Yueyun Hu, Zhan Shi
We consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the walk in the first $n$ steps is $(\log n)^3$.
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In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime.
Random walk is one of the most classical and well-studied model in probability theory. For two correlated random walks on lattice, every step of the random walks has only two states, moving in the same direction or moving in the opposite direction. This paper presents a decomposition method to study the dependency structure of the two correlated random walks. By applying change-of-time technique used in continuous time martingales (see for example [1] for more details), the random walks are decomposed into the composition of two independent random walks $X$ and $Y$ with change-of-time $T$, whe…
We consider the randomly biased random walk on trees in the slow movement regime as in [HS16], whose potential is given by a branching random walk in the boundary case. We study the heavy range up to the $n$-th return to the root, i.e., the number of edges visited more than $k_n$ times. For $k_n=n^θ$ with $θ\in(0,1)$, we obtain the convergence in probability of the rescaled heavy range, which improves one result of [AD20].
Random walk in a rotational environment
本文研究旋转空间中粒子在随机碰撞下的随机游走，证明其极限运动是确定性的，并揭示随机性如何产生渐近确定性行为。
Random walks on regular trees can not be slowed down
本文探讨了正则树上的随机游走能否通过时变顶点置换被减慢，并证明在树上随机游走无法被减慢。
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.
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 $γ< 1/2$.
In this paper we prove that under certain assumptions the transient random walk in random environment with bounded jumps (in $\mathbb{Z}$) grows much slower than the speed $n$. Precisely, there is $0<s<1$, such that although $X_n\rto$ we have $\frac{X_n}{n^{s'}}\rightarrow 0$ for $0<s<s'$ almost surely.
We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.
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 $…
Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $μ$ such that, for any $ε>0$, $\sum|\cdot|^εμ=\infty$. In particular, we provide a sharp lower bound for the return probability in the case when $μ$ has a finite weak-logarithmic moment.
For a random walk in an elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the a ballisticity condition slightly weaker than condition (T'), We consider the probability of linear slowdown. We show an upper bound for this probability which is very close to the lower bound obtained by the "naive trap" analysis. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the same ballisticity condition.
Random walk in slowly changing environments
针对变化环境中随机游走（RWCE）的常返暂留判定难题，提出适用于任意图的慢变化边权有界RWCE的判定条件，证明其常返/暂留性与初始加权图一致。
On the slow phase for fixed-energy Activated Random Walks
研究一维环上高密度下的激活随机游走模型，开发出一种拓扑过程，证明在任意大的睡眠率下，活动能持续很长时间。
We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^3$ in the first $n$ steps. We study the localization problem of $X_n$ and prove that the quenched law of $X_n$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_n|}{(\log n)^2}$ converges in law to some non-degenerate…
Branching random walk and log-slowly varying tails
本文研究了位移分布不属于极值分布最大吸引域的重尾分支随机游走，证明了在尾部对数绝对值为缓慢变化函数的条件下，经非线性变换后过程极值收敛于聚类Cox过程。
We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $αn^{-β}/2$, where $α\geq 0$ and $β\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $β<1$, then it converges to the usual Brownian motion. If $β\in (1,\infty]$, then it converges to the reflected Brownian motion. And at the critical value $β=1$, it converges to the snapping out Brownian motion (SNO…
We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_α>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_ω(X_n < xn)$ with $x \in (0,v_α)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-γ} \log P_ω( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether $γ> 1-1/s$ or $γ< 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_ω( X_n …
Generalized range of slow random walks on trees
本文研究树上随机偏置随机游走的广义值域，在慢速区域证明一般结果，并通过例子展示轨迹与势的相互作用。
We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^κ)$ from the origin, $κ\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ν_0})$ from the origin, $ν_0\in (0,κ)$), and speedup (at time $n$, the particle is at a distance of order $n^{ν_1}$…
We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton--Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erdős and Révész [11] for simple symmetric random walk on …
Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random neighbour, whereas from an excited vertex, she takes one step along some fixed shortest path towards the target vertex. We show, perhaps counterintuitively, that the geodesic-bias can slow the random walker down exponentially: there exist connected, bounded-degree $n$-vertex graphs with excitations where the expected hitting time of a fixed target is at least $\exp (…
We prove an annealed weak limit of the trajectory of the random walks in cooling random environment (RWCRE) under both slow (polynomial) and fast (exponential) cooling. We identify the weak limit when the underlying static environment is recurrent (Sinai's model). Avena and den Hollander have previously proved law of large numbers and Gaussian fluctuation of RWCRE. We find that the weak limit of the trajectory exists as a time-rescaled Brownian motion in the slow cooling case but the limit degenerates to a constant function in the fast cooling case.
A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $ω=(ω_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $κ$ such that $\E[((1-ω_0)/ω_0)^κ]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the …
We consider a random walk $\tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that $\{\frac{1}{\sqrt n} \tilde S(nt)\}$ has no weak limit in $\De$; alternatively, the weak limit is a reflected Brownian motion.
We consider a continuous time random walk $X$ in random environment on $\Z^+$ such that its potential can be approximated by the function $V: \R^+\to \R$ given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such that $b>0$ and $0<\alf<1/2$. We show that $¶$-a.s.\ (where $¶$ is the averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\ln\l…
We consider a random walk $(Z^{(1)}_n, ..., Z^{(K+1)}_n) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor $σ_K^2 = \frac{2}{K+2}$ with respect to the case of the classical simple random walk without constraint.
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
We consider a branching random walk on $\Z$, where the particles behave differently in visited and unvisited sites. Informally, each site on the positive half-line contains initially a cookie. On the first visit of a site its cookie is removed and particles at positions with a cookie reproduce and move differently from particles on sites without cookies. Therefore, the movement and the reproduction of the particles depend on the previous behaviour of the population of particles. We study the question if the process is recurrent or transient, i.e., whether infinitely many particles visit the or…
We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that P(S_n^u=0) decays at polynomial rate n^{-α} where α>0 can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.
We study a biased random walk on the interlacement set of $\mathbb{Z}^d$ for $d\geq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.
We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.
We study the speed of a biased random walk on a percolation cluster on $\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice.
The jump processes W(t) on [0,\infty[ with transitions w -> alpha w at rate b*w^beta (0 =< alpha =< 1, b>0, beta>0) are considered. Their moments are shown to decay not faster than algebraically for t -> \infty, and an equilibrium probability density is found for a rescaled process U = (t + k)^{-beta} W. A corresponding birth process is discussed.
We have shown recently how to calculate the large deviation function of the position $X_{\max}(t) $ of the right most particle of a branching Brownian motion at time $t$. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of $X_{\max}(t)$ has, asymptotically in time, a prefactor characterized by non trivial power law.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d \le 2$, or the environment is ``random enough", then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of "replica overlap"…
We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay.
A particle moves among the vertices of an $(m+1)$-gon which are labeled clockwise as $0,1,...,m$. The particle starts at 0 and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability $p$, or counterclockwise with probability $1-p$. The directions of successive movements are independent. What is the expected number of moves needed to visit all vertices? This and other related questions are answered using recursive relations.
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