Discrete-time Random Walk Research Articles

Cost of excursions until first crossing of the origin for random walk and Lévy flights: An exact general formula

We consider a discrete-time random walk on a line starting at x0≥0 where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy and light tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations. Published by the American Physical Society 2024

Brownian motion in a vector space over a local field is a scaling limit

For any natural number d, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a d-dimensional vector space V over a local field K. Just as the Laplace operator on L2(Rd) is the infinitesimal generator of Brownian motion with state space Rd, the Vladimirov–Taibleson operator on L2(V) is the infinitesimal generator of real-time Brownian motion with state space V. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted V to be the p-adic numbers.

Elephant Random Walk with a Random Step Size and Gradually Increasing Memory and Delays

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. In this work, we investigate the asymptotic normality of the ERW model with a random step size and gradually increasing memory and delays. In particular, we extend some recent results in this subject.

Optimizing leapover lengths of Lévy flights with resetting.

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at b≥0 and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability r of restarting the walk from the initial position. The efficiency of a "myopic search"-in which the search stops upon crossing the target for the first time-is usually characterized in terms of the first-passage time τ. On the other hand, great relevance is encapsulated by the leapover length l=x_{τ}-b, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover ℓ_{b}(r) if the mean jump length is finite. We compute exactly ℓ_{b}(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r^{*} such that ℓ_{b}(r^{*}) is minimal and smaller than the average leapover of the single jump.

Non-uniformly sampled simulated price impact of an order-book

We extend a Discrete Time Random Walk (DTRW) numerical scheme to simulate the anomalous diffusion of financial market orders in a simulated order book. Here using random walks with Sibuya waiting times to include a time-dependent stochastic forcing function with non-uniformly sampled times between order book events in the setting of fractional diffusion. This models the fluid limit of an order book by modelling the continuous arrival, cancellation and diffusion of orders in the presence of information shocks. We study the impulse response and stylised facts of orders undergoing anomalous diffusion for different forcing functions and model parameters. Concretely, we demonstrate the price impact for flash limit-orders and market orders and show how the numerical method generates kinks in the price impact. We use cubic spline interpolation to generate smoothed price impact curves. The work promotes the use of non-uniform sampling in the presence of diffusive dynamics as the preferred simulation method.

P-adic Brownian motion is a scaling limit

A p-adic Brownian motion is a continuous time stochastic process in a p-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1,∞) , and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

Fast Rare Events in Exit Times Distributions of Jump Processes.

Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat-tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology, and finance: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on timescales orders of magnitude smaller than the typical timescale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations.

Fluid flow and solute transport simulations in tight geologic formations: Discrete fracture network and continuous time random walk analyses

Understanding the effect of fracture network heterogeneity on flow and transport in tight formations with low permeability values is of great importance in a wide variety of applications, such as groundwater pollution and storage as well as water production. Although numerous studies have investigated solute transport in fractured media, investigation about scale effects on flow and transport in low-permeability aquifers is limited. More specifically, we are not aware of any study that has addressed how the continuous time random walk (CTRW) model parameters vary with scale and fracture network properties. To this end, we performed a set of numerical simulations in sparse fracture networks of sizes L = 20, 35, and 50 m under two different volumetric fracture counts (p30 = 0.05 and 0.1) using the discrete fracture network approach. The fractures in the networks were elliptical in shape, and their radii followed a truncated power-law distribution with exponents α = 1.5, 2, and 2.5. We simulated fluid flow using the Reynolds equation, and solute transport was simulated using particle tracking. The solute transport behavior was quantified by fitting the solution of the CTRW transport equation to the arrival time distributions. Results showed that as the truncated power law exponent α increased, the permeability of the networks decreased. In all the fracture networks studied, we found non-Fickian solute transport behavior deduced from small values of β(0.69≤β≤1.25), the exponent in the truncated power-law arrival time distribution in the CTRW model. In our fracture networks, the aperture size spanned less than one order of magnitude meaning that the fracture permeability distribution was relatively narrow. Therefore, the observed non-Fickian behavior should be attributed to sparse fracture networks and low connectivity. We found that the effective permeability, keff, at p30 = 0.05 was more scale-dependent compared to that at p30 = 0.1 because the fracture networks with p30 = 0.05 were less connected and, thus, more heterogeneous. Our simulation also disclosed the scale dependence of the CTRW model parameters. We conducted multiple linear regression analysis to link the CTRW model parameters to the network size as well as the geometrical and topological properties of the network. Results of regression analysis showed that all the CTRW model parameters except the average solute velocity were significantly dependent on the system size L (p-value < 0.05). Further investigations using a broader range of fracture network properties, such as broader aperture size distributions, smaller and larger network sizes and volumetric fracture counts are still required.

