We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on {0,1,2,...}, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique - the method of Lyapunov functions.
Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $1$ and ending at $0.$ We study the distribution of $M$ and characterize its asymptotics, which is quite different from those of simple random walks.
Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n,n\geq1\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditione…
Suppose that $\{S_n,\ n\geq0\}$ is an asymptotically stable random walk. Let $g$ be a positive function and $T_g$ be the first time when $S_n$ leaves $[-g(n),\infty)$. In this paper we study asymptotic behaviour of $T_g$. We provide integral tests for function $g$ that guarantee $P(T_g>n)\sim V(g)P(T_0>n)$ where $T_0$ is the first strict descending ladder epoch of $\{S_n\}$
We study the first exit time $τ$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $τ<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-β}$, $β\in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk a…
We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\ldots, S_d(n) )$. Let $τ_x:=\min\{n>0: x_{1}+S_1(n)\le 0\}$ be the first time the random walk $S(n)$ leaves the upper half-space. We obtain the asymptotics of $p_n(x,y):= P(x+S(n) \in y+Δ, τ_x>n)$ as $n$ tends to infinity, where $Δ$ is a fixed cube. From that we obtain the local asymptotics for the Green function $G(x,y):=\sum_n p_n(x,y)$, as $|y|$ and/or $|x|$ tend to infinity.
For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using eac…
Let $V$ be a two sided random walk and let $X$ denote a real valued diffusion process with generator ${1/2}e^{V([x])}\frac{d}{dx}(e^{-V([x])}\frac{d}{dx})$. This process is known to be the continuous equivalent of the one dimensional random walk in random environment with potential $V$. Hu and Shi (1997) described the Lévy classes of $X$ in the case where $V$ behaves approximately like a Brownian motion. In this paper, based on some fine results on the fluctuations of random walks and stable processes, we obtain an accurate image of the almost sure limiting behavior of $X$ when $V$ behaves asy…
For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a strictly stable process with index $α\in (1, 2)$ we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.
It is well known that random walks in one dimensional random environment can exhibit subdiffusive behavior due to presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $\mathbb{Z}^d$, $d\geq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $\varrho_N=C N^{1/(d+2)}$ for any $d\geq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-ε)\varrho_N$ for any $ε>0$, while the case $d\geq 3$ remains open. We complete the picture by showing that for any $d\geq 2$, the random walk range asymptotically contains a ba…
We use generalized Ray-Knight theorems introduced by Bálint Tóth in 1996 together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRWs) to Brownian motions perturbed at extrema (BMPE). Tóth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of …
We construct a coupling between the random walk composed of Lévy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaus…
We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $μ$ and satisfies the Poincaré inequality w.r.t. $μ$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $μ$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for…
We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.
We consider a transient random walk on $Z^d$ which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time $n$.
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some deli…
We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $τ_α$ from a wedge with apex at the origin and interior half-angle $α$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $τ_α< \infty$ a.s. for any $α$ (while for a stronger drift field $τ_α$ is infin…
We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.
We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $γ>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $γ$ large enough. This statement is thus a perturbation of the case $γ=+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i…
By introducing a new measure for the infinite Galton-Watson process and providing estimates for (discrete) Green's functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high dimensions $d\ge 7$, the capacity grows linearly; and in the critical dimension $d=6$, it grows asymptotically proportional to $\frac{n}{\log n}$.
We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is minimized asymptotically by equally spaced p-drifts.
The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e. horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to 2/(πN). As15 pag a consequence we prove strong laws and a limit distribution for the local time.
The longest increasing subsequence of a random walk with mean zero and finite variance is known to be $n^{1/2 + o(1)}$. We show that this is not universal for symmetric random walks. In particular, the symmetric Ultra-fat tailed random walk has a longest increasing subsequence that is asymptotically at least $n^{0.690}$ and at most $n^{0.815}$. An exponent strictly greater than $1/2$ is also shown for the symmetric stable-$α$ distribution when $α$ is sufficiently small.
We study the random walk in random environment on {0,1,2,...}, where the environment is subject to a vanishing (random) perturbation. The two particular cases we consider are: (i) random walk in random environment perturbed from Sinai's regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker will be from the origin after a long time t, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order $(\log t)^β$, for $β\in (1,\infty)$, depending on the perturbation. In addition, in the ergodic ca…
We consider simple random walk on a realization of an Erdős-Rényi graph that is asymptotically almost surely (a.a.s.) connected. We show a Central Limit Theorem (CLT) for the average starting hitting time, i.e. the expected time it takes the random walker on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $π_j$, the invariant measure of the random walk.
We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position.
Consider the time T_oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T_oz in terms of the volume of z and the graph distance between o and z. The bounds are for expected value and large deviations, and are asymptotically sharp. We deduce rate of escape results for random walks on infinite graphs of exponential or polynomial growth, and resolve a conjecture of Benjamini and Peres.
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments.…
In a recent paper, K. Raschel and R. Garbit proved that the exponential decreasing rate of the probability that a random walk (with all exponential moments) stays in a $d$-dimensional orthant is given by the minimum on this orthant of the Laplace transform of the random walk increments, as soon as this minimum exists. In other cases, the random walk is "badly oriented," and the exponential rate may depend on the starting point $x$. We prove here that this rate is nevertheless asymptotically equal to the infimum of the Laplace transform, as some selected coordinates of $x$ tend to infinity.
