Nearest-neighbor Random Walk Research Articles

Full-record statistics of one-dimensional random walks.

We develop a comprehensive framework for analyzing full-record statistics, covering record counts M(t_{1}),M(t_{2}),..., their corresponding attainment times T_{M(t_{1})},T_{M(t_{2})},..., and the intervals until the next record. From this multiple-time distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearest-neighbor random walks, asymmetric run-and-tumble dynamics, and random walks with stochastic resetting.

Speed function for biased random walks with traps

We consider a biased nearest-neighbor random walk on Z which at each step is trapped for some random time with random, site-dependent mean. We derive a simple formula for the speed function in terms of the model parameters.

Large and moderate deviations for record numbers in some non–nearest neighbor random walks

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non–nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of “weak records” (or “ladder points”) in two kinds of one–dimensional non–nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases.

Exact formulas for two interacting particles and applications in particle systems with duality

We consider two particles performing continuous-time nearest neighbor random walk on $\mathbb{Z}$ and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace–Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions and sticky Brownian motions. In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime. We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterise the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.

Analysis of a discrete-time two-class randomly alternating service model with Bernoulli arrivals

We analyze a discrete-time two-class queueing system with a single server which is alternately available for only one customer class. The server is each time allocated to a customer class for a geometrically distributed amount of time. Service times of the customers are deterministically equal to 1 time slot each. During each time slot, both classes can have at most one arrival. The bivariate process of the number of customers of both classes can be considered as a two-dimensional nearest-neighbor random walk. The generating function of this random walk has to be obtained from a functional equation. This type of functional equation is known to be difficult to solve. In this paper, we obtain closed-form expressions for the joint probability distribution for the number of customers of both classes, in steady state.

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

Exponential and Laplace approximation for occupation statistics of branching random walk

We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng [4] showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots $, converges in distribution as $n$ goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation $n$. Here we study the second order fluctuations around this limit, first, by providing a rate of convergence in the Wasserstein metric that holds for all $d\geq 3$, and second, by showing that for $d\geq 7$, the weak limit of the scaled joint differences between the number of occupancy-$j$ sites and appropriate multiples of the total population size converge in the Wasserstein metric to a multivariate symmetric Laplace distribution. We also provide a rate of convergence for this latter result.

Coarse-graining of asymmetric discrete-time random walk on a one-dimensional lattice.

Coarse-graining of an asymmetric nearest-neighbor discrete-time random walk on a one-dimensional lattice allows one to describe this random walk as biased one-dimensional diffusion. The latter is characterized by two parameters: the drift velocity and diffusivity. There is a general expression giving the drift velocity as a function of the parameters determining the random walk. However, a corresponding expression for the diffusivity is known only for the particular case where the random walk escapes from the lattice site at every time step. In this work, we generalize this result and derive an expression for the diffusivity, assuming that the random walk does not necessarily leave the site, and therefore, its mean lifetime on the site can be longer than the time step.

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps ±1 with probability 1∕2 at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.

Nonconvex homogenization for one-dimensional controlled random walks in random potential

We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk $\{X_{i}\}$ on the set of integers. The cost function is the expectation of the exponential of the path sum of a random stationary and ergodic bounded potential plus $\theta X_{n}$. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter $\delta\in[0,1]$. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter $\delta$, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when $\delta=0$. The Bellman equation associated to this control problem is a second-order Hamilton–Jacobi (HJ) partial difference equation with a separable random Hamiltonian which is nonconvex in $\theta$ unless $\delta=0$. We prove that this equation homogenizes under linear initial data to a first-order HJ equation with a deterministic effective Hamiltonian. When $\delta=0$, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in $\theta$. In contrast, when $\delta=1$, the effective Hamiltonian is piecewise linear and nonconvex in $\theta$. Finally, when $\delta\in(0,1)$, the effective Hamiltonian is expressed completely in terms of the tilted free energy for the $\delta=0$ case and its convexity/nonconvexity in $\theta$ is characterized by a simple inequality involving $\delta$ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.

The extinction problem for a distylous plant population with sporophytic self-incompatibility.

In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.

Random walks in a moderately sparse random environment.

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk in a sparse random environment is a nearest neighbor random walk on that jumps to the left or to the right with probability 1/2 from every point of and jumps to the right (left) with the random probability λ k+1 (1 - λ k+1) from the point S k , . Assuming that are independent copies of a random vector and the mean is finite (moderate sparsity) we obtain stable limit laws for X n , properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case (strong sparsity) will be analyzed in a forthcoming paper.

