Ron Doney, Dmitry Korshunov
We consider a transient random walk on $Z^d$ which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time $n$.
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Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely determines the random walk. There are trivial situations where two different random walks have the same return probabilities. However, among random walks in which these situations do not occur, our main result states that the return probabilities do determine the random walk. As a corollary, we will obtain a result dealing with the representation theory of $\math…
Expected First Return Times for Random Walks on Bounded Grids
本文推导了计算有限图上随机游走期望首次返回时间的一般公式，并利用该框架计算了不同边界条件下有界矩形网格上的期望首次返回时间。
A zero-one law of Engelbert--Schmidt type is proven for the norm process of a transient random walk. An invariance principle for random walk local times and a limit version of Jeulin's lemma play key roles.
How often does a critical elephant random walk return to origin
本文研究临界大象随机游走访问原点次数的渐近行为，证明大多数零点发生在最后一次过零前不久，并给出首次返回时间的尾部估计。
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
Transient NN random walk on the line
证明了最近邻瞬态随机游动在无穷远处的局部时的强定理。首先给出了局部时大值的重对数律。然后，我们研究步行穿过（总是从左到右）而不返回的间隔的长度。
On the number of cutpoints of the transient nearest neighbor random walk on the line
我们考虑实数直线的正部分上的瞬时最近邻随机游动。给出了割点和强割点个数有限的判别准则。实例和开放的问题。
On which graphs are all random walks in random environments transient?
无限图G具有如下性质：在G上的随机环境中的一个随机游动定义为i.i.d.具有任何共同分布的电阻几乎必然是瞬态的，当且仅当对于某个p<1，在具有参数p的键渗流下，简单随机游动在G的渗流簇上是瞬态的。
There is a condition (T'), such that it is the necessary condition that a random walk in random environment is ballistic. Under this condition, we show the law of the iterated logarithm for a random walk in random environment.
An intrinsic branching structure within the transient random walk on a strip in a random environment is revealed. As applications, which enables us to express the hitting time explicitly, and specifies the density of the absolutely continuous invariant measure for the "environments viewed from the particle".
Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Csáki, Földers, Révész [J Theor Probab (2010) 23: 624-638].
Return Probability for the Switch--Walk--Switch Lamplighter Walk on a Regular Tree
研究正则树上开关行走的返回概率渐进行为，通过数学方法推导出精确的渐近公式，并展示AI系统生成严格数学证明的能力。
Transient nearest neighbor random walk and Bessel process
证明了瞬态Bessel过程与由瞬态Bessel过程利用停时构造的最近邻随机游动之间强不变原理。同时证明了它们的局部时间足够接近，从而共享相同的强极限定理。此外，它表明，如果两个NN随机游动的分布之间的差异很小，那么步行本身可以构造，使它们足够接近。最后讨论了强极限定理的一些结果。
We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced r…
We study the range of a planar random walk on a randomly oriented lattice, already known to be transient. We prove that the expectation of the range grows linearly, in both the quenched (for a.e. orientation) and annealed ("averaged") cases. We also express the rate of growth in terms of the quenched Green function and eventually prove a weak law of large numbers in the (non-Markovian) annealed case.
We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger.
We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $\mathbb{Z}^d$, $d\ge 3$.
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。
Rate of growth of a transient cookie random walk
我们考虑一维瞬态cookie随机游动。从以前的论文中知道，cookie随机游动$(X_n)$根据某个正参数$α>1$或$\le 1$具有正速度或零速度。本文给出了零速区$(X_n)$的精确增长率，即：对于$0<α<1$，$X_n/n^{\frac{α+1}{2}}$按规律收敛于Mittag-Leffler分布，而对于$α=1$，$X_n(\log n)/n$按概率收敛于某个正常数.
Kesten et al.( 1975) proved the stable law for the transient RWRE (here we refer it as the $κ$-transient RWRE). After that, some similar interesting properties have also been revealed for its continuous counterpart, the diffusion proces in a Brownian environment with drift $κ$. In the present paper we will investigate the connections between these two kind of models, i.e., we will construct a sequence of the $κ$-transient RWREs and prove it convergence to the diffusion proces in a Brownian environment with drift $κ$ by proper scaling. To this end, we need a counterpart convergence for the $κ$-…
An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Brémont (2002).
