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 of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations.
We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time $n$. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e., independent of the step length distribution. We analyze this survival probability at depth, deriving both exact …
We analyze the dynamics of random walks in which the jumping probabilities are periodic {\it time-dependent} functions. In particular, we determine the survival probability of biased walkers who are drifted towards an absorbing boundary. The typical life-time of the walkers is found to decrease with an increment of the oscillation amplitude of the jumping probabilities. We discuss the applicability of the results in the context of complex adaptive systems.
We analyze the dynamics of random walks with long-term memory (binary chains with long-range correlations) in the presence of an absorbing boundary. An analytically solvable model is presented, in which a dynamical phase-transition occurs when the correlation strength parameter μreaches a critical value μ_c. For strong positive correlations, μ> μ_c, the survival probability is asymptotically finite, whereas for μ< μ_c it decays as a power-law in time (chain length).
We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length $\ell$ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance $L$. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate o…
We study a one-dimensional lattice random walk with an absorbing boundary at the origin and a movable partial reflector. On encountering the reflector, at site x, the walker is reflected (with probability r) to x-1 and the reflector is simultaneously pushed to x+1. Iteration of the transition matrix, and asymptotic analysis of the probability generating function show that the critical exponent delta governing the survival probability varies continuously between 1/2 and 1 as r varies between 0 and 1. Our study suggests a mechanism for nonuniversal kinetic critical behavior, observed in models w…
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity v_c of the wall with an essential singularity and we characterize the divergences of the relaxation times for v<v_c and v>v_c. At v=v_c the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distrib…
We consider $N$ Brownian particles moving on a line starting from initial positions ${\bf{u}}\equiv \{u_1,u_2,\dots u_N\}$ such that $0<u_1 < u_2 < \cdots < u_N$. Their motion gets stopped at time $t_s$ when either two of them collide or when the particle closest to the origin hits the origin for the first time. For $N=2$, we study the probability distribution function $p_1(m|{\bf{u}})$ and $p_2(m|{\bf{u}})$ of the maximal distance travelled by the $1^{\text{st}}$ and $2^{\text{nd}}$ walker till $t_s$. For general $N$ particles with identical diffusion constants $D$, we show that the probabili…
We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with fractal dimension $d_a<d$ and are characterized by the absorption rate $w_a$ which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps ($w_a\ll w$), we find that $f(t)$ can be closely approximated by a stretched exponential function over the ini…
Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics
Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert（索邦大学，法国国家科学研究中心，让·勒朗·达朗贝尔研究所）；Departamento de Física, Universidad Nacional de Colombia（哥伦比亚国立大学生物物理系）
We review recent studies demonstrating a nonuniversal (continuously variable) survival exponent for history-dependent random walks, and analyze a new example, the hard movable partial reflector. These processes serve as a simplified models of infection in a medium with a history-dependent susceptibility, and for spreading in systems with an infinite number of absorbing configurations. The memory may take the form of a history-dependent step length, or be the result of a partial reflector whose position marks the maximum distance the walker has ventured from the origin. In each case, a process …
We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival pro…
We calculate the survival probability of a diffusing test particle in an environment of diffusing particles that undergo coagulation at rate lambda_c and annihilation at rate lambda_a. The test particle dies at rate lambda' on coming into contact with the other particles. The survival probability decays algebraically with time as t^{-theta}. The exponent theta in d<2 is calculated using the perturbative renormalization group formalism as an expansion in epsilon=2-d. It is shown to be universal, independent of lambda', and to depend only on delta, the ratio of the diffusion constant of test par…
Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution of organisms' sensor-based information processing and motility. The complexity of the first component is reflected in the statistical characteristics of trajectories produced by an organism -- when it is, e.g., foraging or searching for a mate (or a pathogen) -- and therefore some knowledge about the component can be obtained by analyzing the trajectories wit…
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 …
We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(η)$, characterized by a Lévy index $μ\in (0,2]$, which includes standard random walks ($μ=2$) and Lévy flights ($0<μ<2$). We study the survival probability, $q(x_0,n)$, representing the probability that the RW stays non-negative up to step $n$, starting initially at $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$ varies: (i) for $x_0= O(1)$ ("quantum regime"), the discreteness of the jump process…
The asymptotic behaviour of the survival or reunion probability of vicious walks with short-range interactions is generally well studied. In many realistic processes, however, walks interact with a long ranged potential that decays in $d$ dimensions with distance $r$ as $r^{-d-σ}$. We employ methods of renormalized field theory to study the effect of such long range interactions. We calculate, for the first time, the exponents describing the decay of the survival probability for all values of parameters $σ$ and $d$ to first order in the double expansion in $ε=2-d$ and $δ=2-d-σ$. We show that t…
We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability $0\leq p \leq 1$, while with the complementary probability $1-p$, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution $f_0(η)$. We have shown that, for arbitrary $p$, the statistics of records up to step $N$ is completely universal, i.e., independent of $f_0(η)$ for any $N$. We also compute the connected two-time correlation function $C_p(m_1, m_2)$ of the record-breaking even…
We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line. At each time step, the walker jumps by a length $η$ drawn independently from a symmetric and continuous distribution $f(η)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelate…
The characterization of record events is considered for a discrete-time random walk model with long-term memory arising from correlations between successive steps. An important feature is that the correlations are strong enough to give rise to super-diffusivity and transience. Various quantities related to record statistics are calculated exactly, highlighting important differences in behaviour from the simple random walk with independent steps.
We address the theory of records for integrated random walks with finite variance. The long-time continuum limit of these walks is a non-Markov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated…
In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power-law, in contrast to the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.
We investigate the statistics of three kinds of records associated with planar random walks, namely diagonal, simultaneous and radial records. The mean numbers of these records grow as universal power laws of time, with respective exponents 1/4, 1/3 and 1/2. The study of diagonal and simultaneous records relies on the underlying renewal structure of the successive hitting times and locations of translated copies of a fixed target. In this sense, this work represents a two-dimensional extension of the analysis made by Feller of ladder points, i.e., records for one-dimensional random walks. This…
We revisit the statistics of extremes and records of symmetric random walks with stochastic resetting, extending earlier studies in several directions. We put forward a diffusive scaling regime (symmetric step length distribution with finite variance, weak resetting probability) where the maximum of the walk and the number of its records up to discrete time $n$ become asymptotically proportional to each other for single typical trajectories. Their distributions obey scaling laws ruled by a common two-parameter scaling function, interpolating between a half-Gaussian and a Gumbel law. The exact …
We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful "laboratory" to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statist…
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…
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…
We study survival time statistics in a noisy sample space reducing (SSR) process. Our simulations suggest that both the mean and standard deviation scale as $\sim N/N^λ$, where $N$ is the system size and $λ$ is a tunable parameter that characterizes the process. The survival time distribution has the form $\mathcal{P}_{N}(τ)\sim N^{-θ}J(τ/N^θ)$, where $J$ is a universal scaling function and $θ= 1-λ$. Analytical insight is provided by a conjecture for the equivalence between the survival time statistics in the noisy SSR process and the record statistics in a correlated time series modeled as dr…
The study of record statistics of correlated series is gaining momentum. In this work, we study the records statistics of the time series of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent $α$ lying in the range $1.5 \le α\le 1.8$. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that from the empirical stock data.
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…
It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution grows as the sqrt(4N/pi) while the standard deviation grows as sqrt((2-4/pi) N), so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt(N/pi) and 0.626508... N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.
Random walk on $\mathbb{N}$ with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd) $ν_c$. We study a Fleming-Viot(FV) particle system driven by this process and show that mean normalized densities of the FV unique stationary measure converge to the minimal qsd, $ν_0$, as $N \to \infty$. Furthermore, every other qsd of the random walk ($ν_c$, $c>0$) corresponds to a metastable state of the FV particle system.
We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate $γ$, the (new) velocity of the particle is drawn randomly from a distribution $W(v)$. We study the survival probability $S(x,t)$ of a particle starting from $x \geq 0$ up to time $t$ and obtain an explicit expression for its double Laplace transform (with respect to both $x$ and $t$) for an arbitrary velocity distribution $W(v)$, not necessarily symmetric. This result is obtained as a consequ…
We study the record statistics of random walks after $n$ steps, $x_0, x_1,\ldots, x_n$, with arbitrary symmetric and continuous distribution $p(η)$ of the jumps $η_i = x_i - x_{i-1}$. We consider the age of the records, i.e. the time up to which a record survives. Depending on how the age of the current last record is defined, we propose three distinct sequences of ages (indexed by $α$ = I, II, III) associated to a given sequence of records. We then focus on the longest lasting record, which is the longest element among this sequence of ages. To characterize the statistics of these longest las…
The statistics of records for a time series generated by a continuous time random walk is studied, and found to be independent of the details of the jump length distribution, as long as the latter is continuous and symmetric. However, the statistics depend crucially on the nature of the waiting time distribution. The probability of finding M records within a given time duration t, for large t, has a scaling form, and the exact scaling function is obtained in terms of the one-sided Levy stable law. The mean of the ages of the records, defined as <t/M>, differs from t/<M>. The asymptotic behavio…
In this work we investigate the dynamics of random walk processes on scale-free networks in a short to moderate time scale. We perform extensive simulations for the calculation of the mean squared displacement, the network coverage and the survival probability on a network with a concentration $c$ of static traps. We show that the random walkers remain close to their origin, but cover a large part of the network at the same time. This behavior is markedly different than usual random walk processes in the literature. For the trapping problem we numerically compute $Φ(n,c)$, the survival probabi…
The statistics of persistent events, recently introduced in the context of phase ordering dynamics, is investigated in the case of the 1D lattice random walk in discrete time. We determine the survival probability of the random walker in the presence of an obstacle moving ballistically with velocity v, i.e., the probability that the walker remains up to time n on the left of the obstacle. Three regimes are to be considered for the long-time behavior of this probability, according to the sign of the difference between v and the drift velocity V of the random walker. In one of these regimes (v>V…
Continuous Time Random Walks (CTRW) are widely used to coarse-grain the evolution of systems jumping from a metastable sub-set of their configuration space, or trap, to another via rare intermittent events. The multi-scaled behavior typical of complex dynamics is provided by a fat-tailed distribution of the waiting time between consecutive jumps. We first argue that CTRW are inadequate to describe macroscopic relaxation processes for three reasons: macroscopic variables are not self-averaging, memory effects require an all-knowing observer,and different mechanisms whereby the jumps affect macr…
We present a systematic analytical approach to the trapping of a random walk by a finite density rho of diffusing traps in arbitrary dimension d. We confirm the phenomenologically predicted e^{-c_d rho t^{d/2}} time decay of the survival probability, and compute the dimension dependent constant c_d to leading order within an eps=2-d expansion.
