Central Limit Theorem for the Excited Random Walk in dimension $d \geq 2$
Jean Bérard, Alejandro Ramírez
本文证明了受激随机游动模型在每一维$d \geq 2$中的大数定律和中心极限定理成立。
We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d \geq 2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Excited random walk in three dimensions has positive speed
受激随机游动是一种随机游动，当它到达一个它以前没有到达过的顶点时，向右有一个正漂移。我们表明，在三维的步行漂移到右边的非零速度。
Excited Random Walk
如果Z^d上的随机游动第一次访问一个顶点时，在一个方向上有偏差，但在随后访问该顶点时，游动者均匀随机地选择一个邻居，则该游动被激发。我们证明了Z^d上的受激随机游动是瞬态的当且仅当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 in two dimensions has linear speed
受激随机游动是一个过程，每当遇到一个新的顶点时，它就会向右漂移。该论文表明，在二维空间中，它随时间线性向右漂移。
Recurrence and transience of excited random walks on $\Z^d$ and strips
我们研究了$\Z^d, d\ge 1,$和平面带$\Z\times\{0,1,...,L-1\}$上的受激随机游动，平面带在给定的方向上有漂移。漂移的强度可以取决于随机i.i.d.环境和步行的当地时间。给出了常返性和瞬时性的精确判据，从而推广了Benjamini和Wilson关于$\Z^d$上一次激发随机游动和作者关于$\Z$上多次激发随机游动的结果.
Monotonicity for excited random walk in high dimensions
证明了在$d$维中受激随机游动的漂移$θ(d,β)$对于激励参数$β\in[0, 1]$是单调的，当$d\ge 9$ .
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.
Multi-excited random walks on integers
引入了一类随机和非随机介质中的最近邻整数随机游动，其中包括文献中考虑的受激随机游动。在每个地点，随机步行者都有一个向右的漂移，其强度取决于该地点的环境和步行者之前访问该地点的频率。我们给出了精确的标准，经常性和瞬态，并考虑步行的速度。
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.
Excited random walk against a wall
我们分析了三维晶格上半部分的随机游动，当它遇到一个新的顶点时，它就会下降。受激随机游动。我们表明，它是经常性的期望数量的平方根log n的回报。
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…
Recurrence and transience of a multi-excited random walk on a regular tree
研究了正则树上的多激励随机游动模型，推广了Volkov（2003）提出的一次激励随机游动模型和挖掘随机游动模型.我们证明了存在一个相变的递归/瞬态性质的步行。特别地，我们证明了行走的渐近行为取决于激励的顺序，这与Zerner（2005）研究的一维设置形成对比。我们还考虑了极限速度的步行在瞬态政权和猜想，它不是一个单调的功能的环境。
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.
Maximum occupation time of a transient excited random walk on Z
研究暂态激发随机游走在Z上当前最常访问位点的占据时间的渐近行为，结果表明与随机环境中的随机游走不同，暂态激发随机游走不会在偏好位点（截至某日期最常访问的位点）上花费渐近正的时间比例。
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…
On the speed of the one-dimensional excited random walk in the transient regime
研究了Zerner提出的一类最近邻离散时间整数随机游动，即所谓的多激励随机游动。这种随机步行者的跳跃概率向右漂移，其强度取决于随机或非随机环境，该环境也根据最后访问的站点随时间演变。Zerner在相当一般的环境假设下给出了递归和瞬态阶段的完整描述。我们在本文中贡献的一些结果，使我们能够指出，如果随机步行速度是严格的积极或不在瞬态情况下的一类非随机环境。
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 look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $κ$ for some explicit $κ$. We show that the walk, after renormalization, actually converges to a $κ$-stable completely asymmetric Levy Process.
Positively and negatively excited random walks on integers, with branching processes
考虑整数上的受激随机游动，其独立同分布的个数有界.每个站点的cookie可能会导致向左和向右的漂移。我们推广了M. Zerner和A.- L. Basdevant和A. Singh对这种情况进行了改进，并证明了一个退火的中心极限定理.这些证明基于有关迁移分支过程的文献结果，并利用了一定的更新结构。
A shape theorem for a radially excited random walk in dimensions $d \ge 2$
本文研究d≥2维径向一次激发随机游走，证明其常返性及n步轨迹渐近为欧氏球的球形形状定理，证实了Kozma 2007年的相关猜想。
Recurrence and range of the balanced excited random walk M(2,1,2)
本文证明平面平衡激发随机游走 M(2,1,2) 常返，其访问范围极限与平面简单随机游走相同，并推广至空间非均匀饼干环境。
A note on transience of generalized many-dimensional excited random walks
本文研究广义激发随机游走的一种变体，其漂移下界随时间衰减至零，证明当下界衰减慢于 $n^{-β}$ 时，游走在漂移方向上具有暂态性。
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…
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 study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $\exp ( - |R_N|)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately $ρ_d N^{1/(d+2)}$, for some explicit constant $ρ_d >0$. This proves a conjecture of Bolthausen who obtained this result in the case $d=2$.
