Amine Asselah, Bruno Schapira, Perla Sousi
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.
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We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\Z^d, d\geq5$. Analogous to the regular discrete capacity, branching capacity is closely related to the asymptotics of the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point and the conditional distribution of the hitting point.
Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints
本文研究了在直径约束下随机游走容量的相变现象，并揭示了其容量与几何形状之间的关系。
We continue our study of critical branching random walk and branching capacity. In this paper we introduce branching recurrence and branching transience and prove an analogous version of Wiener's Test.
In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem.
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.
By introducing a new measure for the infinite Galton-Watson process and providing estimates for (discrete) Green's functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high dimensions $d\ge 7$, the capacity grows linearly; and in the critical dimension $d=6$, it grows asymptotically proportional to $\frac{n}{\log n}$.
In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.
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.
Branching capacity of a random walk in $\mathbb Z^5$
本文研究五维整数格点上随机游走值域的分支容量，证明其重整化后依分布收敛于布朗运动值域的布朗蛇容量，关键步骤是分析临界分支随机游走与随机游走值域的相交概率。
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…
We prove that in any finite set of $\mathbb Z^d$ with $d\ge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {\it covering uniformly} a finite set, and characterize some {\it folding} events, under optimal hypotheses. For instance, knowing that a region of space has an {\it atypically high occupation density} by some random walk, we show that this random region is most likely ball-like
Asymptotic Capacity of the Range of Random Walks on Free Products of Graphs
本研究证明了图的自由积上随机游走值域的渐近容量存在且几乎必然为严格正的常数，同时建立了相应的中心极限定理，并揭示了该容量关于有限支撑恒定概率测度的实解析变化性质。
Capacity of loop-erased random walk
论文研究了在Z^d上环擦随机游走（LERW）的容量，证明了四维中强大数定律，并给出了极限表达式；在三维中展示了容量的标度极限是随机的，并给出了容量的标度极限与Kozma的LERW标度极限的关系。
Capacity of the range of random walk: The law of the iterated logarithm
本文研究 $d\ge 3$ 简单随机游走范围容量的极限行为，建立其上、下极限重对数律，并给出三维 Brown 路径结果与四维精确中偏差界。
We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $\mathbb{Z}^d$, in dimension $d\ge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different
Branching capacity of a random walk range
研究 $d\ge5$ 维简单随机游动范围的分支容量，证明其大数定律并确定增长阶，核心贡献是 6 维定律，需引入新要素求树索引游动与双侧游动非相交概率的精确渐近性。
Invariance principle for the capacity and the cardinality of the range of stable random walks
本文针对整数格上的 α-稳定随机游走，在维度与稳定性指数满足特定条件下证明了值域容量和基数的几乎必然不变性原理，并由此导出相应的重对数律。
Moderate Deviations for the Capacity of the Random Walk range in dimension four
本文研究四维随机游走容量的中偏差问题，给出对应二维随机游走值域体积中偏差结果的四维类比，揭示其偏差统计量与广义Gagliardo-Nirenberg不等式常数的关联。
Convergence in law for the capacity of the range of a critical branching random walk
本文证明在三维、四维和五维空间中，临界分支随机游走值域的重整化容量依分布收敛到积分超布朗游标支撑的容量，证明基于相交概率分析。
Consider a branching random walk $(V_u)_{u\in \mathcal T^{IGW}}$ in $\mathbb Z^d$ with the genealogy tree $\mathcal T^{IGW}$ formed by a sequence of i.i.d. critical Galton-Watson trees. Let $R_n $ be the set of points in $\mathbb Z^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in $\mathcal T^{IGW}$. Our main result states that for $d\in \{3, 4, 5\}$, the capacity of $R_n$ is almost surely equal to $n^{\frac{d-2}{2}+o(1)}$ as $n \to \infty$.
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.
Recurrence and capacity of stable branching random walks
本文针对后代分布为临界且吸引α-稳定律的分支随机游走，研究其范围线性增长率，扩展分支容量至α-稳定情形并估计球的对应容量。
Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration
该研究证明整数集上两类相互作用粒子系统为Pfaffian点过程，经扩散重标度得到实数集上的多种极限Pfaffian点过程，含布朗网对应点集过程。
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 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.
Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5
本文研究4维和5维随机游走值域容量的中偏差，在5维中刻画了高斯与非高斯尾部两种机制，并将方法推广到4维几乎全范围，扩展了已有中偏差结果。
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…
Left-continuous random walk on $\mathbb{Z}$ and the parity of its hitting times
该研究聚焦整数集上的左连续随机游走，推导其在奇偶时刻处于负值的概率，进而确定近左连续随机游走最终变负的概率，为分支过程相关问题提供了概率分析支撑。
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.
Asymptotic probability of a fixed edge being on the boundary of the convex hull of a random walk in $\mathbb{Z}^2$
研究二维对称随机游走中固定边在凸包边界多边形中的渐近概率行为，分析步数趋于无穷时的概率渐进行为。
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.
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.
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.
How often can two independent elephant random walks on $\mathbb{Z}$ meet?
本文研究整数格上两个独立象随机游走的相遇频率，发现取决于记忆参数p是否大于3/4，并给出距离渐近结果。
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 $C_y >0$.
Law of the Iterated Logarithm for $p$-Walks on $\mathbb{Z}$
研究 $p$-转子游走的几乎必然渐近边界，通过分解为鞅与运行极值扰动，建立函数型迭代对数定律并求解变分问题得到经典常数。
A zero-one law for random walks in random environments on $\mathbb{Z}^2$ with bounded jumps
本文将两类最近邻随机环境中随机游走的方向暂留性结果推广到有界跳跃情形，包括Zerner-Merkl的平面0-1律在Z^2非平面独立同分布环境中的推广，以及Sabot-Tournier的Dirichlet环境方向暂留性刻画在有界跳跃下的推广。
Branching random walks and Minkowski sum of random walks
本研究证明d≥5维下存活临界分支随机游走值域与两独立简单随机游走值域的闵可夫斯基和交等价，推广了Wiener香肠体积大数律的离散版本，并建立了容量与分支容量的新关系。
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.
A functional limit theorem for lattice oscillating random walk
本文研究了 $\mathbb{Z}$ 上 Kemperman 振荡随机游走的不变性原理，通过利用考虑正负半轴间振荡的 Markov 子过程构造非周期更新算子序列，并结合 S. Gouëzel 的定理，推广了经典与反射随机游走的结果。
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…
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