We study the statistics of increments in record values in a time series $\{x_0=0,x_1, x_2, \ldots, x_n\}$ generated by the positions of a random walk (discrete time, continuous space) of duration $n$ steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of $n$ for large $n$, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability $Q(n)$ that the record increments decrease monotonically up to step $n$. Remarkably, $Q(n)$ is univer…
While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed fro…
"All misfortune of man comes from the fact that he does not stay peacefully in his room", has once asserted Blaise Pascal. In the present paper we evoke this statement as the "Pascal principle" in regard to the problem of survival of an "A" particle, which performs a lattice random walk in presence of a concentration of randomly moving traps "B", and gets annihilated upon encounters with any of them. We prove here that at sufficiently large times for both perfect and imperfect trapping reactions, for arbitrary spatial dimension "d" and for a rather general class of random walks, the "A" partic…
The kinetics of annihilating random walks in one dimension, with the half-line x>0 initially filled, is investigated. The survival probability of the nth particle from the interface exhibits power-law decay, S_n(t)~t^{-alpha_n}, with alpha_n approximately equal to 0.225 for n=1 and all odd values of n; for all n even, a faster decay with alpha_n approximately equal to 0.865 is observed. From consideration of the eventual survival probability in a finite cluster of particles, the rigorous bound alpha_1<1/4 is derived, while a heuristic argument gives alpha_1 approximately equal to 3 sqrt{3}/8 =…
We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate $P_n(c)$, defined as the probability for the $n$th value to be larger than all previous values, for a Gaussian jump distribution with standard deviation $σ$ that is shifted by a constant drift $c$. For…
The problem of how many trajectories of a random walker in a potential are needed to reconstruct the values of this potential is studied. We show that this problem can be solved by calculating the probability of survival of an abstract random walker in a partially absorbing potential. The approach is illustrated on the discrete Sinai (random force) model with a drift. We determine the parameter (temperature, duration of each trajectory, ...) values making reconstruction as fast as possible.
We present a unified framework for first-passage time and residence time of random walks in finite one-dimensional disordered biased systems. The derivation is based on exact expansion of the backward master equation in cumulants. The dependence on initial condition, system size, and bias strength is explicitly studied for models with weak and strong disorder. Application to thermally activated processes is also developed.
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…
We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional nonhomogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum …
We investigate the first passage time t_{j,N} to a given chemical or Euclidean distance of the first j of a set of N>>1 independent random walkers all initially placed on a site of a disordered medium. To solve this order-statistics problem we assume that, for short times, the survival probability (the probability that a single random walker is not absorbed by a hyperspherical surface during some time interval) decays for disordered media in the same way as for Euclidean and some class of deterministic fractal lattices. This conjecture is checked by simulation on the incipient percolation aggr…
We study the distribution of dynamical quantities in various one-dimensional, disordered models the critical behavior of which is described by an infinite randomness fixed point. In the {\it disordered contact process}, the quenched survival probability $\mathcal{P}(t)$ defined in fixed random environments is found to show multi-scaling in the critical point, meaning that $\mathcal{P}(t)=t^{-δ}$, where the (environment and time-dependent) exponent $δ$ has a universal limit distribution when $t\to\infty$. The limit distribution is determined by the strong disorder renormalization group method a…
We study random walk with adaptive move strategies on a class of directed graphs with variable wiring diagram. The graphs are grown from the evolution rules compatible with the dynamics of the world-wide Web [Tadić, Physica A {\bf 293}, 273 (2001)], and are characterized by a pair of power-law distributions of out- and in-degree for each value of the parameter $β$, which measures the degree of rewiring in the graph. The walker adapts its move strategy according to locally available information both on out-degree of the visited node and in-degree of target node. A standard random walk, on the o…
We address the question of distance record-setting by a random walker in the presence of measurement error, $δ$, and additive noise, $γ$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $δ$ and $γ$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $δ$ and $γ$. In the absence of noise ($γ=0$), the pre-factor $S(δ)$ is evaluated explicitly for arbitrary jump distributions …
We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of r…
