Endre Csaki, Antonia Foldes, Pal Revesz
We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers.
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Unlike many particle systems, Activated Random Walk has nontrivial behavior even in one spatial dimension. We prove inner and outer bounds on the spread of n activated random walkers from a single source in Z. The inner bound involves a comparison with the stationary distribution of activated random walkers on a finite interval, while the outer bound involves a comparison with the stabilization of an infinite Bernoulli configuration of activated random walkers on Z.
Suppose that the vertices of the Euclidean lattice Z^d are endowed with a random scenery, obtained by tossing a fair coin at each vertex. A random walker, starting from the origin, replaces the coins along its path by i.i.d. biased coins. For which walks and dimensions can the resulting scenery be distinguished from the original scenery? We find the answer for simple random walk, where it does not depend on dimension, and for walks with a nonzero mean, where a transition occurs between dimensions three and four. We also answer this question for other types of graphs and walks, and raise severa…
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -- m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these …
Interacting Edge-Reinforced Random Walks
研究多个游走者共同影响边权重的边强化随机游走，证明其行为与单个游走者一致，并给出常返性判据。
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.
The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or trans…
We prove a quenched invariance principle for a class of random walks in random environment on $\mathbb{Z}^d$, where the walker alters its own environment. The environment consists of an outgoing edge from each vertex. The walker updates the edge $e$ at its current location to a new random edge $e'$ (whose law depends on $e$) and then steps to the other endpoint of $e'$. We show that a native environment for these walks (i.e., an environment that is stationary in time from the perspective of the walker) consists of the wired uniform spanning forest oriented toward the walker, plus an independen…
We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed.
We prove a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $γ$. First, we establish that if the asymptotic velocity of the walker is non-zero in the limiting case "$γ= \infty$", where the environment gets fully refreshed between each step of the walker, then, for $γ$ large enough, the walker still has a non-zero asymptotic velocity in the same direction. Second, we establish that if the walker is transient in the limiting case $γ= 0$, then, for $γ$ sma…
We consider a random walk in the plane which takes steps uniformly distributed on the unit circle centered around the walker's current position but avoids the convex hull of its past positions. This model has been introduced by Angel, Benjamini and Virag. We show a large deviation estimate for the distance of the walker from the origin, which implies that the walker has positive lim inf speed.
We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration propert…
The ensemble $\CUE^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q.$ $\CUE^{(q)}$ is an exactly solved model of random contraction matrices originally introduced in the context of scattering theory. In this article, we exhibit a connection between $\CUE^{(q)}$ and Fisher's random-turns vicious walker model from statistical mechanics. In particular, we show that the moment generating function of the trace of a random matrix from $\CUE^{(q)}$ is a generating series for the partition function of …
We consider $d$ independent walkers in the same random environment in $\mathbb{Z}$. Our assumption on the law of the environment is such that a single walker is transient to the right but subballistic. We show that - no matter what $d$ is - the $d$ walkers meet infinitely often, i.e. there are almost surely infinitely many times for which all the random walkers are at the same location.
On the first positive position of a random walker
本文研究连续对称步长下一维随机游走者的首个正位置分布，分析其矩与渐近尾部，统一涵盖扩散型随机游走和 Lévy 飞行。
We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger th…
Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps…
We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.
We show propagation of local equilibrium for the symmetric inclusion process (SIP) after diffusive rescaling of space and time, as well as the local equilibrium property of the non-equilibrium steady state in the boundary driven SIP. The main tool is self-duality and a coupling between $n$ SIP particles and $n$ independent random walkers.
