Tianyi Bai, Yijun Wan
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 $\frac{n}{\log n}$.
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We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.
We consider random walks indexed by arbitrary finite random or deterministic trees. We derive a simple sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump. This criterion is given in terms of four quantities : the tail and the expectation of the random walk steps, the height of the tree and the number of its vertices. The results are applied to critical Galton--Watson trees with offspring distributions in the domain of attraction of a stable law.
We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To…
We provide asymptotics for the range R(n) of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that R(n)/n converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4 and in the case of a symmetric random walk with exponential moments, we prove that R(n) grows like n/(log n). We apply our results to asymptotics for the range of branching random walk when the initial size of the population tends to infi…
We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree Gamma are i.i.d.random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on Gamma is transient, but not vice versa. We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish re…
Let (S_n)_{n\in\N} be a Z-valued random walk with increments from the domain of attraction of some α-stable law and let (ξ(i))_{i\in\Z} be a sequence of iid random variables. We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\sum_{1\leq i<j\leq n}h(ξ(S_i),ξ(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n.
We study the range $R_n$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_n$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching ran…
Intersections of branching random walks on $\mathbb{Z}^8$
本文研究$\u2124^8$上以无穷不变树为索引的分支随机游走的相交问题，得到两类不同索引的独立游走间不相交概率的精确阶数，并证明分支随机游走值域的分支容量满足弱大数定律。
A central limit theorem for the range of random walk indexed by an infinite tree
本文研究了临界分支随机游走范围大小的二阶波动，证明在不同维度下其范围满足中心极限定理，并结合多种数学工具进行证明。
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.
We study the capacity of the range of a transient simple random walk on $\mathbb{Z}^d$. Our main result is a central limit theorem for the capacity of the range for $d\ge 6$. We present a few open questions in lower dimensions.
In this paper, we 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.
We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_σ)_{σ\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(Σ_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $Σ_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{Σ_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empir…
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.
Paths maximize the expected range of graph-indexed random walks
本文证明路径最大化图索引随机游走的期望值域，确立BHM猜想期望形式，并由此推导出LNR猜想，证明经Lean 4验证。
Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $τ$. Given a local event $E(i,τ)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $τ$, we prove the existence of a Lipschitz connected surface of cells $(i,τ)$ that separates the origin from infinity on w…
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 give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Z\times \N$) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.
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。
We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.
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。
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.
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。
We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$ and $\tilde Z_n(t)=\int e^{tx}Z_n(dx)$ be its Laplace transform. We show the $L^p$ convergence rate and the uniform convergence of the martingale $\tilde Z_n(t)/\mathbb E[\tilde Z_n(t)|ξ]$, and establish a moderate deviation principle for the measures $Z_n$.
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 consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. For the case where the corresponding branching process $\{Z_n(\mathbb{R})\}$ $ (n\in\mathbb{N})$ is supercritical, we establish large deviation principles, central limit theorems and a local limit theorem for the sequence of counting measures $\{Z_n\}$, and prove that the position $R_n$ (resp. $L_n$) of rightmost (resp. leftmost) particles of generation $n$ satisfies a law of large numbers.
We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This index…
We consider a branching random walk with immigration in a random environment, where the environment is a stationary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures $(Z_n)$ that count the number of particles of generation $n$ located in a Borel set of real line. In the present work, a series of limit theorems related to the above counting measures are established, including a central limit theorem, a moderate deviation principle and a large deviation result as well as a convergence theorem of the free energy.
We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{γ\mathcal R^N_z}$ for $γ>0$. We show that for $γ>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - γ^2/10})$.
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。
Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d > 4 there almost surely do not exist such t. Hoffman showed that for d=2 there almost surely exists t such that the random walk at time t visits the origin only finitely many times. We refine the results of Hoffman for dynamical random walk on Z^2, showing that with probability one there are times when the or…
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 work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the $d$-dimensional integer lattice and consider balanced replacement schemes associated with bounded increment random walks on it. We prove central and local limit theorems for the random color of the $n$-th selected ball and show that irrespective of the null recurrent or transient behavior of the underlying random walks, the asymptotic distribution is Gaussian after appropriate centering and scaling. We show tha…
The extinction rate of a branching random walk with a barrier in a time-inhomogeneous random environment
本文研究时间非齐次随机环境中带随机吸收屏障的超临界分支随机游走，给出屏障导致系统灭绝时在 $L^p$ 意义下的灭绝速率，并证明随机环境下随机游走小偏差概率的 $L^p$ 收敛性。
Recurrence and transience of the critical random walk snake in random conductances
本文研究临界Galton-Watson树上分支随机游走在随机电导或陷阱环境中的常返性与暂留性，证明在有限三阶矩及适当技术条件下，过程在四维及以下常返、更高维暂留，证明基于截断二阶矩方法。
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。
Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $Λ(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\bar S_v=\max\{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}\}$, define $L_n=\min_{v\in \mathbb{D…
We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the pro…
Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central…
We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covaria…
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…
We use the concept of unimodular random graph to show that the branching simple random walk on $\mathbb{Z}^{d}$ indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if $d \leq 4$.
Vertex reinforced branching random walks and generalized time-dependent Polya urns
研究整数集上受顶点强化的无限临界树索引随机游走，关注强强化情形。通过分析时间依赖广义波利亚瓮过程，对每步抽取次数由序列规定、选球概率与球数函数相关的情况进行研究，有界序列时恢复鲁宾颜色固定特征描述。
In this note, we establish a functional central limit theorem for the capacity of the range for a class of $α$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5α/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3α/ 2$.
Drazin-Inverse and heat capacity for driven random walks on the ring
本文利用单树与双树表示及后向生成元 Drazin 逆的图形表示，精确计算环上驱动随机游走的非平衡热容与过剩热，并处理扩散极限。
Extremal process of the local time of simple random walk on a regular tree
本文研究正则有根树上连续时间简单随机游走在两种起止条件下叶节点局部时的极值极限，证明其收敛为随机强度测度下的装饰 Poisson 点过程，并借助同构定理通过 Lindeberg 型交换确定装饰普适性。
We show that any stationary symmteric $α$-stable ($SαS$) random field indexed by a countable amenable group $G$ is weakly mixing if and only if it is generated by a null action, extending works of Samorodnitsky and Wang-Roy-Stoev for abelian groups to all amenable groups. This enables us to improve significantly the domain of a recently discovered connection to von Neumann algebras. We also establish ergodicity of stationary $SαS$ fields associated with boundary and double boundary actions of a hyperbolic group $G$, where the boundary is equipped with either the Patterson-Sullivan or the hitti…
Derivative formula for capacities
本文获得了多种容量概念的导数公式，确定了两个集合并集容量渐近展开的二阶项，适用于牛顿容量、Bessel-Riesz容量及Branching capacity，但渗流设定中的容量问题仍待解决。
Let $X$ be a simple random walk on $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected cover time. We consider the set of points $\mathcal{U}_α$ of $\mathbb{Z}_n^d$ that have not been visited by the walk by time $αt_{\rm{cov}}$ for $α\in (0,1)$. It was shown in [MS17] that there exists $α_1(d)\in (0,1)$ such that for all $α>α_1(d)$ the total variation distance between the law of the set $\mathcal{U}_α$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb{Z}_n^d$ with success probability $n^{-αd}$ tends to $0$ as $n \to \infty$. In [MS17] the constant $α_1…
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