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(ξ(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(ξ(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.
A new proof is given for the formula for the expected return time of a random walk on a graph. This proof makes use of known relationships between electric resistance and random walks.
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。
Transient Random Walks in Random Environment on a Galton-Watson Tree
考虑Galton-Watson树上随机环境中的瞬态随机游动$(X_n)$.在相当一般的假设下，我们给出了渐近速度为正的一个尖锐而明确的准则。因此，零速度的情况被揭示发生。在这种情况下，我们证明了$X_n$的数量级$n^Λ$，与$Λ\in (0,1)$。我们还证明了正则树上的线性边强化随机游动总是具有正的渐近速度，改进了Collevecchio \cite{Col06}的一个结果.
We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.
Simple Transient Random Walks in One-dimensional Random Environment: the Central Limit Theorem
我们考虑一个简单的随机游动（一维，最近邻跳跃）在淬火随机环境。这项工作的目标是提供充分条件，规定的环境属性，在此条件下，中心极限定理（CLT）的步行位置。在独立同分布环境和均匀遍历（因此也是弱混合）环境的情况下，这些条件导致问题的完全解。
Strong law of large numbers for a function of the local times of a transient random walk on groups
本文研究群上瞬态随机游走局部时间函数的强大数定律，利用子加性遍历定理在较弱条件下证明其几乎处处和$L^1$、$L^2$收敛。
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite Galton-Watson tree without leaves. We use regeneration times and ergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peres in 1996 to give an explicit construction of the invariant measure for the forward environment seen by the particule at exit times which is absolutely continuous with respect to the joint law of the tree and the path of …
Limit laws for transient random walks in random environment on $\z$
考虑了随机环境中$\z$上渐近速度为零的瞬态随机游动. Kesten，Kozlov和Spitzer的经典结果表明，水平$n$的首中时在适当的归一化后，按规律收敛于正稳定律，但他们没有得到其参数的描述。这一结果的一个不同的证明，导致这个稳定的法律的一个完整的表征。狄利克雷环境的情况是非常明确的。
Transient random walks on a strip in a random environment
考虑随机环境中带状上的瞬态随机游动。该模型由Bolthausen和Goldsheid提出[Comm.Math.Phys.214（2000）429- 447]。在一般遍历环境下，我们得到了随机游动的一个强大数律，并在一致混合环境下得到了一个退火的中心极限定理。此外，我们证明了法律的“环境从步行者的位置来看”收敛到一个极限分布，如果环境是一个独立同分布。顺序
We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1).
In this work, the probability of return for random walks on $\mathbb{Z}$, whose increment is given by the $k$-bonacci sequence, is determined. Also, the Hausorff, packing and box-counting dimensions of the set of these walks that return an infinite number of times to the origin are given. As an application, we study the return for tribonacci random walks to the first term of the tribonacci sequence.
We show that a sequence of birth-and-death chains, given by lazy random walks in a (transient) environment (RWRE) on [0; n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.
We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ has infinitely many cut times almost surely. This condition applies in particular to any graph of spectral dimension strictly larger than $2$. In fact, our proof applies to general (possibly nonreversible) Markov chains satisfying a similar decay condition for the Green's function that is sharp for birth-death chains. We deduce that a conjecture of Diaconis and Freedman (Ann. Probab. 1980) holds for the same class of …
Limit theorems for one-dimensional transient random walks in Markov environments
在随机环境为平稳马尔可夫过程的条件下，得到了一维随机游动的非高斯极限律。这是Kesten，M.独立同分布中随机游动的Kozlov和Spitzer环境.基本的假设是，底层的马尔可夫链是不可约的，或者具有有限的状态空间，或者具有由概率测度支配的上下过渡核。
Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{ξ(k), k \in \mathbb{Z} \}$ is i.i.d.
We examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper we handle the case of transient walks.