Interface between competing random walks on a cycle
该研究探讨环上两独立随机游走者的竞争，证明端点被不同游走者占据的边的期望数量为$\text{O}(\text{ln}(1+N/d))$，证实了相关论文预测的对数依赖关系。
Variations on Reinforced Random Walks
本论文研究边强化随机游走的变体，包括多游走者相互作用和方向偏置，证明在特定条件下出现常返与暂态间的相变，并揭示与标准模型的根本差异。
Lock step walker model is a one-dimensional integer lattice walker model in discrete time. Suppose that initially there are infinitely many walkers on the non-negative even integer sites. At each tick of time, each walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. It is proved that in the large time limit, a certain conditional probability of the displacement of the leftmost walker is identical to the limiting distribution of the properly scaled largest eigenvalue of a random GOE matrix …
Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics
本文研究了网络上具有预算更新机制的随机游走动力学，分析了目标命中统计、生存概率及寿命分布。
Exponential Bounds and Analyticity for the Tree Builder Random Walk
研究树构建随机游走的指数界与解析性，证明首次更新时间的指数尾部性质，并展示游走速度的指数上界及参数p的解析函数关系。
We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{…
In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the on…
Chemotaxis guidance of random walkers modeling self-wiring of neural networks
提出描述生长锥趋化引导的随机游走者模型，耦合随机微分方程与反应扩散方程，证明解的唯一性并通过数值实验分析参数敏感性与非局部正则化影响。
We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μbetween open and closed, following a Glauber dynamics for the random cluster model with parameters p,q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n^2/μ. In our proof we construct of a non-Markovian coupling through a multi-scale analysis of the environment, …
We study an infinite system of particles initially occupying a half-line $y\leq 0$ and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the $k^{\text{th}}$ particle attains the leadership with probability $e^{-2} k^{-1} (\ln k)^{-1/2}$ when $k\gg 1$. This provides a quantitative measure of the correlation between earlier misfortune (represented by $k$) and eternal failure. We also show that the winne…
We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on th…
We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N \geq 1$, for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average val…
Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches
本文研究了球壳内带有吸收边界条件的莱维行走者，证明逆平方莱维行走优化搜索效率，支持其在随机搜索中的最优性。
Cover times of many random walkers on a discrete network
针对离散网络上多随机游走者覆盖时间研究不足的问题，本文推导了其所有矩的显式公式，发现该覆盖时间仅依赖局部网络性质，并通过随机模拟验证了结论。
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0,t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the …
A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two non-colliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) any $d$-regular graph avoiding a fixed subgraph depending on $d$; and (ii) any square-free graph with minimu…
We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disa…
We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/σ=\sqrt{2n/π}+ γ+O(n^{-1/2}), where σ^2 is the variance of the step lengths. While the leading \sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant γ. For the special case of uniform distribution over [-1,1], Coffman…
We consider a generalization of the vicious walker model. Using a bijection map between the path configuration of the non-intersecting random walkers and the hook Young diagram, we compute the probability concerning the number of walker's movements. Applying the saddle point method, we reveal that the scaling limit gives the Tracy--Widom distribution, which is same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.
We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multi-hopper's hitting times between any two nodes in the graph converge to their minimum possible value, given by the hitting times of a norm…
Extending Pólya's random walker beyond probability I. Complex weights
本文在 Pólya 随机游走的组合模型中证明了顶点常返性定理，并利用生成函数的半形式化方法，将这些定理从概率范畴扩展到具有复权重的更一般模型。
We consider $N$ Brownian particles moving on a line starting from initial positions ${\bf{u}}\equiv \{u_1,u_2,\dots u_N\}$ such that $0<u_1 < u_2 < \cdots < u_N$. Their motion gets stopped at time $t_s$ when either two of them collide or when the particle closest to the origin hits the origin for the first time. For $N=2$, we study the probability distribution function $p_1(m|{\bf{u}})$ and $p_2(m|{\bf{u}})$ of the maximal distance travelled by the $1^{\text{st}}$ and $2^{\text{nd}}$ walker till $t_s$. For general $N$ particles with identical diffusion constants $D$, we show that the probabili…
Graph vertex embeddings based on random walks have become increasingly influential in recent years, showing good performance in several tasks as they efficiently transform a graph into a more computationally digestible format while preserving relevant information. However, the theoretical properties of such algorithms, in particular the influence of hyperparameters and of the graph structure on their convergence behaviour, have so far not been well-understood. In this work, we provide a theoretical analysis for random-walks based embeddings techniques. Firstly, we prove that, under some weak a…
We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces for bond percolation on a square lattice. So it is natural to conjecture that the scaling limit is SLE$_6$. We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.
Emergent population dynamics of random walkers with cooperative reproduction and spatial selection
研究随机游走群体在高阶无性繁殖下的动力学行为，发现繁殖阶数增加导致入侵前沿消失及扩散无关速度，揭示自然中二分裂繁殖的主导地位受入侵动态基本约束。
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