti…

Anomalous diffusion in disordered media and random quantum spin chains

Using exact expressions for the persistence probability and for the leading eigenvalue of the Focker-Planck operator of a random walk in a random environment we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many new exact results are obtained from this correspondence including the space and time correlations of surviving random walks and the distribution of the gaps of the corresponding Focker-Planck operator. In turn we derive analytically the dynamical exponent of the random…

Universal survival probability for a correlated random walk and applications to records

We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution o…

从随机游走至热力骑行：在上升飞行中普遍的异常传输

From Random Walks to Thermal Rides: Universal Anomalous Transport in Soaring Flights

研究揭示了上升飞行中普遍的亚弹道传输特性，通过间歇性和持久性机制，发现学习在搜索阶段提升探索效率。

From diffusion in compartmentalized media to non-Gaussian random walks

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the Brownian yet non-Gaussian diffusion that exhibits linear growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly …