Maximal local time of a d-dimensional simple random walk on subsets

Maximal displacement of a branching random walk in time-inhomogeneous environment

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other…

超临界分支随机游走于群自由积上的最大和最小位移

Maximal and minimal displacement of supercritical branching random walks on free products of groups

研究超临界分支随机游走于群自由积上的最大和最小位移增长特性，证明其几乎必然线性增长，通过大偏差率函数与水平线的交点确定速度。

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures' laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its ma…

Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.

Random walk on comb-type subsets of Z^2

We study the path behavior of the simple symmetric walk on some comb-type subsets of Z^2 which are obtained from Z^2 by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences.

Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between t…

Maximal displacement of simple random walk bridge on Galton-Watson trees

We analyze simple random walk on a supercritical Galton-Watson tree, where the walk is conditioned to return to the root at time $2n$. Specifically, we establish the asymptotic order (up to a constant factor) as $n\to\infty$, of the maximal displacement from the root. Our results, which are shown to hold for almost surely every surviving tree $T$ (subject to some mild moment conditions), are broken up into two cases. When the offspring distribution takes a value less than or equal to $1$ with positive probability, the maximal displacement of the bridge is shown to be on the order of $n^{1/3}$.…

径向对称分支随机游走在$\mathbb{R}^d$中的最大位移

The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$

本文研究径向对称分支随机游走的最大位移，证明在生存条件下其增长与分支布朗运动一致，主要工具为带移动障碍的投票定理。

On the Maximal Displacement of a Critical Branching Random Walk

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance $η^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $σ^{2}$ and finite third moment then the di…

关于随机环境中亚临界分支随机游走最大位移的研究

On the maximal displacement of subcritical branching random walk in random environment

本文研究了随机环境中亚临界分支随机游走的最大位移问题，采用独立的分支和步跃过程，分析了在随机环境中分支概率分布对游走位移的影响。

Maximal Displacement for Bridges of Random Walks in a Random Environment

It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\{S_{2n}=0\}$ the maximal displacement $\max_{k\leq 2n} |S_k|$ converges in distribution when scaled by $\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\bf{Z}$. We show that under the quenched law $P_ω$ (conditioned on the environment $ω$), the maximal displacement of the random walk when condit…

On the Maximal Displacement of Near-critical Branching Random Walks

We consider a branching random walk on $\mathbb{Z}$ started by $n$ particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring $1+θ/n$. For $t\geq 0$, we study $M_{nt}$, the rightmost position reached by the branching random walk up to generation $[nt]$. Under certain moment assumptions on the branching law, we prove that $M_{nt}/\sqrt{n}$ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential …

On the Maximal Displacement of Subcritical Branching Random Walks

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation $n$. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $ρ>1$ such that the function \[ g(c,n):=ρ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the following prop…

Extracting subsets maximizing capacity and Folding of Random Walks

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

The infinite valley for a recurrent random walk in random environment

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

有限域内子集被受限随机游走的覆盖

Covering of an inner subset by the confined random walk

本文研究有限域$D_N$内子集$Λ_N$被受限随机游走的覆盖时间与晚点，得出含Gumbel变量的渐近展开式及晚点泊松分布，方法类似环面随机游走研究，球型子集仅得首项。

在Z上的最大熵随机游走：随机与非随机环境

Maximal Entropy Random Walks in Z: Random and non-random environments

本文研究了在Z上带有环的最大熵随机游走，通过分析其渐进行为，证明了几乎所有此类游走具有正的速度。

Almost sure estimates for the concentration neighborhood of Sinai's walk

We consider Sinai's random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is almost surely bounded. As an application we get that i.o. the maximal distance between two favorite sites is almost surely bounded.

