Gaussian estimates for spatially inhomogeneous random walks on ${\mathbf{Z}}^d$
Sami Mustapha
本文证明了${\mathbf{Z}}^d$上的空间非齐次随机游动所对应的转移核具有上、下高斯估计。
It is shown in this paper that the transition kernel corresponding to a spatially inhomogeneous random walk on ${\mathbf{Z}}^d$ admits upper and lower Gaussian estimates.
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Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $σ^2 (\mathbf{u})$ satisfying $\mathbf{u}^\top σ^2 (\mathbf{u}) \mathbf{u} = U$ and $\mathrm{tr}\ σ^2 (\mathbf{u}) = V$ in all in directions $\mathbf{u}\in\mathbb{S}^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.
We consider the small deviation probability for random walk with time-inhomogeneous random environment. Compared with the result in Mogul'ski\uı (1974) for the i.i.d. random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance.
We consider a random walk $\{S_n\}_{n\in \mathbb{N}}$ in time-inhomogeneous random environment $ξ$. For almost each realization of $ξ$, we formulate a quenched harmonic function, based on which we can define the random walk in random environment conditioned to stay positive by the Doob's $h$-transform. Furthermore, we prove a quenched invariance principle for the conditioned random walk for almost each realization of $ξ$.
The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. Our primary interest is whether the walk visits all points of the point process, and we determine sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.
Branching Random Walk in an inhomogeneous breeding potential
考虑非齐次繁殖势$β|.|^p$中的连续时间分支随机游动，其中$β> 0$，$p \geq 0$ .我们证明了当$p > 1$为正整数时，种群几乎必然在有限时间内爆炸，而当$p \leq 1$为负整数时，种群不爆炸.在非爆炸的情况下，我们确定最右边的粒子的渐近行为。
Branching Random Walks in Time Inhomogeneous Environments
研究了一类时间非齐次环境中分枝随机游动的最大位移。具体而言，将考虑具有高斯增量的二进制分支随机游动，其中增量的方差随时间宏观地变化。我们发现的最大值的渐近$O_P(1)$（随机界）的错误，并专注于以下现象：的方差问题的档案，无论是领先的（速度）项和对数校正项，后者表现出相变。
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…
Stochastic dominance of first return times for nearest-neighbor random walks on $\mathbb{Z}^d$
本文证明：在 $\mathbb{Z}^d$ 上，若概率向量 $\mathbf{h}'$ 优超 $\mathbf{h}''$，则最近邻随机游走首次返回时间 $T^{\mathbf{h}'}$ 随机小于 $T^{\mathbf{h}''}$，且简单随机游走的首次返回时间随机最大。
We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbolα=(α_1,\ldots,α_d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite direction or scaling": (i) if $\sum_{i=1}^d α_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$, 1962]); (ii) if a…
Exponential tail bounds for loop-erased random walk in two dimensions
令$M_n$为在$\mathbb{Z}^2$上从原点到半径为$n$的圆的简单随机游动的环擦除的步数。我们将$M_n$的矩与$Es(n)$的矩联系起来，$Es(n)$是一个随机游动和一个独立的环擦除随机游动都从原点开始，直到离开半径为$n$的球为止不相交的概率。这使我们能够证明存在$C$，使得对于所有$n$和所有$k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$，并因此建立$M_n$的指数矩界。这意味着存在$c>0$，使得对于所有$n$和所有$λ\geq0$，\[\mathbf{P}\{M_n>λ\mathbf{E}[M_n]\}\leq2e^{-cλ}.\]使用类似的技术，我们然后建立了特定条件随机游动的二阶矩结果，这使得我们能够证明对于任何$α<4/5$，存在$C$和$c'>0$，使得对于所有$n$和$λ>0$，\[\mathbf{P}\{M_n<λ^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'λ^α}.\]
Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment
本文研究了在时间非齐次随机环境中的分支随机游走导数鞅收敛性，通过Tanaka分解确定了其非平凡性条件。
