We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.
The asymptotic mean number of distinct sites visited by a subdiffusive continuous time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableaux of results. In this tableaux we include the dominant as well as subdominant contributions in all integer dimensions. Other quantities that can be calcula…
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0,t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the …
Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and cover time are known. However, many recent applications such as search engines and recommender systems involve multiple random walkers on complex networks. In this work, based on numerical simulations, we show that the fraction of nodes of scale-free network not visited by $W$ random walkers in time $t$ has a stretched exponential form independent of the de…
We study the linear large $n$ behavior of the average number of distinct sites $S(n)$ visited by a random walker after $n$ steps on a large random graph. An expression for the graph topology dependent prefactor $B$ in $S(n) = Bn$ is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connect…
The evaluation of the average number S_N(t) of distinct sites visited up to time t by N independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for large N, S_N(t) \approx \hat S_N(t) (1-Δ), where \hat S_N(t) is the volume of a hypersphere of radius (4Dt \ln N)^{1/2}, Δ={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N, d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the dimension and time. The first three terms of these series are calculated explicitly a…
We study an Eulerian walker on a square lattice, starting from an initially randomly oriented background using Monte Carlo simulations. We present evidence that, that, for large number of steps $N$, the asymptotic shape of the set of sites visited by the walker is a perfect circle. The radius of the circle increases as $N^{1/3}$, for large $N$, and the width of the boundary region grows as $N^{α/ 3}$, with $α= 0.40 \pm .05$. If we introduce stochasticity in the evolution rules, the mean square displacement of the walker, $<R_{N}^{2}> \sim N^{2ν}$, shows a crossover from the Eulerian ($ν= 1/3$)…
Majumdar and Tamm [Phys. Rev. E 86 021135 (2012), arXiv:1206.6184] recently obtained analytical expressions for the mean number of common sites W_N(t) visited up to time t by N independent random walkers starting from the origin of a d-dimensional lattice. In this short note I show how the different regimes and the corresponding asymptotic power laws can be retrieved using the notion of fractal intersection.
We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disa…
We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large t growth of W_N(t). These three regimes are separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane. For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\sim t^ν where the exponent ν= N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a consta…
Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between rand…
Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding $μ$ steps. We have obtained analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited in this open system, with $N \gg μ\gg 1$. The expression for $P_N(μ)$ evaluated in the mentioned limit is valid even for sm…
We address the problem of evaluating the number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all initially placed on one site of a deterministic fractal lattice. For a wide class of fractals, of which the Sierpinski gasket is a typical example, we propose that, after the short-time compact regime and for large N, $S_N(t) \approx \hat{S}_N(t) (1-Δ)$, where $\hat{S}_N(t)$ is the number of sites inside a hypersphere of radius $R [\ln (N)/c]^{1/ u}$, R is the root-mean-square displacement of a single random walker, and u and c determine how fast $1-Γ_t({\bf r}…
Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point which has not been visited in the preceding $μ$ steps. Using open boundary conditions, we have calculated analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited, with $N \gg μ\gg 1$. This approximated expression for $P_N(μ)$ is reasonable even for sma…
The average number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity $S_N(t)$ is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or…
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}/|…
We study diffusive lattice gases with local injection of particles, namely we assume that whenever the origin becomes empty, a new particle is immediately injected into the origin. We consider two lattice gases: a symmetric simple exclusion process and random walkers. The interplay between the injection events and the positions of the particles already present implies an effective collective interaction even for the ostensibly non-interacting random walkers. We determine the average total number of particles entering into the initially empty system. We also compute the average total number of …
The class of random walks in one dimension, returning to the origin, restricted by the requirement that any site visited (different from the origin) is visited an even number of times, is analyzed in the present note. We call this class the even-visiting random walks and provide a closed expression to evaluate them.
