We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.
Excited random walk is a random walk that has a positive drift to the right when it reaches a vertex it hasn't been to before. We show that in three dimensions the walk drifts to the right in non-zero speed.
A random walk on Z^d is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on Z^d, is transient iff d>1.
An excited random walk is a non-Markovian extension of the simple random walk, in which the walk's behavior at time $n$ is impacted by the path it has taken up to time $n$. The properties of an excited random walk are more difficult to investigate than those of a simple random walk. For example, the limiting speed of an excited random walk is either zero or unknown depending on its initial conditions. While its limiting speed is unknown in most cases, the qualitative behavior of an excited random walk is largely determined by a parameter $δ$ which can be computed explicitly. Despite this, it i…
Excited random walk is a process that has a drift to the right whenever it encounters a new vertex. The paper shows that in two dimensions it drifts to the right linearly in time.
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 $\Z$.
We give a "direct" coupling proof of strict monotonicity of the speed for 1-dimensional multi-excited random walks with positive speed. This reproves (and extends) a recent result of Peterson without using branching processes.
We consider one-dimensional excited random walks with finitely many cookies at each site. There are certain natural monotonicity results that are known for the excited random walk under some partial orderings of the cookie environments. We improve these monotonicity results to be strictly monotone under a partial ordering of cookie environments introduced by Holmes and Salisbury. While the self-interacting nature of the excited random walk makes a direct coupling proof difficult, we show that there is a very natural coupling of the associated branching process from which the monotonicity resul…
In this paper, we give a detailed construction of an example of excited random walk with speed zero in an ergodic random environment that have an infinite average number of cookies in each site. This example confirms that a result of Mountford, Pimentel and Valle (2006), which gives a sufficient condition for excited random walks in deterministic environment to have positive speed, can not be extended to ergodic random environment.
We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n.
We give non-trivial upper and lower bounds on the range of the so-called Balanced Excited Random Walk in two dimensions, and verify a conjecture of Benjamini, Kozma and Schapira. To the best of our knowledge these are the first non-trivial results for this 2-dimensional model
We consider a class of self-interacting random walks in deterministic or random environments, known as excited random walks or cookie walks, on the d-dimensional integer lattice. The main purpose of this paper is two-fold: to give a survey of known results and some of the methods and to present several new results. The latter include functional limit theorems for transient one-dimensional excited random walks in bounded i.i.d. cookie environments as well as some zero-one laws. Several open problems are stated.
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].
The phase transition of $M$-digging random on a general tree was studied by Collevecchio, Huynh and Kious (2018). In this paper, we study particularly the critical $M$-digging random walk on a superperiodic tree that is proved to be recurrent. We keep using the techniques introduced by Collevecchio, Kious and Sidoravicius (2017) with the aim of investigating the phase transition of Once-excited random walk on general trees. In addition, we prove if $\mathcal T$ is a tree whose branching number is larger than $1$, any multi-excited random walk on $\mathcal{T}$ moving, after excitation, like a s…
We study the asymptotics of the probabilities of extreme slowdown events for transient one-dimensional excited random walks. That is, if $\{X_n\}_{n\geq 0}$ is a transient one-dimensional excited random walk and $T_n = \min\{ k: \, X_k = n\}$, we study the asymptotics of probabilities of the form $P(X_n \leq n^γ)$ and $P(T_{n^γ} \geq n )$ with $γ< 1$. We show that there is an interesting change in the rate of decay of these extreme slowdown probabilities when $γ< 1/2$.
We study the large deviations of one-dimensional excited random walks. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $…
We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious w…
We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method of investigations based on a study of the Radon-Nikodym density of the ERW distribution wit…
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.
Let $W$ be an integer valued random variable satisfying $E[W] =: δ\geq 0$ and $P(W<0)>0$, and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $x\in \mathbb{Z}$ the size of the next step is an independent random variable with the same distribution as $W$. We show that this self-interacting random walk is recurrent if $δ\leq 1$ and transient if $δ>1$. This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-near…
The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.
Consider two random walks on $\mathbb{Z}$. The transition probabilities of each walk is dependent on trajectory of the other walker i.e. a drift $p>1/2$ is obtained in a position the other walker visited twice or more. This simple model has a speed which is, according to simulations, not monotone in $p$, without apparent "trap" behaviour. In this paper we prove the process has positive speed for $1/2<p<1$, and present a deterministic algorithm to approximate the speed and show the non-monotonicity.
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.
