Jay Rosen
For the simple random walk in Z^2 we study those points which are visited an unusually large number of times, and provide a new proof of the Erdos-Taylor conjecture describing the number of visits to the most visited point.
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We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erdős-Rényi strong law for the increments.
We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $μ\ll 1/n$, and is then declared open with probability $p = λ/n$ where $λ> 1$. We study a random walk on this dynamical environment which jumps at rate $1/n$ along every open edge. We show that the mixing time of the full system exhibits cutoff at $\log n/μ$. We do this by showing that the random walk component mixes faster than the environment process; along the way, we control the time it takes for the walk to become isolated.
Meeting and coalescence times for random walks in the largest component of the Erdős-Rényi random graph
研究厄多斯-雷尼随机图最大连通分量上随机游走的相遇和合并时间，通过证明相遇时间界限并结合比较不等式，得出预期合并时间和完全选民模型共识在严格超临界、轻微超临界和临界区域都为\(n\)阶。
We revisit the multifractal analysis of $\R^d$-valued branching random walks averages by considering subsets of full Hausdorff dimension of the standard level sets, over each infinite branch of which a quantified version of the Erdös-Rényi law of large numbers holds. Assuming that the exponential moments of the increments of the walks are finite, we can indeed control simultaneously such sets when the levels belong to the interior of the compact convex domain $I$ of possible levels, i.e. when they are associated to so-called Gibbs measures, as well as when they belong to the subset $(\partial{…
Let $G(N,p)=(V,E)$ be an Erdös-Rényi random graph and $(X_n)_{n \in \mathbb{N}}$ be a simple random walk on it. We study the the order of magnitude of $\sum_{i \in V} π_ih_{ij} $ where $π_i=d_i / 2|E|$ for $d_i$ the number of neighbors of node $i$ and $h_{ij}$ the hitting time for $(X_n)_{n \in \mathbb{N}}$ between nodes $i$ and $j$, in a regime of $p=p(N)$ such that $G(N,p)$ is almost surely connected as $N\to\infty$. Our main result is that $\sum_{i \in V} π_ih_{ij} $ is almost surely of order $N(1+o(1))$ as $N\to \infty$, which coincides with previous results in the physics literature \cite…
A multivariate extension of the Erdös-Taylor theorem
该文将Erdös-Taylor定理推广到多元情形，证明多个独立二维随机游走的成对碰撞局部时经对数重标度后联合收敛于独立指数随机变量向量，并讨论与定向聚合物的联系。
We prove expectation and concentration results for the following random variables on an Erdős-Rényi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probabilit…
We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pal Erdos and Pal Revesz in the early eighties.
We prove that, with probability one, eventually there are no more than three favourite (i.e. most visited) sites of simple random walk. This partially answers a relatively long standing question of Pal Erdos and Pal Revesz.
Self-switching random walks on Erdös-Rényi random graphs feel the phase transition
本文研究 Erdös-Rényi 随机图上边概率随每次返回起点而重新采样的自切换随机游走，证明稠密情形下占用测度收敛到先验测度，稀疏情形下收敛到由 Poisson 分支过程生存概率决定的极限测度，且该测度支撑在超临界区域，从而揭示随机游走能感知相变。
For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $\mathbb Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n\mathbb I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erdős), which corresponds to a function $f(i)=\mathbb I\{i\ge 1\}$; (b) $α$-fold self-intersections of the random walk (studied by Becker and König), which corresponds to $f(i)=i^α$; (c) sites visited by the random walk exactly $j$ times (considered by…
On the Asymptotics of the Connectivity Probability of Erdos-Renyi Graphs
本文通过非齐次随机游走分析，将Erdos-Renyi图连通概率问题转化为泊松增量随机游走在桥条件下的游程概率，从而刻画其精确渐近行为。
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
We consider simple random walk on a realization of an Erdős-Rényi graph that is asymptotically almost surely (a.a.s.) connected. We show a Central Limit Theorem (CLT) for the average starting hitting time, i.e. the expected time it takes the random walker on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $π_j$, the invariant measure of the random walk.
A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.
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).
In the classical paper of Dvoretzky-Erdős, asymptotics for the expected value and the variance of the number of distinct sites visited by a Simple Symmetric Random Walk were calculated. Here, these results are generalized for Random Walks with Internal States. Moreover, both weak and strong laws of large numbers are proved. As a tool for these results, the error term of the local limit theorem in of Krámli and Szász is also estimated.
Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions in the cases of constant a…
For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erdős-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest.
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 …
Let R_n be the radius of the largest disk covered after n steps of a simple random walk. We prove that almost surely limsup_{n \to \infty}(log R_n)^2/(log n log_3 n) = 1/4, where log_3 denotes 3 iterations of the log function. This is motivated by a question of Erdős and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.