The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of $s$ and $r$, where $r$ is the number of sites visited only once up to time $t$. Mean values, variances and covariance are deduced from the generating functions and their finite-size-scaling behaviour is studied. Introducing properly centered and scaled varia…
We consider a system of unbiased diffusing walkers (A0 <-> 0A) in one dimension with random initial conditions. We investigate numerically the relation between the fraction of walkers, U(t), which have never encountered another walker up to time t, calling such walkers ``uninfected'' and the fraction of sites, P(t), which have never been visited by a diffusing particle. We extend our study to include the A + B -> 0 diffusion-limited reaction in one-dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) \simeq [P(t)]^γ, with γ\simeq 1.39, in …
Random walks with memory typically involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the statistics of first-passage and subsequent recurrence times through a site; typically, a preference for revisiting sites is expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in [Phys. Rev. Lett. 112, 240601] to explore this effect, tog…
Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in $d$-dime…
We consider a random walk on the fully-connected lattice with $N$ sites and study the time evolution of the number of distinct sites $s$ visited by the walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a record time $t$ when the walker visits a site of the subset for the first time. The record time $t$ is a partial covering time when $v<n$ and a total covering time when $v=n$. The probability distributions for the number of records $s$, the record value $v$ and the record (covering) time $t$, involving $r$-Stirling numbers, are obtained using generating function tech…
Consider a random medium consisting of points randomly distributed so that there is no correlation among the distances. This is the random link model, which is the high dimensionality limit (mean field approximation) for the euclidean random point structure. In the random link model, at discrete time steps, the walker moves to the nearest site, which has not been visited in the last $μ$ steps (memory), producing a deterministic partially self avoiding walk (the tourist walk). We have obtained analitically the distribution of the number $n$ of points explored by a walker with memory $μ= 2$, as …
We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. Th…
We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N \geq 1$, for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average val…
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…
Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random walk properties may be profound. For instance, whereas the average number of distinct sites visited by an immortal walker grows with time without bound, that of a mortal walker may, depending on dimensionality and rate of evanescence, remain finite or keep growing with the passage of time. This number can in turn be used to calculate other classic quantities s…
We study the starvation of a lattice random walker in which each site initially contains one food unit and the walker can travel $\mathcal{S}$ steps without food before starving. When the walker encounters food, the food is completely eaten, and the walker can again travel $\mathcal{S}$ steps without food before starving. When the walker hits an empty site, the time until the walker starves decreases by 1. In spatial dimension $d=1$, the average lifetime of the walker $<τ>\propto \mathcal{S}$, while for $d > 2$, $<τ>\simeq\exp(\mathcal{S}^ω)$, with $ω\to 1$ as $d\to\infty$. In the marginal cas…
We consider random walkers that deform the medium as they move, enabling a faster motion in regions which have been recently visited. This induces an effective attraction between walkers mediated by the medium, which can be regarded as a space metric, giving rise to a statistical mechanics toy model either for gravity, motion through deformable matter or adaptable geometry. In the strong-deformability regime, we find that diffusion is initially described by the porous medium equation, thus yielding subdiffusive behavior of an initially localized cloud of particles. Indeed, while the average wi…
The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for independent walkers, where any number of walkers can occupy the same site, and for walkers with an exclusion interaction, when no site can contain more than one walker. These analytical results are then compared with numerical simulations, showing very good agreement.
A random walk starts from the origin of a d-dimensional lattice. The occupation number n(x,t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sums M(t) of observables defined locally on the field of occupation numbers. Examples are the number S(t) of visited sites; the area E(t) of the (appropriately defined) surface of the set of visited sites; and, in dimension d=3, the Euler index of this surface. In d > 3, the averages <M>(t) all increase linearly with t as t-->infinity. We show that in d=3, to leading order in an as…
Consider a Brownian particle in three dimensions which is attracted by a plane with a strength proportional to some dimensionless parameter $α$. We investigate the fractal spatial structure of the visited lattice sites in a cubic lattice by the particle around and on the attractive plane. We compute the fractal dimensions of the set of visited sites both in three dimensions and on the attractive plane, as a function of the strength of attraction $α$. We also investigate the scaling properties of the size distribution of the clusters of nearest-neighbor visited sites on the attractive plane, an…
We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability p_c, [PRL 99, 180602 (2007)]. The behaviour of rho(l), the probability of formation of fragments of length l is analysed. An exact expression for the generating function of rho(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form rho(l) ~ 3/[l(l…
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…
The Riemann walk is the lattice version of the Levy flight. For the one-dimensional Riemann walk of Levy exponent 0<α<2 we study the statistics of the support, i.e. the set of visited sites, after t steps. We consider a wide class of support related observables M(t), including the number S(t) of visited sites and the number I(t) of sequences of visited sites. For t->\infty we obtain the asymptotic power laws for the averages, variances, and correlations of these observables. Logarithmic correction factors appear for α=2/3 and α=1. Bulk and surface observables have different power laws for 1\le…
