Single Random Walk Research Articles

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

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.

Kardar-Parisi-Zhang Physics and Phase Transition in a Classical Single Random Walker under Continuous Measurement.

We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate γ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a nontrivial effect on the probability distribution of the random walker. At small γ, we show analytically that the time evolution of the latter can be mapped to the stochastic heat equation. In this limit, the width of the log-probability thus follows a Family-Vicsek scaling law, N^{α}f(t/N^{α/β}), with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e., α_{KPZ}^{1D}=1/2 and β_{KPZ}^{1D}=1/3, respectively. When γ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents α_{M}^{1D}≈1 and β_{M}^{1D}≈1.4. In 3D, varying γ beyond a critical value γ_{M}^{c} leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson class to a new universality class with α_{M}^{3D}≈1.

Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

We consider continuous-time random walks on a random locally finite subset of mathbb {R}^d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group mathbb {G}=mathbb {R}^d or mathbb {G}=mathbb {Z}^d. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on mathbb {Z}^d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media

Flow and multicomponent reactive transport in saturated/unsaturated porous media are modeled by ensembles of computational particles moving on regular lattices according to specific random walk rules. The occupation number of the lattice sites is updated with a global random walk (GRW) procedure which spreads the particles from a lattice site with computational costs comparable to those for a single random walk step in sequential procedures. To cope with the nonlinearity and the degeneracy of the Richards equation the GRW flow solver uses linearization techniques similar to the L-scheme developed in finite element/volume approaches. Numerical schemes for reactive transport, coupled with the flow solver via numerical solutions for saturation and water flux, are implemented in splitting procedures. Diffusion–advection steps are solved by GRW algorithms using either biased or unbiased random walk probabilities. Since the number of particles in GRW simulations can be as large as the number of molecules involved in chemical reactions, one avoids the cumbersome problem of rescaling particle densities to approximate concentrations. Reaction steps are therefore formulated in terms of concentrations, as in deterministic approaches. The numerical convergence of the new schemes is demonstrated by comparisons with manufactured analytical solutions. Coupled flow and reactive transport problems of contaminant biodegradation described by the Monod model are further solved and the influence of flow nonlinearity/degeneracy and of the spatial heterogeneity of the medium is investigated numerically.

Exploring the predictability of cryptocurrencies via Bayesian hidden Markov models

In this paper, we consider a variety of multi-state hidden Markov models for predicting and explaining the Bitcoin, Ether and Ripple returns in the presence of state (regime) dynamics. In addition, we examine the effects of several financial, economic and cryptocurrency specific predictors on the cryptocurrency return series. Our results indicate that the non-homogeneous hidden Markov (NHHM) model with four states has the best one-step-ahead forecasting performance among all competing models for all three series. The dominance of the predictive densities over the single regime random walk model relies on the fact that the states capture alternating periods with distinct return characteristics. In particular, the four state NHHM model distinguishes bull, bear and calm regimes for the Bitcoin series, and periods with different profit and risk magnitudes for the Ether and Ripple series. Also, conditionally on the hidden states, it identifies predictors with different linear and non-linear effects on the cryptocurrency returns. These empirical findings provide important benefits for portfolio management and policy implementation.

Robotic Surveillance Based on the Meeting Time of Random Walks

This article analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate optimization problems and design robotic strategies. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of the transition matrices of the moving agents. This novel expression leads to necessary and sufficient conditions for the meeting time to be finite and to insightful graph-theoretic interpretations. Second, based on the closed-form expression, we set up and study the minimization problem for the expected capture time for a pursuer/evader pair. We report theoretical and numerical results on basic case studies to show the effectiveness of the design.

Biased Random Walk in Crowded Environment: Breaking Uphill/Downhill Symmetry of Transition Times.

Various natural processes can be analyzed using the concept of random walks. For a single random walker, the mean waiting times for uphill and downhill transitions between neighboring sites are equal. Here we investigate the uphill/downhill symmetry of waiting times for transitions of a tracer in crowded environment using exactly solvable one-dimensional stochastic models. It is found that, unexpectedly, the time to move in the direction of the bias (downhill) is always longer than the time to move against the bias (uphill). The degree of asymmetry depends on the particle density, the strength of the bias, and the size of the system. The microscopic origin of the symmetry breaking is discussed.

Smoluchowski flux and lamb-lion problems for random walks and Lévy flights with a constant drift

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index ) in the presence of a constant drift c. Initially these random walkers are uniformly distributed over the positive real line with a density . At the origin z = 0 there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles out of the system (the ‘Smoluchowski problem’) and (ii) the survival probability Sc(n) of the trap or lamb (the ‘lamb-lion problem’) until step n. We show that both observables can be expressed in terms of the average maximum of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both and Sc(n) analytically for large n in the two problems, for any value of and . In particular, for c > 0 and , we show that vanishes as , where is a -dependent positive constant, in contrast with the case of standard random walks (i.e. with ) for which . Our analytical results are confirmed by numerical simulations.

