A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-α}$ with $α\sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-β}/|x|$ where $β= 1+ α$ for $t/|x|>> 1$,and ${\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.
We have considered the persistence of unvisited sites of a lattice, i.e., the probability $S(t)$ that a site remains unvisited till time $t$ in presence of mutually repulsive random walkers. The dynamics of this system has direct correspondence to that of the domain walls in a certain system of Ising spins where the number of domain walls become fixed following a zero termperature quench. Here we get the result that $S(t) \propto \exp(-αt^β)$ where $β$ is close to 0.5 and $α$ a function of the density of the walkers $ρ$. The number of persistent sites in presence of independent walkers of dens…
The spatial distribution of unvisited/persistent sites in $d=1$ $A+A\to\emptyset$ model is studied numerically. Over length scales smaller than a cut-off $ξ(t)\sim t^{z}$, the set of unvisited sites is found to be a fractal. The fractal dimension $d_{f}$, dynamical exponent $z$ and persistence exponent $θ$ are related through $z(1-d_{f})=θ$. The observed values of $d_{f}$ and $z$ are found to be sensitive to the initial density of particles. We argue that this may be due to the existence of two competing length scales, and discuss the possibility of a crossover at late times.
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…
We study how the Hurst exponent $α$ depends on the fraction $f$ of the total time $t$ remembered by non-Markovian random walkers that recall only the distant past. We find that otherwise nonpersistent random walkers switch to persistent behavior when inflicted with significant memory loss. Such memory losses induce the probability density function of the walker's position to undergo a transition from Gaussian to non-Gaussian. We interpret these findings of persistence in terms of a breakdown of self-regulation mechanisms and discuss their possible relevance to some of the burdensome behavioral…
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…
We consider a class of inhomogeneous media known as composite media that is often encountered in experimental sciences and investigate the persistence probability of a random walker in such a system. Analytical and numerical results for the crossover time scales has been obtained for a composite system with two homogeneous components and three homogeneous components respectively.
How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with periodic boundaries). We systematically explore dependence of the probability $Π_d(L,u)$ of percolation (existence of a spanning cluster) of sites not removed by the RW on $L$ and $u$. The concentration of unvisited sites decays exponentially with increasing $u$, while the visited sites are highly correlated -- their correlations decaying with the distance $r…
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^(…
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…
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.
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.…
We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value <x(t)> of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a…
The spatial distribution of persistent (unvisited) sites in one dimensional $A+A\to\emptyset$ model is studied. The `empty interval distribution' $n(k,t)$, which is the probability that two consecutive persistent sites are separated by distance $k$ at time $t$ is investigated in detail. It is found that at late times this distribution has the dynamical scaling form $n(k,t)\sim t^{-θ}k^{-τ}f(k/t^{z})$. The new exponents $τ$ and $z$ change with the initial particle density $n_{0}$, and are related to the persistence exponent $θ$ through the scaling relation $z(2-τ)=θ$. We show by rigorous analyt…
In this paper, we present the detailed calculation of the persistence exponent $θ$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $θ$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $θ$ for non-Gaussian processes, by making a strong connection between the problem of pe…
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…
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…
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…
Very persistent random walkers reveal transitions in landscape topology
ICTP South American Institute for Fundamental Research（国际理论物理中心南美基础研究所）；Instituto de Física Teórica, Universidade Estadual Paulista “Júlio de Mesquita Filho”（圣保罗州立大学朱利奥·德·梅斯基塔·菲略教授理论物理研究所）
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)…
We develop an approach for performing scaling analysis of $N$-step Random Walks (RWs). The mean square end-to-end distance, $\langle\vec{R}_{N}^{2}\rangle$, is written in terms of inner persistence lengths (IPLs), which we define by the ensemble averages of dot products between the walker's position and displacement vectors, at the $j$-th step. For RW models statistically invariant under orthogonal transformations, we analytically introduce a relation between $\langle\vec{R}_{N}^{2}\rangle$ and the persistence length, $λ_{N}$, which is defined as the mean end-to-end vector projection in the fi…
We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times -- leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential and an algebraic decay. Analytical calculations and numerical simulations show, that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is d…
The time distribution of relaxation events in an aging system is investigated via molecular dynamics simulations. The focus is on the distribution functions of the first passage time, $p_1(Δt)$, and the persistence time, $p(τ)$. In contrast to previous reports, both $p_1$ and $p$ are found to evolve with time upon aging. The age dependence of the persistence time distribution is shown to be sensitive to the details of the algorithm used to extract it from particle trajectories. By updating the reference point in event detection algorithm and accounting for the event specific aging time, we unc…
We study the dynamics of a deterministic walk confined in a narrow two-dimensional space randomly filled with point-like targets. At each step, the walker visits the nearest target not previously visited. Complex dynamics is observed at some intermediate values of the domain width, when, while drifting, the walk performs long intermittent backward excursions. As the width is increased, evidence of a transition from ballistic motion to a weakly non-ergodic regime is shown, characterized by sudden inversions of the drift velocity with a probability slowly decaying with time, as $1/t$ at leading …
We study the motion of a random walker in one longitudinal and d transverse dimensions with a quenched power law correlated velocity field in the longitudinal x-direction. The model is a modification of the Matheron-de Marsily (MdM) model, with long-range velocity correlation. For a velocity correlation function, dependent on transverse co-ordinates y as 1/(a+|{y_1 - y_2}|)^alpha, we analytically calculate the two-time correlation function of the x-coordinate. We find that the motion of the x-coordinate is a fractional Brownian motion (fBm), with a Hurst exponent H = max [1/2, (1- alpha/4), (1…
Exact collective first-passage statistics of N trail-interacting walkers
Laboratoire Jean Perrin, CNRS/Sorbonne Université（让·佩兰实验室，法国国家科学研究中心/索邦大学）；Collège de France（法兰西公学院）；Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université（凝聚态物质理论物理实验室，法国国家科学研究中心/索邦大学）
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…
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…
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…
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 …
Among observables characterising the random exploration of a graph or lattice, the cover time, namely the time to visit every site, continues to attract widespread interest. Much insight about cover times is gained by mapping to the (spaceless) coupon-collector problem, which amounts to ignoring spatio-temporal correlations, and an early conjecture that the limiting cover time distribution of regular random walks on large lattices converges to the Gumbel distribution in $d \geq 3$ was recently proved rigorously. Furthermore, a number of mathematical and numerical studies point to the robustnes…
