The paper is concerned with a new approach for the recurrence property of the oscillating process on $\mathbb{Z}$ in Kemperman's sense. In the case when the random walk is ascending on $\mathbb{Z}^-$ and descending on $\mathbb{Z}^+$, we determine the invariant measure of the embedded process of successive crossing times and then prove a necessary and sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating process is recurrent under some H{ö}lder-typed moment assumptions.
Considering homogeneous and oscillating random walks on the integers, we simplify classical works on recurrence of Spitzer and Kemperman, respectively. Some links with renewal theory are discussed.
In this note we show that the area of the partitions making up an oscillating tableaux is described by a random walk on the first quadrant of $\mathbb{Z}^2$ with certain position dependent weights. We are able to recursively calculate the moments of the walk. As the length of the oscillating tableaux becomes large we show that this random walk converges to a Gaussian stochastic process.
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…
We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator-stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.
We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z}^{4}$. On $\mathbb{Z}^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion.
Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration
In this paper, we establish the invariance principle and the large deviation for the biased random walk $RW_λ$ with $λ\in [0,1)$ on $\mathbb{Z}^d, d\geq 1$.
We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution $μ_k$. If $χ_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and $α_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all $α_k χ_k \in (0,1)$ is determined by the sign…
We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one. The proof is based on stochastic algorithms techniques.
We study the capacity of the range of a transient simple random walk on $\mathbb{Z}^d$. Our main result is a central limit theorem for the capacity of the range for $d\ge 6$. We present a few open questions in lower dimensions.
In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z}^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2\log |a|)$ up to a constant.
We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on $\mathbb Z^d$, $d\ge 2$. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on $\mathbb Z^d$, this implies that the scaling limit of the randomly trapped random walk is either Brownian motion or the Fractional Kinetic…
Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.
We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_0$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_0$ and almost surely transient if $a<a_0$. This provides the first examples of phase transition for the Once-reinforced random walk.
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…
We investigate a problem suggested by Dembo, Peres, Rosen, and Zeitouni, which states that the growth exponent of favorite points associated with a simple random walk in ${\mathbb Z}^2$ coincides, on average and almost surely, with those of late points and high points associated with the discrete Gaussian free field.
We consider elliptic random walks in i.i.d. random environments on $\mathbb{Z}^d$. The main goal of this paper is to study under which ellipticity conditions local trapping occurs. Our main result is to exhibit an ellipticity criterion for ballistic behavior which extends previously known results. We also show that if the annealed expected exit time of a unit hypercube is infinite then the walk has zero asymptotic velocity.
We consider a $\mathbb{R}^d$-valued branching random walk with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. With the help of the uniform convergence of martingale and the multifractal analysis, we establish a large deviation result for the measures $Z_n$ as well as a moderate deviation principle.
Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementi…
We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^α$, with $α\in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.
In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $α\in (1, 2)$ and obtain the asymptotic form of the distribution of the hitting time of the origin and that of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in the natural domain of the space and time variables.
The subject of this paper is the simple random walk on $\mathbb{Z}$. We give a very simple answer to the following problem: under the condition that a random walk has already spent $α$-percent of the traveling time on the positive side $\mathbb{Z}_{\ge 0}$, what is the probability that the random walk is now on the positive side? The symmetric random walks which step $2n$-times can be decomposed in the following two ways: (1) how many times the walk steps on the positive side, (2) whether the last step is on the positive side or on the negative side. To answer the problem above, we clarify the…
Let $(Y_n)$ be a sequence of i.i.d. $\mathbb Z$-valued random variables with law $μ$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \in \mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $μ$, it is proved that, for any $ y \in \mathbb N_0$, as $n \to +\infty$, one gets $\mathbb P_x[X_n=y]\sim C_{x, y} R^{-n} n^{-3/2}$ when $\sum_{k\in \mathbb Z} kμ(k) >0$ and $\mathbb P_x[X_n=y]\sim C_{y} n^{-1/2}$ when $\sum_{k\in \mathbb Z} kμ(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$.
We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two.
We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $β\geq 0$. When $β= 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $β\in (0,1]$. We also show that these processes …
We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times Γ$, with $Γ$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.
We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting.
It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.