Distinct nodes visited by random walkers on scale-free networks

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 involving 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 number of walkers and the size of the network. This leads to a power-law relation between nodes not visited by W walkers and by 1 walker within time t. Thus the problem of finding the distinct nodes visited by W walkers, effectively, can be reduced to that of a single walker. The robustness of the results is demonstrated by verifying them on four different real-world networks that approximately display scale-free structure.

Models of infiltration into homogeneous and fractal porous media with localized sources.

We study a random-walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t^{D_{F}/D_{W}} (Alexander-Orbach relation), where D_{F} is the fractal dimension of the medium and D_{W} is the random-walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random-walk simulations. In a Menger sponge in which D_{F}>D_{W} (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media.

Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation

We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren.

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density in [J. Beltran, C. Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.

Two-particle problem in comblike structures

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comblike structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comblike architecture can strongly differ according to whether both reactants are moving (no matter their relative diffusivities) or only one is moving and according to the density of shortcuts among the branches.

Randomized interior point methods for sampling and optimization

We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a self-concordant barrier, whose mixing time from a \central is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under ane transformations of the convex set, thus eliminating the need for rst placing the body in an isotropic position. This strengthens and previous results of [11] for polytopes and generalizes these results to arbitrary convex sets. In the case of a convex set K dened by a semidenite constraint of rank at most and at most m additional linear constraints, our results specialize to the following statement. Let s jpj jqj for any chord pq of K passing through a point x2 K. Then, after t = O n(m +n ) n ln((m +n )s) + ln 1 steps are taken by a Dikin walk starting at x, the total variation distance and theL2 distance of the density (xt) of the point to the uniform density are less than . On every convex set of dimensionn, there exists a self-concordant barrier whose \complexity is polynomially bounded. Consequently, a rapidly mixing Markov chain of the kind we describe can be dened on any convex set. We use these results to design an algorithm consisting of a single random walk for optimizing a linear function on a convex set. We show that this random walk reaches an approximately optimal point in polynomial time with high probability and that the corresponding objective values converge with probability 1 to the optimal objective value as the number of steps tends to innity. One technical contribution is a family of lower

The Hitting Time of Multiple Random Walks

This work provides generalized notions and analysis methods for the hitting time of random walks on graphs. The hitting time, also known as the Kemeny constant or the mean first passage time, of a random walk is widely studied; however, only limited work is available for the multiple random walker scenario. In this work we provide a novel method for calculating the hitting time for a single random walker as well as the first analytic expression for calculating the hitting time for multiple random walkers, which we denote as the group hitting time. We also provide a closed form solution for calculating the hitting time between specified nodes for both the single and multiple random walker cases. Our results allow for the multiple random walks to be different and, moreover, for the random walks to operate on different subgraphs. Finally, using sequential quadratic programming, we show that the combination of transition matrices that generate the minimal group hitting time for various graph topologies is oft...

Erosion by a one-dimensional random walk.

We consider a model introduced by Baker etal. [Phys. Rev. E 88, 042113 (2013)] of a single lattice random walker moving on a domain of allowed sites, surrounded by blocked sites. The walker enlarges the allowed domain by eroding the boundary at its random encounters with blocked boundary sites: attempts to step onto blocked sites succeed with a given probability and convert these sites to allowed sites. The model interpolates continuously between the Pólya random walker on the one-dimensional lattice and a "blind" walker who attempts freely, but always aborts, moves to blocked sites. We obtain some exact results about the walker's location and the rate of erosion.

Slow encounters of particle pairs in branched structures.

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent; i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold, and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comblike structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we provide evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one-particle problem.

Random collisions on branched networks: How simultaneous diffusion prevents encounters in inhomogeneous structures

A huge variety of natural phenomena, including prey-predator interaction, chemical reaction kinetics, foraging, and pharmacokinetics, are mathematically described as encounters between entities performing a random motion on an appropriate structure. On homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. We prove here that this property does not hold on general inhomogeneous structures, and introduce the concept of two-particle transience, providing examples of realistic recurrent structures where two particles may never meet if they both move, while an encounter is certain if either stays put. We anticipate that our results will pave the way for the study of the effects of geometry in a wide array of natural phenomena involving interaction between randomly moving agents.

Anisotropic anomalous diffusion modulated by log-periodic oscillations

We introduce finite ramified self-affine substrates in two dimensions with a set of appropriate hopping rates between nearest-neighbor sites where the diffusion of a single random walk presents an anomalous anisotropic behavior modulated by log-periodic oscillations. The anisotropy is revealed by two different random-walk exponents ν(x) and ν(y) in the x and y directions, respectively. The values of these exponents as well as the periods of the oscillations are obtained analytically and confirmed by Monte Carlo simulations.