We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and steps of length n when entering a region that has never been visited. The process provides a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, with delta = v/2n. Our expressi…
The first passage statistics of a continuous time random walker with Poisson distributed jumps on one and two dimensional infinite lattices is investigated. An exact expression for the probability of first return to the origin in one dimension is derived for a symmetric random walker as well as a biased random walker. The Laplace transform of the occupation probability of a site for a symmetric random walker on a two dimensional lattice is identified with the lattice Green's function for a square lattice. This allows computation of the exact first passage distribution to any arbitrary site on …
We study the dynamics of a starving random walk in general spatial dimension $d$. This model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources. The forager depletes its environment by consuming resources and dies if it wanders too long without finding food. In the exactly-solvable case of one dimension, we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation. We also give a heuristic derivation for the ave…
How much time does it take for a fluctuating system, such as a polymer chain, to reach a target configuration that is rarely visited -- typically because of a high energy cost ? This question generally amounts to the determination of the first-passage time statistics to a target zone in phase space with lower occupation probability. Here, we present an analytical method to determine the mean first-passage time of a generic non-Markovian random walker to a rarely visited threshold, which goes beyond existing weak-noise theories. We apply our method to polymer systems, to determine (i) the first…
We study the motion of random walkers with residence time bias between first and subsequent visits to a site, as a model for synthetic molecular walkers composed of coupled DNAzyme legs known as molecular spiders. The mechanism of the transient superdiffusion has been explained via the emergence of a boundary between the new and the previously visited sites, and the tendency of the multi-legged spider to cling to this boundary, provided residence time for a first visit to a site is longer than for subsequent visits. Using both kinetic Monte Carlo simulation and an analytical approach, we model…
A variation of Rosenstock's trapping model in which $N$ independent random walkers are all initially placed upon a site of a one-dimensional lattice in the presence of a {\em one-sided} random distribution (with probability $c$) of absorbing traps is investigated. The probability (survival probability) $Φ_N(t)$ that no random walker is trapped by time $t$ for $N \gg 1$ is calculated by using the extended Rosenstock approximation. This requires the evaluation of the moments of the number $S_N(t)$ of distinct sites visited in a {\em given} direction up to time $t$ by $N$ independent random walke…
We study the effect of a large obstacle on the so called residence time, i.e., the time that a particle performing a symmetric random walk in a rectangular (2D) domain needs to cross the strip. We observe a complex behavior, that is we find out that the residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle--free strip. We explain the residence time behavior by developing a 1D analog …
Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven dimensions, at the percolation threshold with L^7 sites and L < 420, we confirm the expected time-dependence of the end-to-end distance (including the corrections to the asymptotic behavior).
We consider the one-dimensional target search process that involves an immobile target located at the origin and $N$ searchers performing independent Brownian motions starting at the initial positions $\vec x = (x_1,x_2,..., x_N)$ all on the positive half space. The process stops when the target is first found by one of the searchers. We compute the probability distribution of the maximum distance $m$ visited by the searchers till the stopping time and show that it has a power law tail: $P_N(m|\vec x)\sim B_N (x_1x_2... x_N)/m^{N+1}$ for large $m$. Thus all moments of $m$ up to the order $(N-1…
We consider two random walkers starting at the same time $t=0$ from different points in space separated by a given distance $R$. We compute the average volume of the space visited by both walkers up to time $t$ as a function of $R$ and $t$ and dimensionality of space $d$. For $d<4$, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable $R/\sqrt{t}$. We provide general integral formulas for scaling functions for arbitrary dimensionality $d<4$. In contrast, we show that no scaling function exists for higher dimensionalities $d \geq 4$
An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the same site of the substrate. Three classes of problems are considered: (i) the evaluation of the average number <S_N(t)> of distinct sites visited (territory explored) up to time t by the N random walkers, (ii) the statistical description of the first passage time t_{j,N} to a given distance of the first j random walkers (order statistics of exit times), and (…
We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with probability $(1-r)$, it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability $(1-r)/2$. For $r=0$, it reduces to a standard random walk whose typical distance grows as $\sqrt{n}$ for large $n$. In presence of a nonzero resetting rate $0<r\le 1$, we find that both the average maximum and the average position grow ba…
A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the XX Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called "vicious" random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the XX magnet, can be interpreted as the generating function of paths made by a single walker in a med…
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. The extent of this spatial exploration characterizes many important physical, chemical, and ecological phenomena. In spite of its fundamental interest and wide utility, the number of visited sites gives only an incomplete picture of this exploration. In this work, we introduce a more fundamental quantity, the elapsed time $τ_n$ between visits to the $n^{\rm th}$ and the $(n+1)^{\rm st}$ distinct sites, from which the full dynamics abou…
