Kevin Ford
We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the 4th moment is finite.
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We obtain expected number of arrivals, absorption probabilities and expected time before absorption for a discrete random walk on the integers with an infinite set of equidistant multiple function barriers
We continue the line of research of random walks with barrier initiated by Iksanov and M{ö}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-α$, $α\in(0,1)$, we prove that the number $V_n$ of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for $V_n$ proved in Iksanov and Negadailov (2008).
We consider a branching random walk with an absorbing barrier, where the step of the associated one-dimensional random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We shall prove that there is a barrier $an^{\frac{1}{1+α}}$ and a critical value $a_α$ such that if $a<a_α$, then the process dies; if $a>a_α$, then the process survives. The results generalize previous results in literature for the case $α=2$.
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。
We study a branching random walk on $\r$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. \cite{BLSW91} determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a particle in a given generation, we add a second order term $a n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ g…
We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier.
In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of crossing the barrier scales as $ 1/\sqrt{n} $ as we rescale space by $ \sqrt{n} $ and time by $ n $, we obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at the origin reaches an independent exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at …
The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.
We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ram\'ırez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ram\'ırez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension $d=2$ under which the corresponding random walk is ballistic.
We consider self-avoiding walk on a tree with random conductances. It is proven that in the weak disorder regime, the quenched critical point is equal to the annealed one, and that in the strong disorder regime, these critical points are strictly different. Derrida and Spohn, and Baffet, Patrick and Pul$\acute{\rm e}$ give the exact value of the quenched critical point. We give another heuristic approach by the fractional moment estimate.
Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier
该研究针对带随机屏障的独立同分布随机环境中超临界分支随机游走，在可积条件下证明淬灭生存概率的对数依概率、几乎处处或依$L^p$收敛于依赖环境的显式负常数，将已有结果拓展至随机环境情形。
Sharp asymptotics for $N$-point correlation functions of coalescing heavy-tailed random walk
本文研究了 coalescing 重尾随机游走系统的 N 点相关函数的精确渐进行为，通过尾部估计和非碰撞概率分析，推导了重尾随机游走的精细估计。
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。
In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely man…
Sharp asymptotics of disconnection time of large cylinders by simple and biased random walks
本文研究了简单与有偏随机游走对大离散圆柱体的渐近断开时间，推导出匹配的精确下界并改进了有偏游走的界，同时提出了随机游走迹与随机交织间的强耦合工具。
We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $α$ of the right tail lies in the interval $(1, \, 3/2)$.
Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk
本文研究三维回路消除随机游走的标度极限，证明其极限占据测度与Minkowski内容等价，并提供了一点函数和球击概率的精确估计。
Random walk with barriers on a graph
该研究针对带多个函数势垒的图上非对称离散随机游走，推导了期望到达数、吸收概率与吸收前期望时间，且为图的边和顶点势垒定义了特定概率。
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^α-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,α=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,α=\fra…
In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman-Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article, is one of the few…
For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (…
Sharp threshold for the ballisticity of the random walk on the exclusion process
本文研究排斥过程平流下随机游走的弹道性，证明回溯概率在临界密度处发生尖锐转变，速度严格单调且零速度区域坍缩为单点，从而解决一个已有猜想。
Sharp barrier estimates for Bessel bridges
本文推导了贝塞尔桥保持在线性障碍下的概率的精确估计，确定其当$T\to \u221e$时的首阶项及渐近误差，分析$a,j\to \u221e$时首阶项行为，得到误差界，并建立线性障碍被小凹函数扰动的互补结果。
A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
本文研究重尾位移分支随机游走在超线性杀死屏障下的存活相变，证明临界值为4而非一阶矩启发式的2，并分析临界值处的非确定性。
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and the dodecahedron variant. In two dimensions we obtain explicit formula in case of rotated barriers.
We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.
Extremal regime for one-dimensional Mott variable-range hopping
该研究针对呈现严重阻塞的一维Mott随机游走的渐近行为展开，证明经适当重标度的游走位于两个环境可测势垒间且势垒位置有极值标度极限，同时给出了势垒间游走分布的渐近描述。
For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.
The maximum relaxation time of a random walk on regular graphs
本文确定了给定阶数连通简单正则图上随机游走最大松弛时间的渐近精确值，证明了Aldous--Fill谱隙猜想，并给出了极值图的唯一性与稳定性刻画。
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization approximation of Brownian motion to sample barrier-hitting events, i.e. hitting for the first time a deterministic "barrier" function; and to sample extreme events, i.e. attaining a minimum on a given compact time interval or unbounded closed time interval. For each case we study the discretization error between the actual time the event occurs versus the time t…
Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$
研究离散时间简单随机游走在 $\mathbb{Z}^d$（$d\ge 3$）上最大局部时间的向下偏差，通过引入环剪枝随机游走和环剪枝分解，证明了匹配的下界，从而得到了向下偏差概率的精确渐近公式。
Application of simplest random walk algorithms for pricing barrier options
本文展示了一阶算法在障碍期权定价中的有效性，该算法基于弱欧拉近似和线性插值，易于实现且通用，无需触发概率知识。
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.
Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates
本文研究了单向大象随机游走（uERW）在记忆指数β∈(-1, p/(1-p))下的相变行为，通过耦合论证得出了位置Sn的精确渐近增长率和中心极限定理。
Graph distance and effective resistance of the four-dimensional random walk trace
本文研究了四维随机游走轨迹的图距离和有效电阻，建立了前n步轨迹中起始点与终点之间期望距离的精确渐进行为估计。
Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks
针对整数上步长为 +y 和 -b 的双 barrier 游走，提出两阶段修剪方法构造核根展开，在示例中有效排除大量非必要项，提升计算效率。
The sequence of the simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ has a cutoff phenomenon for each integer $q$ greater than or equal to $3$. In this paper, for the sequence of simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ with $q\geq 3$, we give a simple majorant and a sharp minorant function for total variance distances between transition distributions and stationary distributions.
In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.
We prove error bounds in a central limit theorem for solutions of certain convolution equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to weakly self-avoiding random walks in high dimensions. As an application we treat such self-avoiding walks in continuous space. The bounds obtained are sharper than the ones obtained by other methods.
Rate of escape of the conditioned two-dimensional simple random walk
本文利用KMT型耦合，从连续类似物的渐近结果推导出避免固定有限集的二维简单随机游走逃逸速率的精确渐近估计。
We consider a family X^{(n)}, n \in \bbN_+, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result,…
We consider a discrete random walk (RW) in n dimensions . The RW is adapted with a geometric absorption process: at any discrete time there is a constant probability that absorption occurs in the current state. To model the RW with geometric absorption we use the concept of a multiple function barrier (MFB). In a MFB there is a modification of the original RW: each transition probability in the original RW is multiplied by β and there is an additional probability (1-β) of absorption, where 0<β<1. We study three cases: one-dimensional simple asymmetric RW, n-dimensional simple symmetric RW (n>1…
In Robbins' problem of minimizing the expected rank, a finite sequence of $n$ independent, identically distributed random variables are observed sequentially and the objective is to stop at such a time that the expected rank of the selected variable (among the sequence of all $n$ variables) is as small as possible. In this paper we consider an analogous problem in which the observed random variables are the steps of a symmetric random walk. Assuming continuously distributed step sizes, we describe the optimal stopping rules for the cases $n=2$ and $n=3$ in two versions of the problem: a "full …
Very sharp distance and range transitions for random walk bridges on Ramanujan graphs
本文证明在具有对数围长的顶点传递 Ramanujan 图上，长度为对数阶的随机游走桥的最大距离在临界窗口内从根号对数阶突变至对数阶，并给出范围的两点极限定律。
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.
Consider the time T_oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T_oz in terms of the volume of z and the graph distance between o and z. The bounds are for expected value and large deviations, and are asymptotically sharp. We deduce rate of escape results for random walks on infinite graphs of exponential or polynomial growth, and resolve a conjecture of Benjamini and Peres.
Our paper gives bounds for the rate of convergence for a class of random walks on the d-dimensional torus generated by a set of n vectors in R^d/Z^d. We give bounds on the discrepancy distance from Haar measure; our lower bound holds for all such walks, and if the generators arise from the rows of a "badly approximable" matrix, then there is a corresponding upper bound. The bounds are sharp for walks on the circle.
We show that the spectral gap of a random walk on the domain of normal attraction of an $α$-stable law is of order $\mathcal O(n^α)$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be of independent interest. The comparison principle also allows to derive a sharp bound on the spectral gap of exclusion and zero-range processes with long jumps when restricted to finite boxes in terms of the gap on the complete graph.
Meeting times of Markov chains via singular value decomposition
本文利用奇异值分解和矩阵扰动理论，给出大型图上随机游走期望相遇时间的非渐近精确界，并在稠密 Erdős-Rényi 随机图上验证了方法的有效性。
We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.
We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for independent identically distributed X,Y is used. In part II we shall discuss the connection of this result to 'polygonal recurrence' of higher-dimensional walks and some conjectures on directionally random walks in the sense of Mauldin, Monticino and v.Weizsaecker [5].
We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em pocket-extension} of a group $G$---this leads to the study of certain finitely generated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$. Some sharp estimates are obtained while many challenging questions remain.
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.
We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better …
Return Probability for the Switch--Walk--Switch Lamplighter Walk on a Regular Tree
研究正则树上开关行走的返回概率渐进行为，通过数学方法推导出精确的渐近公式，并展示AI系统生成严格数学证明的能力。
True self-repelling motion above a general barrier
本文研究了在一般屏障上构造的真实自避运动，并探讨了其在不规则屏障下的证明扩展难题。
We develop the theory of the $W$ and $Z$ scale functions for right-continuous (upwards skip-free) discrete-time discrete-space random walks, along the lines of the analogue theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one and two-parameter scale functions $Z$, which appear for example in the joint problem of deficit at ruin and time of ruin, and in problems concerning the walk reflected at an upper barrier. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous. The theory is …
We consider a transient random walk $(X_n)$ in random environment on a Galton--Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that $X_n$ is of order of magnitude $n^Λ$, with $Λ\in (0,1)$. We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio \cite{Col06}.
This paper considers non-backtracking random walks on random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph tends to infinity. Subject to mild general conditions, we link two mixing times: one for a static version of the random graph, the other for a class of dynamic versions of the random graph in which the edges are randomly rewired but the degrees are preserved. The link is provided by the probability that the random walk has not yet stepped along a previously …
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