We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment assuming that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with a finite state space or with the transition kernel dominated above and below by a probability measure.
A zero-one law of Engelbert--Schmidt type is proven for the norm process of a transient random walk. An invariance principle for random walk local times and a limit version of Jeulin's lemma play key roles.
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
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 $n$.
We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning.
We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.
An infinite graph G has the property that a random walk in random environment on G defined by i.i.d. resistances with any common distribution is almost surely transient, if and only if for some p<1, simple random walk is transient on a percolation cluster of G under bond percolation with parameter p.
There is a condition (T'), such that it is the necessary condition that a random walk in random environment is ballistic. Under this condition, we show the law of the iterated logarithm for a random walk in random environment.
An intrinsic branching structure within the transient random walk on a strip in a random environment is revealed. As applications, which enables us to express the hitting time explicitly, and specifies the density of the absolutely continuous invariant measure for the "environments viewed from the particle".
Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Csáki, Földers, Révész [J Theor Probab (2010) 23: 624-638].
We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed.
We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced r…
We study the range of a planar random walk on a randomly oriented lattice, already known to be transient. We prove that the expectation of the range grows linearly, in both the quenched (for a.e. orientation) and annealed ("averaged") cases. We also express the rate of growth in terms of the quenched Green function and eventually prove a weak law of large numbers in the (non-Markovian) annealed case.
We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger.
We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $\mathbb{Z}^d$, $d\ge 3$.
We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $α>1$ or $\le 1$. In this article, we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely: for $0<α<1$, $X_n/n^{\frac{α+1}{2}}$ converges in law to a Mittag-Leffler distribution whereas for $α=1$, $X_n(\log n)/n$ converges in probability to some positive constant.
Kesten et al.( 1975) proved the stable law for the transient RWRE (here we refer it as the $κ$-transient RWRE). After that, some similar interesting properties have also been revealed for its continuous counterpart, the diffusion proces in a Brownian environment with drift $κ$. In the present paper we will investigate the connections between these two kind of models, i.e., we will construct a sequence of the $κ$-transient RWREs and prove it convergence to the diffusion proces in a Brownian environment with drift $κ$ by proper scaling. To this end, we need a counterpart convergence for the $κ$-…
An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Brémont (2002).
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(ξ(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(ξ(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.
We study a class of nearest-neighbor discrete time integer random walks introduced by Zerner, the so called multi-excited random walks. The jump probabilities for such random walker have a drift to the right whose intensity depends on a random or non-random environment that also evolves in time according to the last visited site. A complete description of the recurrence and transience phases was given by Zerner under fairly general assumptions for the environment. We contribute in this paper with some results that allows us to point out if the random walker speed is strictly positive or not in…
We consider a transient random walk $(X_n)$ in random environment on a Galton--Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that $X_n$ is of order of magnitude $n^Λ$, with $Λ\in (0,1)$. We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio \cite{Col06}.
We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.
We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central Limit Theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.
This is the story of the encounter between two worlds: the world of random walks and the world of Variable Length Markov Chains (VLMC). The meeting point turns around the semi-Markov property of underlying processes.
We describe a new construction of a family of measures on a group with the same Poisson boundary. Our approach is based on applying Markov stopping times to an extension of the original random walk.
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite Galton-Watson tree without leaves. We use regeneration times and ergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peres in 1996 to give an explicit construction of the invariant measure for the forward environment seen by the particule at exit times which is absolutely continuous with respect to the joint law of the tree and the path of …
We consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
We consider transient random walks on a strip in a random environment. The model was introduced by Bolthausen and Goldsheid [Comm. Math. Phys. 214 (2000) 429--447]. We derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments. In addition, we prove that the law of the "environment viewed from the position of the walker" converges to a limiting distribution if the environment is an i.i.d. sequence.
We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1).
We show that a sequence of birth-and-death chains, given by lazy random walks in a (transient) environment (RWRE) on [0; n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.
We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ has infinitely many cut times almost surely. This condition applies in particular to any graph of spectral dimension strictly larger than $2$. In fact, our proof applies to general (possibly nonreversible) Markov chains satisfying a similar decay condition for the Green's function that is sharp for birth-death chains. We deduce that a conjecture of Diaconis and Freedman (Ann. Probab. 1980) holds for the same class of …
Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{ξ(k), k \in \mathbb{Z} \}$ is i.i.d.
We examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper we handle the case of transient walks.
Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $α\in[0,\infty)$ and let $L_n(α)$ be the spatial sum of the $α$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $α$, $L_n(α)$ is the number of the $α$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(α)$ as $n\to\infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimens…
We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. \cite{Sz01}). This weakens by first time the Kalikow ballistic assumption in mixing and proves finite moments of arbitrary order for the approximate regeneration time of \cite{CZ02}. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.
For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $\mathbb Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n\mathbb I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erdős), which corresponds to a function $f(i)=\mathbb I\{i\ge 1\}$; (b) $α$-fold self-intersections of the random walk (studied by Becker and König), which corresponds to $f(i)=i^α$; (c) sites visited by the random walk exactly $j$ times (considered by…
In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the…
We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1) ... X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n \rightarrow \infty$ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.
We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate last passage decomposition. Secondly, we obtain exponential moment bounds for functionals of the branching random walk, under optimal condition. As a corollary we obtain an approximate variational characterisation of the branching capacity. We finally derive upper bounds involving the branching capacity for the tail of the time spent in an arbitrary finite coll…
We consider a time-homogeneous Markov chain $X_n$, $n\ge0$, valued in ${\bf R}$. Suppose that this chain is transient, that is, $X_n$ generates a $σ$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift.
In this paper, we investigate the properties of recurrent planar Markov random walks. More precisely, we study the set of recurrent points with the use of local limit theorems. The Nagaev-Guivarc'h spectral method provides several examples for which these local limit theorems are satisfied as soon as the (standard or non-standard) central limit theorem holds.
We prove two propositions with conditions that a system, which is described by a transient Markov chain, will display local stability. Examples of such systems include partly overloaded Jackson networks, partly overloaded polling systems, and overloaded multi-server queues with skill based service, under first come first served policy.
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, …
Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(ξ(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(ξ(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(ξ(S_n))_{n\geq 0}$.
In this article we continue the study of the quenched distributions of transient, one-dimensional random walks in a random environment. In a previous article we showed that while the quenched distributions of the hitting times do not converge to any deterministic distribution, they do have a weak weak limit in the sense that - viewed as random elements of the space of probability measures - they converge in distribution to a certain random probability measure (we refer to this as a weak weak limit because it is a weak limit in the weak topology). Here, we improve this result to the path-valued…
A mixing Markov chain is proved to be realized as a random walk in a directed graph subject to a synchronizing road coloring. The result ensures existence of appropriate random mappings in Propp--Wilson's coupling from the past. The proof is based on the road coloring theorem. A necessary and sufficient condition for approximate preservation of entropies is also given.
This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk's probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over…
We consider excited random walks (ERWs) on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [KZ08] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta>4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta in (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parame…
In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.
We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ>0$ that determines the fluctuations of the process. When $0<κ<2$, the averaged distributions of the hitting times of the random walk converge to a $κ$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and sca…
We consider a nearest-neighbor, one-dimensional random walk $\{X_n\}_{n\geq 0}$ in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an $s\in(1,2)$ such that the annealed law of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched …
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{ξ_{S_k}\leq s\}}-s\big) \end{equation*} where $(ξ_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $ξ$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1α}S_n)_{n\geq 1}$ converges in distr…