Regularity of biased 1D random walks in random environment

We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ ∈ R. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+. When λ− and λ+ are distinct, v(λ) might fail to be continuous. We refine the assumptions in [?] for having a recentered CLT with diffusivity σ 2 (λ) and give explicit conditions for σ 2 (λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ 2 (λ) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of Lam and Depaw (2016).

Residence time near an absorbing set

We determine how long a diffusing particle spends in a given spatial range before it dies at an absorbing boundary. In one dimension, for a particle that starts at x0 and is absorbed at x = 0, the average residence time of the particle in the range is for x < x0 and for x > x0, where D is the diffusion coefficient. We extend our approach to biased diffusion, to a particle confined to a finite interval, and to general spatial dimensions. We then use the generating function technique to derive parallel results for the average number of times that a one-dimensional symmetric nearest-neighbor random walk visits site x when the walk starts at x0 = 1 and is absorbed at x = 0. We also determine the distribution of times when the random walk first revisits x = 1 before being absorbed.

Fractional and High Order Asymptotic Results of the MFPT

This work discusses the Mean First Passage Time (MFPT) and Mean Residence Time (MRT) of continuous-time nearest-neighbor random walks in a finite one-dimensional system with a trap at the origin and a reflecting barrier at the other end. The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point, have a variety of dependencies with respect to its size N. For example, for the case of birth and death processes the MFPT~N; for the case of symmetric random walks the MFPT~N2 and for the case of biased random walks the MFPT~αN where α is a constant that depends on the system’s rates. In this work a transition matrix is derived in such a way that the MRT of the system is equal to (m+1)d where m is the site number, and d is any arbitrary number satisfying the condition d>0. Since the MFPT is the sum of MRTs then the corresponding MFPT for such a transition matrix is MFPT~N(1+d). Thus, one can determine the asymptotic result of the MFPT to be N1+d for any arbitrary d>0, and based on it, obtain the corresponding transition matrix. Several examples of fractional and high order asymptotic results of the MFPT such as N3.5, N5, N6, are presented.

Matrix geometric approach for random walks: Stability condition and equilibrium distribution

ABSTRACTIn this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions[30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions.[13] Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix R appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix R.

Excited random walks with Markovian cookie stacks

We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $\omega_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(\omega_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the strength of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie strengths within the stack $(\omega_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.

Range of a random walk in random environment

Consider a random walk in random environment $\{X_n\}_{n\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, let $N(x)=\#\{i\in[0,x]:$\linebreak $ \exists\, n\ge0, X_n= i \}$ be the number of sites in $[0,x]$ ever being visited by the walk. It is proved that almost surely $\lim_{x\rightarrow\infty}\frac{N(x)}{x}=\theta$ for some $0＜\theta＜1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance.

A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and maximal inequalities

There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the \(L_{2}\) mixing time, \(\tau _{2}\) (while there are sophisticated analytic tools to bound \( \tau _2\), in general they do not determine \(\tau _2\) up to a constant factor and they lack a probabilistic interpretation). In this work we show that \(\tau _2\) can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, \(c_{{\mathrm {LS}}}\), as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of \(c_{{\mathrm {LS}}}\) in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, \(\tau _2\) is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, \(\tau _2 \) is robust under bounded perturbations of the edge weights.

Invariance principle for variable speed random walks on trees

We consider stochastic processes on complete, locally compact tree-like metric spaces $(T,r)$ on their “natural scale” with boundedly finite speed measure $\nu$. Given a triple $(T,r,\nu)$ such a speed-$\nu$ motion on $(T,r)$ can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all $x,y\in T$ and all positive, bounded measurable $f$, \begin{equation}\label{eabstract}\mathbb{E}^{x}[\int^{\tau_{y}}_{0}\mathrm{d}sf(X_{s})]=2\int_{T}\nu(\mathrm{d}z)r(y,c(x,y,z))f(z)<\infty,\end{equation} where $c(x,y,z)$ denotes the branch point generated by $x,y,z$. If $(T,r)$ is a discrete tree, $X$ is a continuous time nearest neighbor random walk which jumps from $v$ to $v'\sim v$ at rate $\frac{1}{2}\cdot(\nu(\{v\})\cdot r(v,v'))^{-1}$. If $(T,r)$ is path-connected, $X$ has continuous paths and equals the $\nu$-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-$\nu_{n}$ motions on $(T_{n},r_{n})$ converge weakly in path space to the speed-$\nu$ motion on $(T,r)$ provided that the underlying triples of metric measure spaces converge in the Gromov–Hausdorff-vague topology introduced in [Stochastic Process. Appl. 126 (2016) 2527–2553].