Independent Random Walks Research Articles

On a class of stochastic fractional heat equations

For the fractional heat equation ∂ ∂ t u ( t , x ) = − ( − Δ ) α 2 u ( t , x ) + u ( t , x ) W ˙ ( t , x ) \frac {\partial }{\partial t} u(t,x) = -(-\Delta )^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise W ˙ \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d > 2 + α d>2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.

Complete integrability of the problem of full statistics of nonstationary mass transfer in the simple inclusion process.

The simple inclusion process (SIP) interpolates between two well-known lattice gas models: the independent random walkers and the Kipnis-Marchioro-Presutti model. Here we study large deviations of nonstationary mass transfer in the SIP at long times in one dimension. We suppose that N≫1 particles start from a single lattice site at the origin, and we are interested in the probability P(M,N,T) of observing M of the particles, 0≤M≤N, to the right of the origin at a specified time T≫1. At large times, the corresponding full probability distribution has a large-deviation behavior, -lnP(M,N,T)≃sqrt[T]s(M/N,N/sqrt[T]). We determine the rate function s exactly by uncovering and utilizing complete integrability, by the inverse scattering method, of the underlying equationsof the macroscopic fluctuation theory. We also analyze different asymptotic limits of the rate function s.

First-passage time for many-particle diffusion in space-time random environments.

The first-passage time for a single diffusing particle has been studied extensively, but the first-passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many-particle diffusion-one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first-passage time of a single diffusing particle under both models shows the same statistics and scaling behavior. However, for many-particle diffusions, the first-passage time among all particles (the extreme first-passage time) is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate the impact of the random environment with that of the sampling trajectories within it. We identify a power law describing the impact of the environment on the variance of the extreme first-passage time. Through numerical simulations, we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first-passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring.

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks.

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

A multivariate extension of the Erdős–Taylor theorem

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if LN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N$$\\end{document} is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then πlogNLN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N} \\,\ extsf{L}_N$$\\end{document} converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if LN(1,2)=∑n=1N1{Sn(1)=Sn(2)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N^{(1,2)}=\\sum _{n=1}^N \\mathbb {1}_{\\{S_n^{(1)}= S_n^{(2)}\\}}$$\\end{document}, then πlogNLN(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N}\\, \ extsf{L}^{(1,2)}_N$$\\end{document} converges in distribution to an exponential random variable of parameter one. We prove that for every h⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\geqslant 3$$\\end{document}, the family {πlogNLN(i,j)}1⩽i<j⩽h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big \\{ \\frac{\\pi }{\\log N} \\,\ extsf{L}_N^{(i,j)} \\big \\}_{1\\leqslant i<j\\leqslant h}$$\\end{document}, of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

ANOPOW FOR REPLICATED NONSTATIONARY TIME SERIES IN EXPERIMENTS.

We propose a novel analysis of power (ANOPOW) model for analyzing replicated nonstationary time series commonly encountered in experimental studies. Based on a locally stationary ANOPOW Cramér spectral representation, the proposed model can be used to compare the second-order time-varying frequency patterns among different groups of time series and to estimate group effects as functions of both time and frequency. Formulated in a Bayesian framework, independent two-dimensional second-order random walk (RW2D) priors are assumed on each of the time-varying functional effects for flexible and adaptive smoothing. A piecewise stationary approximation of the nonstationary time series is used to obtain localized estimates of time-varying spectra. Posterior distributions of the time-varying functional group effects are then obtained via integrated nested Laplace approximations (INLA) at a low computational cost. The large-sample distribution of local periodograms can be appropriately utilized to improve estimation accuracy since INLA allows modeling of data with various types of distributions. The usefulness of the proposed model is illustrated through two real data applications: analyses of seismic signals and pupil diameter time series in children with attention deficit hyperactivity disorder. Simulation studies, Supplementary Materials (Li, Yue and Bruce, 2023a), and R code (Li, Yue and Bruce, 2023b) for this article are also available.

The Kuramoto model on dynamic random graphs

We propose a Kuramoto model of coupled oscillators on a time-varying graph, whose dynamics are dictated by a Markov process in the space of graphs. The simplest representative is considering a base graph and then the subgraph determined by N independent random walks on the underlying graph. We prove a synchronisation result for solutions starting from a phase-cohesive set independent of the speed of the random walkers, an averaging principle and a global synchronisation result with high probability for sufficiently fast processes. We also consider Kuramoto oscillators in a dynamical version of the random conductance model.

Infinite collision property for the three-dimensional uniform spanning tree

Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text].

Yule’s “nonsense correlation” for Gaussian random walks

This paper provides an exact formula for the second moment of the empirical correlation (also known as Yule’s “nonsense correlation”) for two independent standard Gaussian random walks, as well as implicit formulas for higher moments. We also establish rates of convergence of the empirical correlation of two independent standard Gaussian random walks to the empirical correlation of two independent Wiener processes.

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.

Local and global survival for infections with recovery

We establish two open problems from Kesten and Sidoravicius (Kesten and Sidoravicius, 2006). Particles are initially placed on Zd with a given density and evolve as independent continuous-time random walks. Particles initially placed at the origin are declared as infected. Infection transmits instantaneously to healthy particles on the same site and infected particles become healthy with a positive rate. We prove that, for small enough recovery rates, the infection process survives and visits the origin infinitely many times on the event of survival. Second, we establish the existence of density parameters for which the infection survives for all choices of the recovery rate.