Moments and distribution of the local times of a transient random walk on $\Z^d$
考虑具有$d\in\N$的$\Z^d$上的任意瞬态随机游动。选取$α\in[0,\infty)$，并令$L_n(α)$为行走的$n$步长局部时间的$α$次幂的空间和。因此，$L_n(0)$是范围$L_n(1)=n+1$，对于整数$α$，$L_n(α)$是行走的$α$倍自相交的数目。证明了$L_n(α)$的一个强大数定律为$n\to\infty$ .此外，我们确定了局部时的渐近律在一个随机站点上均匀分布的范围。这些结果的补充和对比类似的结果经常走动在两个维度最近派生的Černalov \cite{Ce07}。虽然这些断言肯定是众所周知的专家，我们不能找到任何证据，在文献中，在这种普遍性。
A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random waker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we present a limit of the return probability for a discrete-time 3-state quantum walk on a hexagonal lattice.
We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. \cite{Sz01}). This weakens by first time the Kalikow ballistic assumption in mixing and proves finite moments of arbitrary order for the approximate regeneration time of \cite{CZ02}. The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.
Axis-Driven Random Walks on $\mathbb{Z}^2$ (transient cases)
本文研究 Z^2 上带轴排斥力的随机游走，证明即使参数 α 很小，游走仍具有瞬态超扩散行为，并给出分布尾部估计。
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…
Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks
在第一部分（数学。PR/0406392）中，我们证明了一个任意的一维随机游动与独立的增量，在一个给定的时间n，穿越一个水平的概率是极大序模平方根的n。在更高的维度中，我们称之为随机游动“周期性递归”（分别为：瞬变），如果a.s.无穷多个（resp.两个连续点之间的直线中的许多条都到达给定的有界集。上述估计意味着具有独立分量的三维随机游动是时间瞬变的。类似地，在Mauldin、Monticino和v.Weizsaecker [1]意义下，Z ^3上的定向强化随机游动也是瞬态的。另一方面，我们构造了一个在Z ^2上具有独立分量的瞬时但连续递归的随机游动的例子。
We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1) ... X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n \rightarrow \infty$ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.
We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate last passage decomposition. Secondly, we obtain exponential moment bounds for functionals of the branching random walk, under optimal condition. As a corollary we obtain an approximate variational characterisation of the branching capacity. We finally derive upper bounds involving the branching capacity for the tail of the time spent in an arbitrary finite coll…
Scaling limit for local times and return times of a randomly biased walk on a Galton-Watson tree
本文研究超临界 Galton-Watson 树上零常返随机游走（次扩散区间）的局部时间与第 n 次返回时间，证明经重整化后分别依分布收敛到稳定 Lévy 过程的最大值与击中时间，推广了 Hu 的结果。
Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment
本文证明在亚弹道随机环境中，一个常返简单随机游走与任意多个暂态随机游走几乎必然在无穷多个时刻同时碰撞于同一位置。
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, …
Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(ξ(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(ξ(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(ξ(S_n))_{n\geq 0}$.
Asymptotic estimates for multiple point ranges of transient random walks on graphs
本文通过用关于返回起始点时间的均匀尾部条件替代代数和平移不变性假设，研究了图上随机游走的多重点范围，得到了访问次数至少为给定次数和恰好为给定次数的位点数期望的线性阶上下界，并在更强条件下证明了几乎必然界。
In this article we continue the study of the quenched distributions of transient, one-dimensional random walks in a random environment. In a previous article we showed that while the quenched distributions of the hitting times do not converge to any deterministic distribution, they do have a weak weak limit in the sense that - viewed as random elements of the space of probability measures - they converge in distribution to a certain random probability measure (we refer to this as a weak weak limit because it is a weak limit in the weak topology). Here, we improve this result to the path-valued…
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 $C_y >0$.
Geometric approximation for the number of returns of a transient Markov chain to its origin
针对离散时间暂态马尔可夫链，建立了前n步内返回起始状态次数的几何分布近似的显式全变差误差界，并用热核表示，应用于几何分支过程和多种随机游走模型。
This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk's probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over…
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…
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…
Heat Kernel Estimates for Random Walks with Killing on Transient Graphs
针对暂态图上带杀伤的随机游走，研究人员对加权图上薛定谔算子Δ+W的热核给出匹配的双侧估计，通过两种方法证明结果，适用于分形型图等示例。
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