Majid Hosseini, Krishnamurthi Ravishankar
We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special cases. With minor changes the same argument can be used to prove the scaling limit of the corresponding walk in Z^d.
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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 the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, namely, for some critical value $λ_{\mathrm{c}} >0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk is strictly positive if the bias $λ$ is strictly smaller than $λ_{\mathrm{c}}$, whereas $\overline{\mathrm{v}}=0$ if $λ\geq λ_{\mathrm{c}}$. We show that at the critical bias $λ= λ_{\mat…
Consider uniformly elliptic random walk on $\bbZ^d$ with independent jump rates across nearest neighbour bonds of the lattice. We show that the infinite volume effective diffusion matrix can be almost surely recovered as the limit of finite volume periodized effective diffusion matrices.
An infinite graph G has the property that a random walk in random environment on G defined by i.i.d. resistances with any common distribution is almost surely transient, if and only if for some p<1, simple random walk is transient on a percolation cluster of G under bond percolation with parameter p.
We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of T. Franco, A. Neumann and P. Gonçalves (2013), which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel's principl…
We consider a stationary and ergodic random field {ω(b)} parameterized by the family of bonds b in Z^d, d>1. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming that the set of bonds with positive conductance has a unique infinite cluster C, we prove homogenization results for the random walk among random conductances on C. As a byproduct, applying the general criterion of \cite{F} leading to the hydrodynamic limit of exclusion processes with bond-dependent transition rates, for almost all realizations of the environment we p…
Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.
We give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs. In particular, this condition is verified by the random conductance model on $\Z^d$, $d\geq 3$, when the conductances are i.i.d. and the bonds with positive conductance percolate. The harmonic measure from infinity also exists for random walks on supercritical clusters of $\Z^2$. This is proved using results of Barlow (2004).
A central limit theorem for unbalanced step-reinforced random walks
本文研究一类统一了象随机游走等模型的不平衡步长强化随机游走，通过建立与随机递归树键渗流的联系将其表示为随机加权和，推导出正态与稳定中心极限定理，进而获得该类游走的统一中心极限定理。
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.
We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias $λ> 0$, then its asymptotic linear speed $\overline{\mathrm{v}}$ is continuous in the variable $λ> 0$ and differentiable for all sufficiently small $λ> 0$. In the paper at hand, we complement this result by proving that $\overline{\mathrm{v}}$ is differentiable at $λ= 0$. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at $λ= 0$ equ…
We present a coupled decreasing sequence of random walks on $ \mathbb Z $ that dominates the edge process of oriented-bond percolation in two dimensions. Using the concept of "random walk in a strip ", we construct an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented-bond percolation. Numerical calculations of the first ten lower bounds thereby generated lead to an improved,i.e. higher, rigorous lower bound to this critical probability, viz. $p_{c} \geq 0.63328 $. Finally a computer simulation technique is presented; the use …
We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value $λ_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $λ\in (0,λ_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $λ>λ_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $λ$ on the interval $(0,λ_{\mathrm{c}})$ and differentiable on $(0,λ_{\mathrm{c}}/2)$. Mo…
Ladder costs for random walks in Lévy random media
本文研究 Lévy 随机介质中随机游走的首达阶梯高度与长度分布律，通过构建键上随机环境随机游走（RWRSB）模型，推导广义 Spitzer-Baxter 恒等式并利用局部时间表示，刻画了首出时间前环境值之和的尾部分布，且结果可推广至整个阶梯变量序列。
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。
Anomalous Heat Kernel for Random Walks in Random Environments of Conductances
针对网格$\mathbb{Z}^d$上的随机电导模型，研究$[0,1]$多项式尾电导下$d\ge5$反常热核衰减阶与$d=4$正常行为，及$[1,\infty)$多项式电导的相反返回概率行为。
Let X= {X_t, t \ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {μ_e \in [1, \infty), e \in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\ge 3. Let R = {x \in Z^d: X_t = x for some t \ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily large …
Continuity for random walks in hyperbolic percolation: drift, entropy, and dimension
本文证明了双曲群Cayley图渗流无限簇上随机游走的漂移和渐近熵随渗流参数连续变化，进而推出调和测度Hausdorff维数的连续性。
Let $\mathcal{C}^n$ be the largest open cluster for supercritical Bernoulli bond percolation in $[-n, n]^d \cap \mathbb{Z}^d$ with $d \ge 2$. We obtain a sharp estimate for the effective resistance on $\mathcal{C}^n$. As an application we show that the cover time for the simple random walk on $\mathcal{C}^n$ is comparable to $n^d (\log n)^2$. Noting that the cover time for the simple random walk on $[-n, n]^d \cap \mathbb{Z}^d$ is of order $n^d \log n$ for $d \ge 3$ (and of order $n^2 (\log n)^2$ for $d = 2$), this gives a quantitative difference between the two random walks for $d \ge 3$.
