We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension $1/2$ almost surely, and give bounds on the rate at which the walk diverges at such times.
We also show noise sensitivity of the event that our random walk is positive after $n$ steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $\varepsilon_n$ such that $n\varepsilon_n\to\infty$. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.
A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non-zero limiting speed).
For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
We consider recurrence versus transience for models of random walks on domains of $\mathbb{Z}^d$, in which monotone interaction enforces domain growth as a result of visits by the walk (or probes it sent), to the neighborhood of domain boundary.
We establish recurrence and transience criteria for critical branching processes in random environment with immigration. These results are then applied to discuss recurrence and transience of a recurrent random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1.
We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that \emph{Condition L} of Comets and Popov is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.
We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between Branching Markov Chains and the spectral radius. We prove properties of the spectral radius of the Random Walk in Random Environment with the help of appropriate Branching Markov Chains.
The notion of degree and related notions concerning recurrence and transience for a class of L'evy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed.
We investigate excited random walks on $\Z^d, d\ge 1,$ and on planar strips $\Z\times\{0,1,...,L-1\}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of the walk. We give exact criteria for recurrence and transience, thus generalizing results by Benjamini and Wilson for once-excited random walk on $\Z^d$ and by the author for multi-excited random walk on $\Z$.
The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner [8].
Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.
We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.
We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is,formally speaking, the Doob's $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.
We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to …
Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $ω(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$, from which time onward the probability of jumping to the right is $\frac12$. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments $\{ω(x)\}_{x\in Z}$. In deterministic environments, we also study the speed of the process.
We consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on point processes and conductances, we show that, for almost any realization of the point process, these random walks are recurrent if $d=2$ and transient if $d\geq 3$. These results hold for a large variety of point processes including Poisson point processes, Mat{é}rn cluster and Mat{é}rn hardcore processes which have clustering or repulsive properties. In order to prove them, we…
The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or trans…
This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classica…
We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the pro…
We consider excited random walk (ERW) on $\mathbb{Z}$ in environments with identical stacks of infinitely many cookies at each site, subject to the constraint that the total drift per site $δ= \sum (2p_j - 1)$ is finite. Building on the methods of Kozma, Orenshtein, and Shinkar (arXiv:1311.7439), we show that ERW in finite-drift environments is recurrent when $|δ|<1$ and transient when $|δ|>1$. In the case $|δ|=1$ we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when $|δ|=1$, an interesting new behavior that was not presen…
We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$. We prove a sharp cutoff for right/left transience of …
Consider a sequence of i.i.d. random variables $X_n$ where each random variable is refreshed independently according to a Poisson clock. At any fixed time $t$ the law of the sequence is the same as for the sequence at time 0 but at random times almost sure properties of the sequence may be violated. If there are such \emph{exceptional times} we say that the property is \emph{dynamically sensitive}, otherwise we call it \emph{dynamically stable}. In this note we consider branching random walks on Cayley graphs and prove that recurrence and transience are dynamically stable in the sub-and superc…
A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in question is not regular.
We study the frog model on $\mathbb{Z}^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and …
We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $μ$ and $ν$ are supercritical offspring distributions with means $\bar μ< \bar ν$ then every space-time set that is recurrent with respect to the offspring distribution $μ$ is also recurrent with …
We consider a random walk on a second countable locally compact topological space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has escape of mass for almost every starting point. We then apply this result in the context of homogeneous random walks on infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence theorem.
The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. co…
We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+δ)}, δ>-1$, with $c_k=C_0+C_1\log k+C_2k^α$, non-negative constants $C_0, C_1, C_2$, and $α>0$. Percolation was proved in Dawson and Gorostiza (2013) for $δ<1$, and for the critical case, $δ=1,C_2>0$, with $α>2$. In this paper we improve the result for the critical case by showing percolation for $…
Recently Simenhaus proved that for any elliptic random walk in random environment, transience in the neighborhood of a given direction is equivalent to the a.s. existence of a deterministic asymptotic direction and to transience in any direction in the open half space defined by this asymptotic direction. Here we prove an improved version of this result and review some open problems.
General characterizations of ergodic Markov chains have been developed in considerable detail. In this paper, we study the transience for discrete-time Markov chains on general state spaces, including the geometric transience and algebraic transience. Criteria are presented through establishing the drift condition and considering the first return time. As an application, we give explicit criteria for the random walk on the half line and the skip-free chain on nonnegative integers.
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 a random interacting particle system, known as the frog model, on infinite Galton-Watson trees allowing offspring zero and one. The system starts with one awake particle (frog) at the root of the tree and a random number of sleeping particles at the other vertices. Awake frogs move according to simple random walk on the tree and as soon as they encounter sleeping frogs, those will wake up and move independently according to simple random walk. The frog model is called transient, if there are almost surely only finitely many particles returning to the root. In this paper we prove a …
We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.
We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with uniformly bounded above and below, independently time-varying edge conductances, of (effectively) non-decreasing in time vertex conductances (i.e. reversing measure), thereby affirming part of [ABGK, Conj. 7.1].
We study the transience of algebraic varieties in linear groups. In particular, we show that a "non elementary" random walk in SL_2(R) escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk is on the real points of a semi-simple split algebraic group and show such a result for a wide family of random walks. As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.
The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree.
We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\simβ/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is transient, and when $s\geq 2d$, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension $d$, if $d<s<2d$, then critical percolation has no infinite clusters. This result is extended to the free random cluster model. A second corollary is that when $d\…
We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.
We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a …
In the present paper we define conservative and semiconservative random walks in $\mathbb{Z}^d$ and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (Pólya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in $\mathbb{Z}^d$ by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in $\mathbb{Z}^d$ for any $d\geq3$ and transient random walks in $\mathbb{…
One dimensional excited random walk has been extensively studied for bounded, i.i.d. cookie environments. In this case, many important properties of the walk including transience or recurrence, positivity or non-positivity of the speed, and the limiting distribution of the position of the walker are all characterized by a single parameter $δ$, the total expected drift per site. In the more general case of stationary ergodic environments, things are not so well understood. If all cookies are positive then the same threshold for transience vs. recurrence holds, even if the cookie stacks are unbo…
We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567--592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229--1241] and [Ann. Probab. 18 (1990) 931--958] as part of…
For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the random walk with increments $\log M_{1},\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.
We study the asymptotic behaviour of random walks in i.i.d. random environments on $\Z^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) o…
We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such walks for various sequences of ${\bf a}$. In particular, we establish the classification in the cases where $a_k=\lfloor k^β\rfloor$, $β>0$, as well as in the case $a_k=\lceil \log_γk \rceil$ or $a_k=\log_γk$ for $γ>1$.
We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on $\mathbb{Z}^d$. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on $\mathbb{Z}^d$. Its proof uses the general zero-one law.