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.
A Central Limit Theorem for the average target hitting time for a random walk on a random graph
本文在特征向量离域猜想下，证明了渐近连通的Erdős-Rényi图上简单随机游走的平均目标命中时间满足中心极限定理。
For any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on (g_n)_{n>0} and the distribution of S_1 determining the set of accumulation points for (g_n S_n)_{n>0}. This extends, with a simpler proof, a result of K.L. Chung and P. Erdos. Finally, for recurrent, symmetric random walks, we give a criterion characterizing the increasing sequences (g_n)_{n>0} of positive numbers for which liminf g_n|S_n|=0.
Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in random environment. It is known for a phenomenon of strong localization, namely, the walk spends almost all time at or near the bottom of deep valleys of the potential. Our main result shows a weakness of this localization phenomenon: with probability one, the zones where the walk stays for the most time can be far away from the sites where the walk spends the most time. In particular, this gives a negative answer to a problem of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modelli…
We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdos-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal g…
Exponential moments of truncated branching random walk martingales
本文研究漂移至无穷远的分支随机游走的马尔萨斯鞅，证明原点以下粒子被清除时该鞅极限呈指数尾（无清除时为多项式尾），临界情形下截断导数鞅也有相同结论，该研究受Erdős–Rényi图首次通过渗流相关工作启发。
Limit theorems for walks and triangles on Erdös-Rényi random graphs with large interaction radius
本文研究了Erdös-Rényi随机图中q步行走数量的累积量，在顶点数N、浓度c和相互作用半径R趋于无穷时，通过树型图示来关联这些累积量，并证明了非闭合行走和三角形数量的极限定理，同时指出了三角形数量在随机图中的渐近阈值。
We study the simple random walk on the giant component of a supercritical Erdős-Rényi random graph on $n$ vertices, in particular the so-called vacant set at level $u$, the complement of the trajectory of the random walk run up to a time proportional to $u$ and $n$. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter $u_{\star}$: For $u<u_{\star}$ the vacant set has with high probability a unique giant component of order $n$ and all other components small, of order at most $\log^{7}n$, whereas for $u>u_{\star}$ it has with high probability…
Favorite sites for simple random walk in two and more dimensions
针对d≥2时ℤᵈ上简单随机游走的偏好位点演化问题，证明d=2时三个偏好位点几乎必然同时出现无穷多次且最终无四个同时出现，推导d≥3时偏好位点数量的精确渐近性质，回答了Erdős和Révész（1987）的开放问题。
A random walk among random graphs
本文为奥赛硕士课程讲义，系统介绍一维随机游走、随机树、Erdös–Rényi随机图巨分量出现的三种证明，以及随机递归树、随机排列与连续时间嵌入技术。
We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-m…
We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph G, with any feasible number of particles. Our estimate is proportional to $\mathsf{T}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon)$, where |V| is the number of vertices in G, and $\mathsf{T}_{\mathsf{RW}(G)}$ is the 1/4-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdos-Renyi random graph and Poisson point processes in $\mathbb{R}^d$. Our technical tools include a …
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。
There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint work with A. Dembo, J. Rosen and O. Zeitouni. As a consequence, we proved two conjectures about simple random walk in two dimensions: The first, due to Erdős and Taylor (1960), involves the number of visits to the most visited lattice site in the first $n$ steps of the walk. The second, due to Aldous (1989), concerns the number of steps it takes a simple ra…
A Method for Generating Connected Erdos-Renyi Random Graphs
本文提出将图探索过程与非齐次 Poisson 随机游走耦合的精确采样算法，可在线性时间内生成稀疏连通 $G(n,p)$，并扩展至 $G(n,M)$。
The harmonic measure $ν$ on the boundary of the group $Sol$ associated to a discrete random walk of law $μ$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $μ$ with singular harmonic measure $ν$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed…
The reciprocal class of a Markov path measure is the set of all mixtures of its bridges. We give characterizations of the reciprocal class of a continuous-time Markov random walk on a graph. Our main result is in terms of some reciprocal characteristics whose expression only depends on the jump intensity. We also characterize the reciprocal class by means of Taylor expansions in small time of some conditional probabilities. Our measure-theoretical approach allows to extend significantly already known results on the subject. The abstract results are illustrated by several examples.