We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.
Importance sampling algorithms for heavy-tailed random walks are considered. Using a specification with algorithms based on mixtures of the original distribution with some other distribution, sufficient conditions for obtaining bounded relative error are presented. It is proved that mixture algorithms of this kind can achieve asymptotically optimal relative error. Some examples of mixture algorithms are presented, including mixture algorithms using a scaling of the original distribution, and the bounds of the relative errors are calculated. The algorithms are evaluated numerically in a simple …
We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the $n$-th generation behaves asymptotically like $\frac{3}{2}\ln n$, provided the non-extinction of the system. The main goal of this paper, is to prove that the path from the root to the leftmost particle, after a suitable normalizatoin, converges weakly to a Brownian excursion in $D([0,1],\r)$.
We study the set of directions asymptotically explored by a spatially homogeneous random walk in $d$-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We also explore links to the asymptotics of one-dimensional projections, and to the growth of the convex hull of the random walk.
We study how memory impacts passages at the origin for a so-called elephant random walk in the diffusive regime. We observe that the number of zeros always grows asymptotically like the square root of the time, despite the fact that, depending on the memory parameter, first return times to $0$ may have a finite expectation or a fat tail with exponent less than $1/2$. We resolve this apparent paradox by recasting the questions in the framework of scaling limits for Markov chains and self-similar Markov processes.
We consider a special case of random walk in random environment (RWRE) on Z^d where the environment is periodic (RWPE). Under natural conditions, we show that law of large numbers and central limit theorem holds. In the ballistic nearest neighbour reversible case, we prove that the angle between the asymptotic direction of the RWPE and the average negative gradient of the potential function of the reversible environment is less than pi/2, that is, the potential cannot increase asymptotically along the trajectory of the RWPE. But this angle can be close to pi/2.
We show the minimum spectral gap of the normalized Laplacian over all simple, connected graphs on $n$ vertices is $(1+o(1))\tfrac{54}{n^3}$. This minimum is achieved asymptotically by a double kite graph. Consequently, this leads to sharp upper bounds for the maximum relaxation time of a random walk, settling a conjecture of Aldous and Fill. We also improve an eigenvalue-diameter inequality by giving a new lower bound for the spectral gap of the normalized Laplacian. This eigenvalue lower bound is asymptotically best possible.
We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on Z^d, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e^{-λ_1\sqrt{t}} for d=1, as e^{-λ_2 t/\log t} for d=2, and as e^{-λ_d t} for d>= 3, where λ_1 and λ_2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e^{-\tilde λ_d t}, with \tilde λ_d>0 for all d>= 1. A ke…
We study an homogeneous irreducible markovian random walk in a square lattice of arbitrary dimension, with an antisymmetric perturbation acting only in one point. We compute exactly spatial correction to the diffusive behaviour in the asympotics of probability, in the spirit of local limit theorems for random walks.
In this article, we consider the speed of the random walks in a (uniformly elliptic and i.i.d.) random environment (RWRE) under perturbation. We obtain the derivative of the speed of the RWRE w.r.t. the perturbation, under the assumption that one of the following holds: (i) the environment is balanced and the perturbation satisfies a Kalikow-type ballisticity condition, (ii) the environment satisfies Sznitman's ballisticity condition. This is a generalized version of the Einstein relation for RWRE. Our argument is based on a modification of Lebowitz-Rost's argument developed in [Stochastic Pro…
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature o…
In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Kni…
We consider the time evolution of a lattice branching random walk with local perturbations. Under certain conditions, we prove the Carleman type estimation for the moments of a particle subpopulation number and show the existence of a steady state.
Real networks are often dynamic. In response to it, analyses of algorithms on {\em dynamic networks} attract more and more attentions in network science and engineering. Random walks on dynamic graphs also have been investigated actively in more than a decade, where in most cases the edge set changes but the vertex set is static. The vertex sets are also dynamic in many real networks. Motivated by a new technology of the analysis of random walks on dynamic graphs, this paper introduces a simple model of graphs with increasing the number of vertices, and presents an analysis of random walks ass…
We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -δ/2x + o(1/x) with δ> -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}ϕ(n), where c = (3+δ)/2 and ϕis a slowly varying function given explicitly in terms of the o(…
Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by rand…
We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.
We study discrete-time stochastic processes $(X_t)$ on $[0,\infty)$ with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at $x$ is about $c/x$. Our focus is the recurrent case (when $c$ is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form…
We consider a symmetric random walk on the $ν$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.
We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. For this model, Fribergh and Hammond showed the existence of an exponent $γ$ such that: for $γ<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by $γ$; whereas for $γ>1$, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of th…
This article studies vertex reinforced random walks that are non-backtracking (denoted VRNBW), i.e. U-turns forbidden. With this last property and for a strong reinforcement, the emergence of a path may occur with positive probability. These walks are thus useful to model the path formation phenomenon, observed for example in ant colonies. This study is carried out in two steps. First, a large class of reinforced random walks is introduced and results on the asymptotic behavior of these processes are proved. Second, these results are applied to VRNBWs on complete graphs and for reinforced weig…
We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condi…