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…
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…
A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is homeomorphic to the closure of the set $\{w={z}/{(1+|z|)}:z\in{\mathbb{N}}^*\times{\mathbb{N}}^*\}$ in ${\mathbb{R}}^2$. The method is based on a ratio limit theorem for local processes and large deviation techniques.
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 $(X_n)_{n\in \mathbb{N}\cup\{0\}}$ in a sparse random environment $(S_k,λ_k)_{k\in\mathbb{Z}}$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z}\setminus \{\ldots,S_{-1},S_0=0,S_1,\ldots\}$ and jumps to the right (left) with the random probabilit…
In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a non-homogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk in random environment with bounded jumps and specify the explicit invariant density for the Markov chain of ``the environment viewed from the particle" .The invariant density and the limit velocity could be expressed explicitly in te…
We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results f…
We study the persistence exponent for random walks in random sceneries (RWRS) with integer values and for some special random walks in random environment in $\mathbb Z^2$ including random walks in $\mathbb Z^2$ with random orientations of the horizontal layers.
We consider simple random walks on random graphs embedded in $\mathbb{R}^d$ and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we show an annealed invariance principle for these random walks. These results hold for a large variety of point processes including Poisson point processes, Matérn cluster and Matérn hardcore processes which have respectively clustering and repulsiveness properties. The proof relies on the use the process of the environment seen from the particle. It allows to…
We examine a class of random walks in random environments on $\mathbb{Z}$ with bounded jumps, a generalization of the classic one-dimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For this model, we characterize recurrence and transience, and in the transient case we characterize ballisticity. For ballisticity, we give two parameters, $κ_0$ and $κ_1$. The parameter $κ_0$ governs finite trapping effects, and $κ_1$ governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is right-transie…
We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. The approach allows us to prove that with high probability, the $π/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we extend the "escape" phenomenon to a broad class of random matrices; as an appli…
On the complete graph ${\cal{K}}_M$ with $M \ge3$ vertices consider two independent discrete time random walks $\mathbb{X}$ and $\mathbb{Y}$, choosing their steps uniformly at random. A pair of trajectories $\mathbb{X} = \{ X_1, X_2, \dots \}$ and $\mathbb{Y} = \{Y_1, Y_2, \dots \}$ is called {\it{non-colliding}}, if by delaying their jump times one can keep both walks at distinct vertices forever. It was conjectured by P. Winkler that for large enough $M$ the set of pairs of non-colliding trajectories $\{\mathbb{X},\mathbb{Y} \} $ has positive measure. N. Alon translated this problem to the l…
We study the asymptotic behavior of a Markov chain on $\mathbb{Z}^2$ that corresponds to the two-dimensional marginals of a reinforcement process on $\mathbb{Z}^{\mathbb{N}}$. Three distinct asymptotic regimes are identified, depending on the scaling of the reinforcement parameter $Δ$ with respect to the number of steps performed by the chain.
We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\log \log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.
In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${\mathbb{Z}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${\mathcal{C}_{\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We…
We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. For the case where the corresponding branching process $\{Z_n(\mathbb{R})\}$ $ (n\in\mathbb{N})$ is supercritical, we establish large deviation principles, central limit theorems and a local limit theorem for the sequence of counting measures $\{Z_n\}$, and prove that the position $R_n$ (resp. $L_n$) of rightmost (resp. leftmost) particles of generation $n$ satisfies a law of large numbers.
We study massive (reccurent) sets with respect to a certain random walk $S_α$ defined on the integer lattice $\mathbb{Z} ^d$, $d=1,2$. Our random walk $S_α$ is obtained from the simple random walk $S$ on $\mathbb{Z} ^d$ by the procedure of discrete subordination. $S_α$ can be regarded as a discrete space and time counterpart of the symmetric $α$-stable Lévy process in $\mathbb{R}^d$. In the case $d=1$ we show that some remarkable proper subsets of $\mathbb{Z}$ , e.g. the set $\mathcal{P}$ of primes, are massive whereas some proper subsets of $\mathcal{P}$ such as Leitmann primes $\mathcal{P}_h…
Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $ξ=(ξ_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $(Σ_n)_{n\geq 0}$ where $$Σ_n=\sum_{k=0}^n ξ_{S_k}, n\in\mathbb{N}.$$ Under a weak dependence assumption on the scenery $ξ$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).