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.
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…
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 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…
Consider the real Markov walk $S_n = X_1+ \dots+ X_n$ with increments $\left(X_n\right)_{n\geq 1}$ defined by a stochastic recursion starting at $X_0=x$. For a starting point $y>0$ denote by $τ_y$ the exit time of the process $\left( y+S_n \right)_{n\geq 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event $τ_y \geq n$ and of the conditional law of $y+S_n$ given $τ_y \geq n$ as $n \to +\infty$.
Limit shapes for centrally excited walks
本文证明了 Kozma 关于中心激发随机游走在所有维度 $d\geq2$ 中访问集收敛到欧几里得球的猜想，并给出访问次数均匀收敛到锥体的结果，同时推广到一般范数情形。
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…
Excited Random Walk in One Dimension
我们研究了兴奋的随机游动，在这个游动中，一个在含有饼干的站点上的游动吃了一个饼干，然后以概率p向右跳，以概率q=1-p向左跳。如果游动跳到一个空站点上，就没有偏差。对于半线上的1-兴奋行走（每个站点最初有一个cookie），在时间t首次返回起点的概率按比例缩放为t^{-（2-p）}。虽然对于所有p，返回原点的平均时间都是无限的，但当p<1/2时，在第一次返回之前，步行平均只吃有限数量的饼干。对于无限长的直线，1-激发行走的概率分布在原点有一个不寻常的异常。最左边和最右边未吃的饼干的位置可以通过概率参数精确估计，并且它们对应的分布在原点附近具有幂律奇点。无限直线上的2-激发行走在p>3/4的区域中表现出独特的特征，其中行走是短暂的，包括随t^{nu}增长的平均位移，其中nu>1/2取决于p，以及缩放的分解步行的概率分布。
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…
Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products of the increments of $X$ and study the two-dimensional walk $(X,Y)=((X_n,Y_n))_{n\geq0}$. We show that it is recurrent and when suitably normalised converges to a two-dimensional Brownian motion with independent components; this independence occurs despite the functional dependence between the pre-limit processes. The process of recycling increments in this way is repeated and a multi-dimensional analog of this limit theorem together with a …
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $\mathbb{Z}^d$, $d\geq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $\varrho_N=C N^{1/(d+2)}$ for any $d\geq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-ε)\varrho_N$ for any $ε>0$, while the case $d\geq 3$ remains open. We complete the picture by showing that for any $d\geq 2$, the random walk range asymptotically contains a ba…
Tail asymptotics for the maximum of perturbed random walk
考虑一个随机游动$S=(S_n:n\geq 0)$，它被平稳序列$(ξ_n:n\geq 0)$“扰动”以产生过程$(S_n+ξ_n:n\geq0)$。本文研究了具有负漂移的扰动随机游动的$M_{\infty}=\max \{S_k+ξ_k:k\geq0\}$的分布。这种最大值出现在几种不同的应用环境中，包括生产系统、通信网络和保险风险。我们的主要结果描述了$\mathbb{P}(M_{\infty}>x)$作为$x\to\infty$的渐近性。尾部渐近性很大程度上取决于$ξ_n$是轻尾还是重尾。在轻尾条件下，尾渐近与标准随机游动的Cramér-Lundberg渐近密切相关.
Let $\{S_n,n\geq 0\} $ be a random walk whose increments belong without centering to the domain of attraction of an $α$-stable law $\{Y_t,t\geq 0\}$, i.e. $S_{nt}/a_n\Rightarrow Y_t,t\geq 0,$ for some scaling constants $a_n$. Assuming that $S_0=o(a_{n})$ and $S_n\leq φ(n)=o(a_n),$ we prove several conditional limit theorems for the distribution of $S_{n-m}$ given $m=o(n)$ and $\min_{0\leq k\leq n}S_k\geq 0$. These theorems complement the statements established by F. Caravenna and L. Chaumont in 2013. The obtained results are applied for studying the population size of a critical branching proc…
Excited against the tide: A random walk with competing drifts
我们研究一个随机游动，有一个漂移$\fracβ{d}$的权利时，位于一个以前未访问的顶点和漂移$\fracμ{d}$的左边，否则。我们证明，在高维中，对于每个$μ$，向右漂移是一个严格增加和连续函数$β$，并恰好有一个值$β_0(μ,d)$的速度为零。
We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $ω_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(ω_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。