随机环境中临界分支随机游走的最大位移

On the maximal displacement of critical branching random walk in random environment

本文研究随机环境中临界分支随机游走的最大位移，证明条件极限定理与尾概率渐近界，发现其位移增长速度与尾概率衰减均慢于恒定环境情形，方法基于约化分支随机游走的路径大偏差。

Optimal co-adapted coupling for a random walk on the hyper-complete-graph

The problem of constructing an optimal co-adapted coupling for a pair of symmetric random walks on $Z_2^d$ was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such co-adapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal co-adapted coupling for the continuous-time symmetric random walk on $K_n^d$, where $K_n$ is the complete graph with $n$ vertices. Moreover, we show that although this coupling is not maximal for any $n$ (i.e. it does not achieve equality in the coup…

On Maximal Ranges of Vector Measures for Subsets and Purification of Transition Probabilities

Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $σ$-field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional…

关于具有多项式增长的群上随机游走最大位移的集中不等式

Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

本文研究了具有多项式增长的群上随机游走最大位移的集中不等式，证明了在特定群类上随机游走的扩散性质。

The Spread of a Catalytic Branching Random Walk

We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -αn$ where $α$ is a deterministic constant.

The maximal flow from a compact convex subset to infinity in first passage percolation on Z^d

We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity $ϕ$(nA)/n^ (d--1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary $\partial$A of A and is the integral of a deterministic funct…

On massive sets for subordinated random walks

We study massive (reccurent) sets with respect to a certain random walk $S_α$ defined on the integer lattice $\mathbb{Z} ^d$, $d=1,2$. Our random walk $S_α$ is obtained from the simple random walk $S$ on $\mathbb{Z} ^d$ by the procedure of discrete subordination. $S_α$ can be regarded as a discrete space and time counterpart of the symmetric $α$-stable Lévy process in $\mathbb{R}^d$. In the case $d=1$ we show that some remarkable proper subsets of $\mathbb{Z}$ , e.g. the set $\mathcal{P}$ of primes, are massive whereas some proper subsets of $\mathcal{P}$ such as Leitmann primes $\mathcal{P}_h…

关于带或不带杀死机制的次临界分支随机游走的最大位移

On the maximal displacement of subcritical branching random walks with or without killing

本文在最优矩假设下给出次临界分支随机游走最大位移的尾部渐近式，证实了 Neuman-Zheng 猜想，并将结果推广到带杀死机制的情形。

Asymptotic behaviour of the simple random walk on the 2-comb

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical…

Branching random walks with uncountably many extinction probability vectors

Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf q(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probab…

分支随机游动最大值的精确上偏差估计

Precise upper deviation estimates for the maximum of a branching random walk

本文研究了分支随机游动最大位移的精确上大偏差估计，描述了该大偏差事件条件下的极值过程，并引入了类似分支布朗运动装饰测度的点测度族。

Branching Random Walks in Time Inhomogeneous Environments

We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change over time macroscopically. We find the asymptotics of the maximum up to an $O_P(1)$ (stochastically bounded) error, and focus on the following phenomena: the profile of the variance matters, both to the leading (velocity) term and to the logarithmic correction term, and the latter exhibits a phase transition.

Tail asymptotic of the stationary distribution for the state dependent (1,R)-reflecting random walk: near critical

In this paper, we consider the $(1,R)$ state-dependent reflecting random walk (RW) on the half line, allowing the size of jumps to the right at maximal $R$ and to the left only 1. We provide an explicit criterion for positive recurrence and the explicit expression of the stationary distribution based on the intrinsic branching structure within the walk. As an application, we obtain the tail asymptotic of the stationary distribution in the "near critical" situation.

Three favorite sites occurs infinitely often for one-dimensional simple random walk

For a one-dimensional simple random walk $(S_t)$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth (2001), and disproves a conjecture of Erdös and Révész (1984) and of Tóth (2001).

Random walks on the two-dimensional K-comb lattice

We study the path behavior of the symmetric walk on some special comb-type subsets of ${\mathbb Z}^2$ which are obtained from ${\mathbb Z}^2$ by generalizing the comb having finitely many horizontal lines instead of one.

On the critical branching random walk III: the critical dimension

In this paper, we study the critical branching random walk in the critical dimension, $Z^4$. We provide the asymptotics of the probability of visiting a fixed finite subset and the range of the critical branching random walk conditioned on the total number of offsprings. We also prove that conditioned on visiting, the first visiting point converges in distribution.

The maximal drawdown of the Brownian meander

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

Reflecting random walk in fractal domains

In this paper, we show that reflecting Brownian motion in any bounded domain D can be approximated, as $k\to\infty$, by simple random walks on "maximal connected" subsets of $(2^{-k}\mathbb{Z}^d)\cap D$ whose filled-in interiors are inside of D.

