Martin T. Barlow, Xinghua Zheng
We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^ω_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for $p^ω_{n^2t}(x,y)$ for all $x,y$ in a ball.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.
Results have been moved to a published article, see arXiv:0812.2669v4[math.PR]
We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.
We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.
We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$ with a power law with an exponent $γ$ near 0. We are interested in estimating the quenched decay of the return probability $P_ω^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for $γ> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. As an expected concequence, the same result holds for the discrete-time case.
This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
We study models of discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $γ>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $γ$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the par…
Quenched scaling limit for biased random walks on random, heavy tailed conductances: low dimensions
本文研究 $\mathbb{Z}^d$ 上重尾随机电导中有偏随机游走的次弹道行为，将已有退火标度极限推广并证明为对所有 $d\ge2$ 成立的淬火极限。
Aging and sub-aging for one-dimensional random walks amongst random conductances
研究一维随机电导中随机游走的长时间行为，证明在电导于0处重尾时过程最大值的 aging 现象，以及在无穷远处重尾时游走者位置的 aging 与 sub-aging 结论。
A surprising result of Pillai and Meng (2016) showed that a transformation $\sum_{j=1}^n w_j X_j/Y_j$ of two iid centered normal random vectors, $(X_1,\ldots, X_n)$ and $(Y_1,\ldots, Y_n)$, $n>1$, for any weights $0\leq w_j\leq 1$, $ j=1,\ldots, n$, $\sum_{j=1}^n w_j=1$, has a Cauchy distribution regardless of any correlations within the normal vectors. The correlations appear to lose out in the competition with the heavy tails. To clarify how extensive this phenomenon is, we analyze two other transformations of two iid centered normal random vectors. These transformations are similar in spiri…
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq 2$) with zero Dirichlet condition. We assume that the conductances $w$ are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If $γ=\sup \{ q\geq 0\colon \mathbb{E} [w^{-q}]<\infty \}<1/4$, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold $γ_{\rm c} = 1/4…
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in $\Z^d$. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent …
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first $k$ eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the $k$th eigenvector is close to the principal eigenvector of an auxiliary spectral problem.
We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\Z^d$ in the spirit of Donsker-Varadhan \cite{DV75}. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify th…
We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that the scaling limit of the process is the inverse of an $α$-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [Enriquez, Sabot, Zindy, Bull. Soc. Math. 2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], some `traps' a…
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.
On the Tail Transition of First Arrival Position Channels: From Cauchy to Exponential Decay
研究探讨了首次到达位置通道在非零漂移影响下的尾部过渡，从重尾代数 regime 转向指数正则化，通过分析精确的 FAP 密度确定了传播距离边界，验证了零漂移柯西分布在低漂移环境中的鲁棒性。
A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.
We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0, and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
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。
Quenched local limit theorem for random conductance models with long-range jumps
该研究针对带任意长度跳跃的平稳遍历随机电导模型，利用弱抛物Harnack不等式与对角热核估计，建立了d≥2维整数格点上可逆随机游走的淬火局部极限定理，部分解决了相关开放问题，还得到带额外尾项的极大不等式与校正子的处处次线性性质。
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in $\mathbb{Z}^d$. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, th…
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit la…
The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techn…
A scheduled arrival model is one in which customers are scheduled to arrive at constant interarrival times, but each customer actual arrival time is perturbed from her scheduled arrival time by a random perturbation. The sequence of perturbations is independent and identically distributed. It has previously been shown that the arrival counting process for scheduled traffic obeys a functional central limit theorem (FCLT) with fractional Brownian motion (fBM) with Hurst parameter H strictly between 0 and 1/2 when the pertubratons have a Pareto-like tail with tail exponent lying in (0,1). Such li…
Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime
研究具有 Pareto 势的抛物型 Anderson 模型在重尾区域 $\alpha\in(d,2d)$ 中解的总质量随时间趋于无穷时的完全渐近行为，发现维度 $d=1$ 与 $d\ge 2$ 存在定性差异。
We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on $\Z^4$) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability $\cmss P_ω^{2n}(0,0)$ after $2n$ steps is at most $C(ω) n^{-2} \log n$, but the best lower bound till now has been $C(ω) n^{-2}$. Here we will show that the $\log n$ term marks a real phenomenon by constructing an environment, for each sequence $λ_n\to\infty$, such that $$ \cmss P_ω^{2n}(0,0)\ge C(ω)\log(n)n^{-2}/λ_n, $$…
Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model
本文研究了Galton-Watson树的非线性电导及其在近临界随机簇模型中的应用，揭示了临界点下根磁化率的衰减规律。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。