It is known that a random walk on $\Z^d$ among i.i.d. uniformly elliptic random bond conductances verifies a central limit theorem. It is also known that approximations of the covariance matrix can be obtained by considering periodic environments. Here we estimate the speed of convergence of this homogenization result. We obtain similar estimates for finite volume approximations of the effective conductance and of the lowest Dirichlet eigenvalue. A lower bound is also given for the variance of the Green function of a random walk in a random non-negative potential.
Coupling between Brownian motion and random walks on the infinite percolation cluster
本研究针对超临界伯努利边渗流构建无穷团随机游走与布朗运动的耦合，证明路径最大距离均值阶为$T^{1/3+o(1)}$，部分解决公开问题并恢复重对数律、确定极限常数。
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.
Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return …
We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d.\ site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations $χ'$ are spoiled by reading errors that occur with probability $p<1$. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct t…
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $ω$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk ret…
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $ω_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $ω_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_ω^{2n}(0,0)$. We prove that $P_ω^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)…
Collisions of random walks on comb graphs with a planar base
本文研究以递归平面图为基底的梳状图上两个独立随机游走的碰撞次数，证明其随齿长轮廓函数的增长发生相变，并解决了 Barlow 等人关于 $\mathbb{Z}^2$ 基底上对数增长轮廓的碰撞有限性与无限性的临界问题。
We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffus…
We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-α}$ with $α>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv2(α\wedge2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriente…
We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov-Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.
We study continuous-time (variable speed) random walks in random environments on $\mathbb{Z}^d$, $d\ge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space-time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The…
In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The lat…
In this article, we consider random walk on the infinite cluster of bond percolation on $\Z^d (d \geq 2)$. We show that the Laplace transformation of the number of visited points $N\_n$, has a behaviour as the random walk was on $\Z^d$. More precisely, for all $0<α<1$, we proved that there exist constants $C\_i$ and $C\_s$ such that for all infinite cluster that contains the origin, we have: $$ e^{-C\_i n^{\frac{d}{d+2}}} \leq \E\_0^ω (α^{N\_n}) \leq e^{-C\_sn^{\frac{d}{d+2}}}.$$ Our approach is based on finding an isoperimetric inequalities on the infinite cluster, lifted on a wreath product …
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process …
We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\simβ/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is transient, and when $s\geq 2d$, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension $d$, if $d<s<2d$, then critical percolation has no infinite clusters. This result is extended to the free random cluster model. A second corollary is that when $d\…
We prove a formula for the speed of distance stationary random sequences generalizing the law of large numbers of Karlsson and Ledrappier. A particular case is the classical formula for the largest Lyapunov exponent of i.i.d.\ matrix products, but our result has applications in various different contexts. In many situations it gives a method to estimate the speed, and in others it allows to obtain results of dimension drop for escape measures related to random walks. We show applications to stationary reversible random trees with conductances, Bernoulli bond percolation of Cayley graphs, and r…
We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e the behavior of the "ant in the labyrinth". It is natural to conjecture (see [16] and [8]) that the scaling limit for random walks on large critical random graphs exists in high dimensions, and is universal. This scaling limit is simply the natural Brownian Motion on the Integrated Super-Brownian Excursion. We give here a set of four natural sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task…
Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n) and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random properties, then critical bond-percolation on G_n has a scaling window of width n^{-1/3}, as it would on a random graph. A consequence of our theorems is that if G_n is a transitive expander family with girth at least (2/3 + eps) \log_{d-1} n, then the size of the largest component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is roughly n^{2/3}. In particu…
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$…
Localization induced by Sinai disorder in mass-exchange models
该研究引入带淬火无序的保守连续质量交换模型，通过生灭链求解其平稳质量分布，揭示β√N参数对分布类型的区分，表征涨落与局域相关定律，还分析了极值质量的标度特性。
From quenched invariance principle to semigroup convergence with applications to exclusion processes
本文基于淬火不变原理，推导了半群的$L^1$收敛，并应用于简单对称排斥过程的水动力极限研究。
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