Mixing of fast random walks on dynamic random permutations
研究动态随机排列上快速随机游走的混合行为，证明全变差距离在缩放后以单次跳跃下降，跳跃时间分布和跳跃后衰减函数与Erdős-Rényi图最大分量相关。
Some results for range of random walk on graph with spectral dimension two
本文研究了谱维数为二的图上随机游走范围的性质，提出了一种强大数定律形式，并在特定均匀条件下证明了其在整数平方格子和分形图上的适用性，同时给出了新的证明方法及递归图的构造。
Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk
针对一维简单随机游走最小局部时间f(n)的上极限增长问题，本文确定其精确渐近增长率为几乎必然有limsup_{n→∞}f(n)/log log n=1/log 2，回答了Révész（2013）提出的开放问题。
Convergence of blanket times for sequences of random walks on critical random graphs
本文在图序列于 Gromov-Hausdorff 拓扑下收敛的条件下，证明随机游走 ε-覆盖时间的渐近界与收敛性，并将其用于临界 Galton-Watson 树、Erdős-Rényi 图和 configuration model。
Concentration of Hitting Times in Erdös-Rényi graphs
本文证明Erdős-Rényi图中随机游走击中时间高度集中，可由边数、度与距离的简单公式近似，误差极小。
We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into and overcome these by providing Taylor expansions of the inverse Riemannian exponential map, while also comparing the differential of the exponential map to parallel transport. Finally, we obtain the analogue of Cramér's theorem for geodesic random walks by showing that the curva…
Last-Coverage Probabilities of Random Walks on Cycle Graphs for Symmetric, Biased, and Time-Dependent Walks
本文针对环图上的对称、有偏和时变随机游走，利用连通弧性质推导最后覆盖概率公式，通过模拟验证结果，并对比不同图结构的最后覆盖分布。
We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^ε$, where $n$ is the number of vertices in the random graph, and $ε$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős-Rényi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a co…
Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-β<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms of Taylor series about $β=0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $β\downarr…
We study random walks on the giant component of the Erdős-Rényi random graph ${\cal G}(n,p)$ where $p=λ/n$ for $λ>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(ν{\bf d})^{-1}\log n \pm (\log n)^{1/2+o(1)}$, where $ν$ and ${\bf d}$ are the speed …
Mixing cutoff for simple random walks on the Chung-Lu digraph
本文研究Chung-Lu有向图上简单随机游走的混合行为，证明在平均度随图大小对数增长的弱稠密状态下，总变差距离在熵时间处呈现截断行为且收敛于高斯尾函数，将已有硬约束模型的结果扩展至软约束模型。
We prove that in sparse Erdős-Rényi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lovàsz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erdős-Rényi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic…
A central limit theorem for the giant in a stochastic block model
本文利用广度优先游走方法，为超临界随机块模型中巨分量顶点数给出了中心极限定理的简洁证明，并将其归结为 Erdős-Rényi 模型的经典结果。
We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time, while on the other hand many distributed networks are inherently dynamic, and their topology is subjected to potentially drastic changes. In this work we study the mixing (i.e., converging) properties of random …
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on $n$ elements. Consider this walk in continuous time starting at the identity and let $D_t$ be the minimum number of transpositions needed to go back to the identity from the location at time $t$. $D_t$ undergoes a phase transition: the distance $D_{cn/2} \sim u(c)n$, where $u$ is an explicit function satisfying $u(c)=c/2$ for $c \le 1$ and $u(c)1$…
The Power of Two Choices for Random Walks
本文研究图上的随机游走中，通过二选一策略（从两个随机邻居中选择）可显著加速击中与覆盖时间，并在多种图类上给出线性或近线性的覆盖时间上界，同时证明计算最优策略的效率二分性。
Let $(X,d)$ be a geodesic Gromov-hyperbolic space, $o \in X$ a basepoint and $μ$ a countably supported non-elementary probability measure on $\operatorname{Isom}(X)$. Denote by $z_n$ the random walk on $X$ driven by the probability measure $μ$. Supposing that $μ$ has finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence $\frac{1}{n}d(z_n,o)$ and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence $d(z_n,o)$. This provides a positive answer to a question rais…
Tournaments and random walks
本文利用积分随机游走桥的极限理论验证了关于竞赛图计数的Moser猜想，并给出了随机得分序列的标度极限及随机聚合物模型中持续概率的精确渐近性。
Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $α$-stable law, $0<α<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dime…
A non-backtracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The non-backtracking matrix of a graph is indexed by its directed edges and can be used to count non-backtracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the non-backtracking matrix of the Erdos-Renyi random graph and of the Stochastic Block Model in the regime where the…
Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space
本文研究无限维相空间中基于 Weyl 表示的量子随机游走，分析高斯平均平移形成的自伴压缩半群，并证明其收敛到以无限维谐振子 Hamilton 量为生成元的量子振子演化半群。
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