Two repelling random walks on [formula omitted

We consider two interacting random walks on Z such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter β≥0. When β=0 both processes are independent symmetric random walks on Z, and hence recurrent. We show that both random walks are further recurrent if β∈(0,1]. We also show that these processes are transient and diverge in opposite directions if β>2. The case β∈(1,2] remains widely open. Our results are obtained by considering the dynamical system approach to stochastic approximations.

Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble L=(L1,…,LN) consists of N independent random walks on the integers conditioned not to cross one another, i.e., L1≥⋯≥LN. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles fN=(f1N,…,fNN) as the number of curves N tends to infinity. Assuming log-concavity and a KMT-type coupling for the random walk jump distribution, we prove that under mild control of the one-point marginals of the top curves with a global parabolic shift, the full sequence (fN) is tight in the topology of uniform convergence over compact sets, and moreover any weak subsequential limit possesses the Brownian Gibbs property. If in addition the top curves converge in finite-dimensional distributions to the parabolic Airy2 process, then a result of Dimitrov (2021) implies that (fN) converges to the parabolically shifted Airy line ensemble. These results apply to a broad class of discrete jump distributions, including geometric as well as any log-concave distribution whose support forms a compact integer interval.

Branching random walk in a random time-independent environment

ABSTRACT In a lattice population model, particles move randomly from one site to another as independent random walks, split into two offspring, or die. If duplication and mortality rates are equal and take the same value over all lattice sites, the resulting model is a critical branching random walk (characterized by a mean total number of offspring equal to ). There exists an asymptotical statistical equilibrium, also called steady state. In contrast, when duplication and mortality rates take independent random values drawn from a common nondegenerate distribution (so that the difference between duplication and mortality rates has nonzero variance), then the steady state no longer exists. Simultaneously, at all lattice sites, if the difference between duplication and mortality rates takes strictly positive values with strictly positive probability, the total number of particles grows exponentially. The lattice includes large connected sets where the duplication rate exceeds the mortality rate by a positive constant amount, and these connected sets provide the growth of the total population. This is the supercritical regime of branching processes. On the other hand, if the difference between duplication and mortality rates is almost surely negative or null except when it is almost surely zero, then the total number of particles vanishes asymptotically. The steady state can be reached only if the difference between duplication and mortality rates is almost surely zero.

Statistical properties of sites visited by independent random walks

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 hyper-cubic lattices 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 ordering probabilities in spatial dimensions d = 1, 2, 3, 4. We also study the time evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as a 1 ln t with a 1 = 0.970 508 in one dimension and as (ln t)2 in two dimensions.

Price levels in the European Monetary Union: Even tradables follow independent random walks

This study analyzes the time series behavior of real exchange rates between member states of the European Monetary Union. The analysis is based on the Harmonized Index of Consumer Prices and eight sub-components for tradable and non-tradable goods and services. Stationarity of the real exchange rates is a necessary condition for long-term convergence of price levels. The results of pairwise tests show that the real exchange rates, resulting from these price levels are typically random walks and their components are generally not cointegrated. There is no evidence of the existence of country-specific convergence clusters. A systematic difference in the results for tradable and non-tradable goods cannot be detected.

Optimal searcher distribution for parallel random target searches.

We consider a problem of finding a target located in a finite d-dimensional domain, using N independent random walkers, when partial information about the target location is given as a probability distribution. When N is large, the first-passage time sensitively depends on the initial searcher distribution, which invokes the question of the optimal searcher distribution that minimizes the first-passage time. Here, we analytically derive the equationfor the optimal distribution and explore its limiting expressions. If the target volume can be ignored, the optimal distribution is proportional to the target distribution to the power of one third. If we consider a target of a finite volume and the probability of the initial overlapping of searchers with the target cannot be ignored in the large N limit, the optimal distribution has a weak dependence on the target distribution, with its variation being proportional to the logarithm of the target distribution. Using Langevin dynamics simulations, we numerically demonstrate our predictions in one and two dimensions.

Percolation of worms

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v∈(0,∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then Sv undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that Sv percolates for all v>0 if d≥5. E.g., if the probability mass function of the length distribution is m(ℓ)=c⋅ln(ln(ℓ))ɛ/(ℓ3ln(ℓ))1[ℓ≥ℓ0]for some ℓ0>ee and ɛ>0 then Sv percolates for all v>0. Note that the second moment of this length distribution is only “barely” infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being “extremal” in this family of models.

Stochastic sandpile on a cycle

In the stochastic sandpile (SS) model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability 0 < p < 1 of not moving. These interactions continue until each site has no more than one particle on it. We provide a formal coupling between the SS and the activated random walk models, and we use the coupling to show that for the SS with n particles on the cycle graph , the system stabilizes in O(n 3) time for all initial particle configurations, provided that p(n) tends to 1 sufficiently rapidly as n → ∞.

Broadcasting on paths and cycles

Consider the following broadcasting process run on a connected graph G=(V,E). Suppose that k≥2 agents start on vertices selected from V uniformly and independently at random. One of the agents has a message that she wants to communicate to the other agents. All agents perform independent random walks on G, with the message being passed when an agent that knows the message meets an agent that does not know the message. The broadcasting time ξ(G,k) is the time it takes to spread the message to all agents. We provide tight bounds for ξ(Pn,k) and ξ(Cn,k) that hold asymptotically almost surely for the whole range of the parameter k.