Some inequalities for reversible Markov chains and branching random walks via spectral optimization

We present results relating mixing times to the intersection time of branching random walk (BRW) in which the logarithm of the expected number of particles grows at rate of the spectral-gap $\mathrm{gap}$ . This is a finite state space analog of a critical branching process. Namely, we show that the maximal expected hitting time of a state by such a BRW is up to a universal constant larger than the $L_{\infty}$ mixing-time, whereas under transitivity the same is true for the intersection time of two independent such BRWs. Using the same methodology, we show that for a sequence of reversible Ma…

流形上的极大 $\Lambda(p)$ 子集

Maximal $Λ(p)$-subsets of manifolds

本文在弯曲流形上构造了最优指数范围内的极大 $\Lambda(p)$ 子集，结合限制性估计、解耦与概率方法。

具有拉伸指数尾的次临界分支随机游动的最大位移

On the maximal displacement of subcritical branching random walks with stretched exponential tail

研究一维次临界分支随机游动中最大位移的渐近行为，在步长具有拉伸指数尾条件下，得到了最大位移尾概率的精确极限。

Estimates of random walk exit probabilities and application to loop-erased random walk

We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.

On the critical branching random walk I: Branching capacity and visiting probability

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.

最大熵随机游走与中心马尔可夫链

Maximal entropy random walks and central Markov chains

本文在加权布拉特利图上提出最大熵随机游走（MERW）框架，证明其为中心马尔可夫链的特例，并借此推广二叉树钩长公式、证明BST过程为MERW，同时提出基于Knuth算法的蒙特卡洛近似方法。

分支随机游走的胚单调族的临界参数

Critical parameters of germ-monotone families of branching random walks

提出胚单调分支随机游走（GMBRW）族，定义与子集相关的临界参数，统一并扩展了全局和局部临界参数，并研究繁殖律修改对临界参数的影响。

Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant.

超立方体递增子集上的函数不等式与随机游走

Functional inequalities and random walks on increasing subsets of the hypercube

该研究针对超立方体递增子集，通过限制归纳法证明两个离散函数不等式，改进删失随机游走混合时间上界，建立$p$-偏置边等周不等式并给出概率解释。

Continuous time random walk as a random walk in a random environment

We show that for a weakly dense subset of the domain of attraction of a positive stable random variable of index $0<α<1$($DOA\left(α\right))$ the functional stable convergence is a time-changed renewal convergence of distribution of finite mean. Applied to Continuous Time Random Walk(CTRW) á la Montroll and Wiess we show that CTRW with renewal times in a weakly dense set of $DOA\left(α\right)$ can be realized as random walk in a random environment. We find the quenched limit and give a bound on the error of the approximation.

有线平面区域覆盖时间的紧性

Tightness for the Cover Time of Wired Planar Domains

本文研究有线边界平面区域上连续时间简单随机游走的覆盖时间，证明归一化覆盖时间平方根在特定对数尺度附近紧，证实 Bramson–Zeitouni 猜想的有线类比。

\(\mathbb{R}^d\) 中临界分支随机游走的渐近行为

Asymptotic behaviours of critical branching random walk in $\mathbb{R}^d$

研究\(\mathbb{R}^d\)中临界分支随机游走渐近行为，在特定假设下，建立全时最大位移尾概率衰减率，证明第\(n\)代最大位移条件极限定理，研究总后代数渐近行为，揭示最大位移与总后代数关系，扩展一维结果至\(d\geq2\) 。

Approximating critical parameters of branching random walks

Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the…

Convex hulls of stable random walks

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.

The probabilities of extinction in a branching random walk on a strip

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative…

Random walk on discrete point processes

We consider a model for random walks on random environments (RWRE) with random subset of the d-dimensional Euclidean lattice as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We prove that the velocity of such random walks is almost surely 0, and give partial characterization of transience and recurrence for the different dimensions. Finally we prove Central Limit Theorem for such random walks, under a condition on the distance between nearest coordinate nearest points.

The maximum of a tree-indexed random walk in the big jump domain

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.