Probing the large deviations for the beta random walk in random medium.

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

Large deviations at level 2.5 and for trajectories observables of diffusion processes: the missing parts with respect to their random-walks counterparts

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension d involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing b; (ii) discrete-time random walks in continuous space with a small time-step τ. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits b → 0 and τ → 0 respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits b → 0 and τ → 0 for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

Optimizing the random search of a finite-lived target by a Lévy flight.

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

Gaussian fluctuations of the elephant random walk with gradually increasing memory

The elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p∈[0,1] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmüller (2022 Stat. Probab. Lett. 189 109598).

Optimizing search processes with stochastic resetting on the pseudofractal scale-free web.

The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.

On Random Walks with Geometric Lifetimes

We consider the following stochastic process. Initially, there is a single particle placed at the origin of Z . This particle moves as a discrete-time simple random walk on Z , which takes one step to the right with probability p. Furthermore, the particle has a probability of death ( 1 − α ) before each step. Whenever this particle returns to the origin, it gives birth to a new particle, which performs the same dynamics. For which values of the pair ( p , α ) does this model survive with positive probability? Can we obtain the survival probability as a function of ( p , α ) ? By answering these questions, we review some fundamental concepts and tools in probability, such as the return times of a simple random walk on the integers, probability generating functions, branching processes, and coupling.

Parrondo-Like Behavior in Continuous-Time Random Walks with Periodically Alternating Jumps

As a natural generalization of the discrete-time random walk, the continuous-time random walk (CTRW) has been applied to stochastic models with random dynamics in various fields. In this paper, we show that the deterministic alternation of two unbiased CTRWs can lead to a phenomenon similar to the Parrondo paradox, in which the asymptotic mean drift of the combined CTRW becomes positive or negative depending on the parameter values. This extends the case in which the paradox occurs due to the random combination of two CTRWs with memory shown by Montero [Phys. Rev. E 84 (2011) 051139].

Harmonic functions for singular quadrant walks

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions (i,j) with i+j≥0 and small negative jumps, i.e., i,j≥−1. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.

Inverse Problems for Discrete Heat Equations and Random Walks for a Class of Graphs

We study the inverse problem of determining a finite weighted graph from the source-to-solution map on a vertex subset for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on . In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on , or from the observation on of one realization of the random walk.

On Coalescence Time in Graphs: When Is Coalescing as Fast as Meeting?

Coalescing random walks is a fundamental distributed process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as that of Cooper et al., the coalescence time for graphs, such as binary trees, d -dimensional tori, hypercubes, and, more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log 2 n ), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. Finally, we prove a tight worst-case bound for the coalescence time of O(n 3 ) . By duality, our results yield identical bounds on the voter model. Our techniques also yield a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin, as well as those by Aldous and Fill.

Rate of moment convergence in the central limit theorem for the elephant random walk

The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Schütz and Trimper (2004 Phys. Rev. E 70 045101). It has a parameter : the case α = 0 corresponds to the simple symmetric random walk, and when α > 0 (resp. α < 0), the mean displacement of the walker at time n grows (resp. vanishes) like n α . The walk admits a phase transition at from the diffusive behavior to the superdiffusive behavior. In this paper, we study the rate of the moment convergence in the central limit theorem for the position of the walker when . We find a crossover phenomenon in the rate of convergence of the 2mth moments with inside the diffusive regime .

Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.