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…
Local limit theorem for the maximum of a random walk
考虑一族具有增量$X_i^{(a)}$和非正漂移$-a$的$Δ$-格化非周期随机游动$\{S^{(a)},0\le a\le a_0\}$.假设某些$\varepsilon>0$的$\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$和$\sup_{a\le a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$。假设$X^{(a)}\xrightarrow[]{w} X^{(0)}$为$a\to 0$，并用$M^{(a)}=\max_{k\ge 0} S_k^{(a)}$表示随机游动$S^{(a)}$的最大值。在本文中，我们提供了渐近的$\mathbf{P}(M^{(a)}=yΔ)$作为$a\to 0$的情况下，当$y\to \infty$和$ay=O(1)$。这种渐近性来自于$\mathbf{P}(M^{(a)}=yΔ)$的一种表示，通过几何和和一致更新定理，本文也证明了这一点。
Asymptotics for first-passage times of Lévy processes and random walks
本文研究了Lévy过程$X_t$第一次穿过负水平$-x$时$τ_x$分布的精确渐近性。我们证明了对于某个函数$V(x)$，$\mathbf P(τ_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$是$t\to\infty$.利用已知的结果为大偏差的随机游动，我们得到的渐近$\mathbf P(τ_x>t)$明确在轻尾和重尾的情况下。我们也应用我们的结果，找到渐近分布的忙期在M/G/1排队。
Martingale approach to subexponential asymptotics for random walks
考虑具有独立同分布增量和负均值$\mathbf Eξ=-m<0$的随机游动$S_n=ξ_1+...+ξ_n$。设$M=\sup_{0\le i} S_i$是随机游动的上确界。在这篇笔记中，我们提出了推导的渐近$\mathbf P(M>x), x\to\infty$长尾分布。这种推导是基于鞅参数，不需要任何先验知识的理论长尾分布。此外，相同的方法允许获得$\mathbf P(M_τ>x)$的渐近性，其中$M_τ=\max_{0\le i<τ}S_i$和$τ=\min\{n\ge 1: S_n\le 0 \}$。
Tightness for branching random walk in a space-inhomogeneous random environment
针对整数集上空间非齐次随机环境中的分支随机游走，证明存在仅依赖环境的中心化项，使最大值经中心化后在退火意义下具有紧性。
Conditions for recurrence and transience for time-inhomogeneous random walks
本文扩展了Abramov关于时间非齐次生灭过程的结果至时间非齐次随机游走，完整解决了Menshikov和Volkov提出的开放问题。
We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the ICRTs (scaling limits of $\mathbf p$-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation. The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree.
Transition phenomena for ladder epochs of random walks with small negative drift
对于一族满足$\mathbf{E}S_1^{(a)}=-a<0$的随机游动$\{S^{(a)}\}$，我们考虑梯历元$τ^{(a)}=\min\{k\geq1: S_k^{(a)}<0\}$。我们研究的渐近，作为$a\to0$，行为的$\mathbf{P}(τ^{(a)}>n)$的情况下，当$n=n(a)\to\infty$。作为结果，我们还获得了$τ^{(a)}$的时刻的增长率。
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。
Let $ξ_1,ξ_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|ξ_1|]=\infty.$ Consider a random walk $S_n=ξ_1+\cdots+ξ_n$, a stopping time $τ=\min\{n\ge 1: S_n\le 0\}$ and let $M_τ=\max_{0\le i\le τ} S_i$. We study the asymptotics for $\mathbf P(M_τ>x),$ as $x\to\infty$.
We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the cen…
Uniformity of the uncovered set of random walk and cutoff for lamplighter chains
我们证明了对$\mathbf {Z}_n^d$，$d\geq3$的标记的测量，其中${0,1}$的元素由i.i.d.公平硬币在随机游动$X$运行的范围$\mathcal {R}$上翻转，直到时间$T$和0，否则在阈值$T=1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$处变得与这种标记上的均匀测量不可区分。作为我们的方法的结果，我们表明，总变差混合时间的随机游动的点灯图$\mathbf {Z}_2\wr \mathbf {Z}_n^d$，$d\geq3$，有一个截止阈值$1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$。我们给出了一个一般的标准下，这两个结果持有;其他的例子，这适用于包括有界度扩展家庭，一个无限的超临界渗流集群的交叉点，一个不断增加的家庭球，超立方体和卡利图的对称群所产生的换位。证明也产生精确的渐近性的衰减相关性在未覆盖的设置。
We provide a characterization for anti-concentration of inhomogeneous random walks in non-abelian groups. In application we extend the classical bounds by Erdos-Littlewood-Offord and Sarkozy-Szemeredi to non-abelian settings.
Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods
研究流形上一类与时间相关度量有关的扩散过程的时间离散测地线随机游动的逼近问题。我们假设的度量条件是一个自然的时间非齐次的里奇曲率下界的扩展。特别地，它包括向后里奇流的情况，并且不需要进一步的先验曲率界。作为一个应用，我们构造了一个耦合反射产生一个很好的估计的耦合时间，从而梯度估计相关的半群。
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.
In this work, we consider a modification of time \emph{inhomogeneous} branching random walk, where the driving increment distribution changes over time macroscopically. Following Bandyopadhyay and Ghosh (2021), we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the last generation. We call this process \emph{last progeny modified time inhomogeneous branching random walk (LPMTI-BRW)}. Under very minimal assumptions on the underlying point processes of the displacements, we show that the maximum displacement converges to a limit after only an a…
This paper continues the discussion on the stability of time-inhomogeneous Markov chains. In particular, this paper defines a time-inhomogeneous, discrete-time Markov chain governed by a continuous evolution in the appropriate martrix space. This matrix space, $\mathcal{P}_{n}^{ia}$, is the space of all stochastic matrices that are irreducible and aperiodic. For this new type of evolution there is a definition of a specific type of stability called the stable adiabatic time. This measure is bounded by a function of the optimal mixing time over the evolution. Namely, for a time-inhomogeneous, d…
Consider a family of random walks $S_n^{(a)}=X_1^{(a)}+\cdots+X_n^{(a)}$ with negative drift $\mathbf E X_1^{(a)}=-a<0$ and finite variance $\mbox{var}(X_1^{(a)})=σ^2<\infty$.Let $M^{(a)}=\max_{n\ge 0} S_n^{(a)}$ be the maximums of the random walks. The exponential asymptotics $\mathbf P(aM^{(a)}>x)\sim e^{-2x/σ^2}$, as $a\to 0$, were found by Kingman and are known as heavy traffic approximation in the queueing theory. For subexponential random variables the large deviation asymptotics for $\mathbf P(M^{(a)}>x)\sim \frac{1}{a}\overline F^I(x)$ hold for fixed $a$ as $x\to\infty$. In this paper …
On Partition Functions for Time-Inhomogeneous Branching Random Walks
研究时间非齐次分支随机游走的配分函数相变与普遍性，通过改进GMC框架证明其临界值与时间齐次一致，并在亚临界区域证明配分函数L1收敛，扩展临界参数下的衰减率，提出GMC框架在非对数相关领域的新路径。
This article studies discrete height functions on the discrete hypertorus. These are functions on the vertices of this hypertorus graph for which the derivative satisfies a specific condition on each edge. We then perform a random walk on the set of such height functions, in the spirit of Diffusivity of a random walk on random walks, a work of Boissard, Cohen, Espinasse, and Norris. The goal is to estimate the diffusivity of this random walk in the mesh limit. It turns out that each height functions is characterised by a number of so-called fractures of the hypertorus. These fractures are then…
Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion
本文提出并分析了一种新的二维不均匀随机游走模型，结合空间衰减的转移概率与每次游走的时序重置结构，研究了空间不均匀性对返回原点的影响以及随机环境对转移概率参数的影响。
Minima in branching random walks
给定一个分支随机游动，设$M_n$是第$n$代中任何成员的最小位置。在分支随机游动的一般条件下，我们计算$\mathbf{E}M_n$到O（1）以内，并证明了$\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$的指数尾界。特别是结合Bramson [Z.华什弗鲁Gebiete 45（1978）89- 108]，我们的结果充分地描述了当分支随机游动具有有界分支和步长时$\mathbf {E}M_n$的可能行为。
Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\sup_{n\geqslant0}\sum_{i=1}^nξ_i>x)$ can be bounded above by $\varrho_1\exp\{-\varrho_2x\}$ with some positive constants $\varrho_1$ and $\varrho_2$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the ne…
Inhomogeneous central limit theorems for the voter model occupation times
本文研究了投票模型在空间非均匀初始分布下的占用时间中心极限定理，利用对偶关系和Donsker原理推导主要结论。
Stable Adiabatic Times for Markov Chains
在本文中，我们继续我们在Kovchegov（2010）和Bradford and Kovchegov（2011）中首次引入的时间非齐次马尔可夫链的绝热时间的工作。我们的研究类似于著名的量子绝热（QA）定理，该定理描述了量子系统演化的量子绝热时间，作为应用一系列Hamilton算子的结果，每个Hamilton算子是两个给定的初始和最终Hamilton算子的线性组合，即$\mathbf{H}(s) = (1-s)\mathbf{H_0} + s\mathbf{H_1}$ .非正式地，量子系统的量子绝热时间指定了哈密顿算符改变的速度，使得系统的基态在任何时间$s$将总是保持$ε$ -接近于由哈密顿算符$\mathbf{H}(s)$在时间$s$诱导的。类似地，我们通过应用一系列转移概率矩阵来指定时间非齐次马尔可夫发展的稳定绝热时间的充分条件，每个转移概率矩阵是两个给定的不可约和非周期转移概率矩阵的线性组合，即，$\mathbf{P_{t}} = (1-t)\mathbf{P_{0}} + t\mathbf{P_{1}}$。特别是，我们表明，稳定的绝热时间$t_{sad}(\mathbf{P_{0}}, \mathbf{P_{1}}, ε) = O (t_{mix}^{4}(ε\slash 2) \slash ε^{3}), $其中$t_{mix}$表示的最大混合时间超过所有的$\mathb…
Local time, upcrossing time and weak cutpoints of a spatially inhomogeneous random walk on the line
本文研究了正半轴上具有渐近零漂移的空间非齐次随机游走，给出了具有相同局部时、上穿时间及弱割点的点数有限性判别准则，证明了其期望阶数及缩放后依分布收敛于标准指数分布，并由此解决了三个相关猜想。
Tightness for branching random walk in time-inhomogeneous random environment
该研究针对时间非齐次随机环境下的分支随机游走，证明存在仅依赖环境的中心化序列，使最大粒子位置与该序列的差序列以高概率具有紧性，推进了相关随机过程的极限理论研究。
On the barrier problem of branching random walk in time-inhomogeneous random environment
本文研究带随机吸收障碍的时间非齐次随机环境分支随机游走，给出以 α 和临界常数 a_c 划分的灭绝与存活判据及灭绝速率，将既有确定性结果推广至随机环境。
Boundaries from inhomogeneous Bernoulli trials
研究了带权边的正方形格${\mathbb Z}_+^2$上非齐次增随机游动的边值问题.给出了一些与经典和广义数三角形有关的例子的显式解。
Expansions for random walks conditioned to stay positive
本文研究零均值有限方差一维随机游走在保持为正条件下的首达时尾分布与局部概率渐近展开，并由此得到离散多调和函数及其更新定理类似物。
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigen…
Time-inhomogeneous random walks on finite groups and cokernels of random integer block matrices
研究有限群上非时齐随机游走的收敛性，通过谱方法给出收敛速率，并应用于随机整数矩阵余核的普适性定理。
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a uniformly random infinite word on the alphabet $\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk $\mathcal{W}_G$ starts at the origin and follows the directions specified by $\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk $\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset \mathbb{Z}^2$, the walk $\…
We consider the Pólya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.
