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 returns to the starting point at a given time (cf math.PR/0611666).
Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.
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.
In this note, we establish a functional central limit theorem for the capacity of the range for a class of $α$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5α/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3α/ 2$.
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.
In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.
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…
In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup i…
The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well known result with a long history. In this paper we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from arXiv:1812.01915. We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.
We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d\ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in $d\ge4$. In $d=3$, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for $d=3$ al…
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a 2d-random walk in different situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.
We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In a second part of the paper, we show that the (exponential) deviation inequality holds for me…
In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the n…
Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, μ)$ with a finite system of generators $(A_1, ..., A_d)$. Let $A^{\el}$ denote $A_1^{\ell_1} ... A_d^{\ell_d}$, for ${\el}= (\ell_1, ..., \ell_d)$. If $(Z_k)$ is a random walk on $\Z^d$, one can study the asymptotic distribution of the sums $\sum_{k=0}^{n-1} \, f \circ A^{\,{Z_k(ω)}}$ and $\sum_{\el \in \Z^d} \PP(Z_n= \el) \, A^\el f$, for a function $f$ on $X$. In particular, given a random walk on commuting matrices in $SL(ρ, \Z)$ or in ${\Cal M}^*(ρ, \Z)$ acting on the torus $\T^ρ$, $ρ\geq 1$, what is th…
We prove a Central Limit Theorem for the proportion of infected individuals for an epidemic model by dealing with a discrete time system of simple random walks on a complete graph with n vertices. Each random walk makes a role of a virus. Individuals are all connected as vertices in a complete graph. A virus duplicates each time it hits a susceptible individual, dying as soon as it hits an already infected individual. The process stops as soon as there is no more viruses. This model is closely related to some epidemiologial models like those for virus dissemination in a computer network.
We study random walk among random conductance (RWRC) on complete graphs with N vertices. The conductances are i.i.d. and the sum of conductances emanating from a single vertex asymptotically has an infinitely divisible distribution corresponding to a Lévy subordinator with infinite mass at 0. We show that, under suitable conditions, the empirical spectral distribution of the random transition matrix associated to the RWRC converges weakly, as N goes to infinity, to a symmetric deterministic measure on [-1,1], in probability with respect to the randomness of the conductances. In short time scal…
Let $(X_n)$ be a Markov chain on a standard borelian space $\mathbb{X}$. Any stopping time $τ$ such that $\mathbb{E}_xτ$ is finite for all $x\in\mathbb{X}$ induces a Markov chain in $\mathbb{X}$. In this article, we show that there is a bijection between the invariant measures for the original chain and for the induced one. We then study drift functions and prove a few relations that link the Markov operator for the original chain and for the induced one. The aim is to use this drift function and the induced operator to link the solution to Poisson's equation $(I\_d-P)g=f$ for the original cha…
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…
Let $ν\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in\b N$, let $(X_n^p)_{n\ge1}$ be i.i.d. $\b R^p$-valued radial random variables with radial distribution $ν$. We derive two central limit theorems for $ \|X_1^p+...+X_n^p\|_2$ for $n,p\to\infty$ with normal limits. The first CLT for $n>>p$ follows from known estimates of convergence in the CLT on $\b R^p$, while the second CLT for $n<<p$ will be a consequence of asymptotic properties of Bessel convolutions. Both limit theorems are considered also for $U(p)$-invariant random walks on the space of $p\times q$ matri…
We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. …
We study variable-speed random walks on $\mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.
We prove an averaged CLT for a random walk in a dynamical environment where the states of the environment at different sites are independent Markov chains.
We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.
Self-attractive random walks undergo a phase transition in terms of the applied drift: If the drift is strong enough, then the walk is ballistic, whereas in the case of small drifts self-attraction wins and the walk is sub-ballistic. We show that, in any dimension at least 2, this transition is of first order. In fact, we prove that the walk is already ballistic at critical drifts, and establish the corresponding LLN and CLT.
Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $λ$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $λ/(λ+d_v)$, and moves to each of the offspring with probability $1/(λ+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $ 0<λ<m$ and $\v=0$ for $λ…
We derive laws of the iterated logarithm for random walks on random conductance models under the assumption that the random walks enjoy long time sub-Gaussian heat kernel estimates.
In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We study the regularity of the speed as a function of the distribution of conductances, in particular when the distribution of conductances converges to a non-elliptic limit.
We consider random walk among random conductances where the conductance environment is shift invariant and ergodic. We study which moment conditions of the conductances guarantee speed zero of the random walk. We show that if there exists α>1 such that E[log^α(ω_e)]<\infty, then the random walk has speed zero. On the other hand, for each α>1 we provide examples of random walks with non-zero speed and random walks for which the limiting speed does not exist that have E[log^α(ω_e)]<\infty.
We prove an almost sure invariance principle for a random walker among i.i.d. conductances in $\Z^d$, $d\geq 2$. We assume conductances are bounded from above but we dot require they are bounded from below.
We explain a unified approach to a study of ballistic phase for a large family of self-interacting random walks with a drift and self-interacting polymers with an external stretching force. The approach is based on a recent version of the Ornstein-Zernike theory developed in earlier works. It leads to local limit results for various observables (e.g. displacement of the end-point or number of hits of a fixed finite pattern) on paths of n-step walks (polymers) on all possible deviation scales from CLT to LD. The class of models, which display ballistic phase in the "universality class" discusse…
We prove an invariance principle for continuous-time random walks in a dynamically averaging environment on $\mathbb Z$. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.
We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos Santos, Sidoravicius. We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.
We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds…
We consider the random walk among random conductances on Z^d. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t^{-1/10} for d < 3, and speed t^{-1/5} otherwise, up to logarithmic corrections.
We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for the position of the walker, properly rescaled. Moreover, we state a functional Central Limit Theorem where an atypical process, related to the Fractional Kinetics, appears in the limit.
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.
We consider random walks among random conductances on $\mathbb{Z}^2$ and establish precise asymptotics for the associated potential kernel and the Green's function of the walk killed upon exiting balls. The result is proven for random walks on i.i.d. supercritical percolation clusters among ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau $\nabla ϕ$-interface model.
We consider a weighted lattice $Z^d$ with conductance $μ_e=|e|^{-α}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$ and $α\in (-1,0)$, two independent random walks on such weighted lattice will collide infinite many times while they are transient.
We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is…
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)…
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 consider two models of one-dimensional random walks among biased i.i.d. random conductances: the first is the classical exponential tilt of the conductances, while the second comes from the effect of adding an external field to a random walk on a point process (the bias depending on the distance between points). We study the case when the walk is transient to the right but sub-ballistic, and identify the correct scaling of the random walk: we find $α\in[0,1]$ such that $\log X_n / \log n \to α$. Interestingly, $α$ does not depend on the intensity of the bias in the first case, but it does i…
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 show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a …
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…
We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann.\ Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.
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 the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting.
Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of GLd (R), such as the Marcinkiewicz-Zygmund strong law of large numbers, the CLT (with rates in Wasserstein's distances) and almost sure invariance principles with rates.
We consider the random conductance model, where the underlying graph is an infinite supercritical Galton--Watson tree, the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that, if the mean conductance is finite, there is a deterministic, strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$ a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a formula for $v$ in terms of the laws of certain effective conductances and show that, if the conductances share the same expected value, the speed i…