Excited random walks (ERWs) are a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as $V = \lim_{n \rightarrow \infty} \frac{X_n}{n}$ where $X_n$ is the state of the walk at time $n$. While results exist that indicate when the speed is non-zero, there exists no explicit formula for the speed. It is difficult to solve for the speed directly due to complex dependencies in the walk since the next step of the walker depends on how many ti…
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…
Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter delta. For recurrence/transience the critical threshold is |delta|=1, for ballisticity it is |delta|=2 and for diffusivity |delta|=4. In this paper we establish a phase transition at |delta|=3. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff |delta|>3. This result follows from an explicit description of the tail behaviour of the return time as a function of delta, which is achieved by diffusion approximatio…
Using renewal times and Girsanov's transform, we prove that the speed of the excited random walk is infinitely differentiable with respect to the bias parameter in $(0,1)$ for the dimension $d\ge 2$. At the critical point $0$, using a special method, we also prove that the speed is differentiable and the derivative is positive for every dimension $2\leq d\neq 3.$ However, this is not enough to imply that the speed is increasing in a neighborhood of $0.$ It still remains to prove the derivative is continuous at $0$. Moreover, this paper gives some results of monotonicity for $m-$excited random …
We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.
Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $\mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to…
We study a natural continuous time version of excited random walks, introduced by Norris, Rogers and Williams about twenty years ago. We obtain a necessary and sufficient condition for recurrence and for positive speed. This is analogous to results for excited (or cookie) random walks.
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 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…
We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say $ϕ$, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by Tóth and his coauthors in multiple occasions, wh…
We study a model of multi-excited random walk with non-nearest neighbour steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either $x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton-Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh [Probab. Theory Related Fields (2008), 141 (3-4)], we extend their result (w.r.t the case $L=1$) to our model. More specif…
We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $e_1-$exchangeable cookie environment and prove the law of large numbers for this case.
We describe scaling limits of recurrent excited random walks (ERWs) on integers in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, delta, belongs to the interval [-1,1]. We show that if |delta|<1 then the diffusively scaled ERW under the averaged measure converges to a (delta,-delta)-perturbed Brownian motion. In the boundary case, |delta|=1, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens…
We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero.
The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in $2011$, is defined as a discrete time stochastic process in $\mathbb Z^d$, depending on two integer parameters $1\le d_1,d_2\le d$, which whenever it is at a site $x\in\mathbb Z^d$ at time $n$, it jumps to $x\pm e_i$ with uniform probability, where $e_1,\ldots,e_d$ are the canonical vectors, for $1\le i\le d_1$, if the site $x$ was visited for the first time at time $n$, while it jumps to $x\pm e_i$ with uniform probability, for $1+d-d_2\le i\le d$, if the site $x$ was already visited before time $n$. Here we give…
In this paper we consider an excited random walk on $\mathbb{Z}$ in identically piled periodic environment. This is a discrete time process on $\mathbb{Z}$ defined by parameters $(p_1,\dots p_M) \in [0,1]^M$ for some positive integer $M$, where the walker upon the $i$-th visit to $z \in \mathbb{Z}$ moves to $z+1$ with probability $p_{i\pmod M}$, and moves to $z-1$ with probability $1-p_{i \pmod M}$. We give an explicit formula in terms of the parameters $(p_1,\dots,p_M)$ which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case…
We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed.
We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle \cite{Pem000} on trees.
The paper considers excited random walks (ERWs) on integers in i.i.d. environments with a bounded number of excitations per site. The emphasis is primarily on the critical case for the transition between recurrence and transience which occurs when the total expected drift $δ$ at each site of the environment is equal to 1 in absolute value. Several crucial estimates for ERWs fail in the critical case and require a separate treatment. The main results discuss the depth and duration of excursions from the origin for $|δ|=1$ as well as occupation times of negative and positive semi-axes and scalin…
According to Comets, Gantert and Zeitouni on the one hand and to Derriennic on the other hand, some functionals associated to the hitting times of random walks in random environment on the integer line coincide, for the walk itself and for the walk in the reversed environment. We show that these two duality principles are algebraically equivalent, that they both stem from the Markov property of the walk in a fixed environment, and not of the ergodicity of the model, and that there exists finitist and almost sure versions of this duality.
In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by Benjamini and Wilson. We consider a discrete-time stochastic process $(X_n,n=0,1,2,...)$ taking values on ${\mathbb{Z}}^d$, $d\geq2$, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction $\ell$; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove th…
We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive `step' or a multiplicative `jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.
We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the proce…
We consider the range of a one-parameter family of self-interacting walks on the integers up to the time of exit from an interval. We derive the weak convergence of an appropriately scaled range. We show that the distribution functions of the limits of the scaled range satisfy a certain class of de Rham's functional equations. We examine the regularity of the limits.
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 consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the "core" process. Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional rando…
For a symmetric, homogeneous and irreducible random walk on d-dimensional integer lattice Z^d, having zero mean and a finite variance of jumps, we study the passage times (with possible infinite values) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contr…
We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^{-(2-p)}. Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p<1/2. For the infinite line,…
We study certain self-interacting walks on the set of integers, that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighbourhood of their current position (in the present paper, typically, we will discuss the case where one considers the neighbouring edges and the next-to-neighbouring edges). We survey a variety of possible behaviours, including some where the walk is eventually confined to an interval of large length. We also focus on certain "asymmetric" drifts, where we prove that with posi…