Limit theorems for inhomogeneous random walks on $GL(d,\mathbb R)$
本文研究了不相容随机游走的极限定理，证明了Berry-Esseen定理、几乎不变原理速率和大偏差，同时刻画了乘积对数范数方差的发散。
We work under the A\"ıdékon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm const}+o(1)$ as $x\to\infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the …
Stationary Measures of Space-Inhomogeneous Three-State Quantum Walks on Line: Revisited
本文通过约化矩阵方法，在温和条件下研究了线上空间非齐次三态量子游走两个模型的平稳测度，推广了文献中的已有结果。
A local limit theorem for a transient chaotic walk in a frozen environment
本文研究了一维非均匀介质中粒子的传播，其中粒子的运动规律由混沌和确定性局部映射产生。假设粒子的初始位置是随机的且均匀分布的，这个动力系统可以归结为一维非均匀环境中的随机游动，其方向是禁止的.我们的主要结果是一个局部极限定理，详细解释了为什么从长远来看，随机游动的概率质量函数不收敛到高斯密度，虽然相应的限制分布在一个粗糙的扩散空间尺度是高斯。
Random Lie bracket on $\mathfrak{sl}_2(\mathbf{F}_p)$
研究在$\mathfrak{sl}_2(\mathbf{F}_p)$上通过随机李括号的随机游走，揭示其极限分布和收敛速度，并通过引入随机扭实现对整个李代数的覆盖。
Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outc…
A continuous-time random walk in the quarter plane with homogeneous transition rates is considered. Given a non-negative reward function on the state space, we are interested in the expected stationary performance. Since a direct derivation of the stationary probability distribution is not available in general, the performance is approximated by a perturbed random walk, whose transition rates on the boundaries are changed such that its stationary probability distribution is known in closed form. A perturbed random walk for which the stationary distribution is a sum of geometric terms is consid…
Annealed Survival Probability of Random Walk in an Inhomogeneous Poisson Environment of Mobile Traps
该研究确定了1、2维移动陷阱非齐次泊松环境中随机游走退火生存概率的渐近行为，推广了齐次情形的相关结论，还建立了d≥1时非齐次环境下的大数定律等极限定理，并给出了两类特殊环境的实例。
Painting a graph with competing random walks
令$X_1,X_2$是在$\mathbf{Z}_n^d$，$d\geq3$上的独立随机游动，每个从均匀分布开始。最初，$\mathbf{Z}_n^d$的每个站点都是未标记的，并且每当$X_i$访问这样的站点时，它都被不可逆地设置为$i$。一旦所有的$\mathbf{Z}_n^d$都被访问，则$|\mathcal{A}_i|$的平均值，即由$i$绘制的站点的集合$\mathcal{A}_i$的基数，通过对称性为$\frac{1}{2}n^d$。证明了Pemantle和Peres的猜想：对于每个$d\geq3$，存在一个常数$α_d$，使得$\lim_{n\to\infty}\operatorname{Var}(|\mathcal {A}_i|)/h_d(n)=\frac{1}{4}α_d$，其中$h_3(n)=n^4$，$h_4(n)=n^4(\log n)$和$h_d(n)=n^d$是$d\geq5$的.我们还将明确识别$α_d$，并显示$α_d\to1$为$d\to\infty$。这是一个特殊情况下，一个更一般的定理，使渐近的$\operatorname{Var}(|\mathcal{A}_i|)$的一大类瞬态，顶点传递图;其他例子包括超立方体和卡利图的对称群所产生的换位。
In a Markov chain started at a state $x$, the hitting time $τ(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(τ(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{n}{t}$. 2) In a reversible chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{\sqrt{2n}}{t}$ for $t \ge 4n + 4$. 3) For random wal…
Bounds on Mixing Time for Time-Inhomogeneous Markov Chains
针对动态随机环境中的随机游走，提出非时齐马尔可夫链混合时间概念，扩展演化集方法并证明动态 Erdős-Rényi 图上近似匹配的上下界。
Local limit theorem and Edgeworth expansions for inhomogeneous random walks on $GL(d,\mathbb R)$
本文针对可逆独立随机矩阵乘积的范数对数，证明了非格型局部中心极限定理与埃奇沃思展开，还为文献[MatBE]的中心极限定理最优速率提供了新证明，并给出收缩的若干充分条件。
In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Fre…
Inhomogeneous Branching Random Walks: Incorporating Genealogy and Density Effects
本文提出非均匀分支随机游走模型，结合基因谱和时空因素，研究细菌菌落生长模式的长期行为及结构特征。
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