Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in ${\mathbb{Z}^d}$ , d?≥ 2. We first prove an invariance principle (for d?≥ 2) and the transience of the random walks when d?≥ 3 (recurrence when d?=?2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.     

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z^d${\mathbb{Z}}^d$. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z^d${\mathbb{Z}}^d$, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.     

Quenched invariance principles for walks on clusters of percolation or among random conductances

In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ^d. We prove a quenched functional central limit theorem for the walk when d4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ^d, when d1.V. Sidoravicius would like to thank the FIM for financial support and hospitality during his multiple visits to ETH. His research was also partially supported by FAPERJ and CNPq.     

Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials

 This article describes the almost sure infinite volume asymptotics of the ground state energy of random Schr?dinger operators with scaled Gibbsian potentials. The random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered Gibbs measure with a superstable pair interaction. There is no restriction on the strength of the pair interaction: it may be taken, e.g., at a critical point. The potential is scaled with the box size in a critical way, i.e. the scale is determined by the typical size of large deviations in the Gibbsian cloud. The almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities. The qualitative behaviour of the ground state energy asymptotics is analysed: Depending on the dimension and on the H?lder exponents of the free energy density, it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy. Received: 24 June 2002 / Revised version: 17 February 2003 Published online: 12 May 2003 Mathematics Subject Classification (2000): Primary 82B44; Secondary 60K35 Key words or phrases: Gibbs measure – H?lder exponents – Random Schr?dinger operator – Ground state – Large deviations     

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

Finitely additive random walks on infinitely generated free groups

We consider any purely finitely additive probability measure supported on the generators of an infinitely generated free group and the Markov strategy with stationary transition probability . As well as for the case of random walks (with countably additive transition probability) on finitely generated free groups, we prove that all bounded sets are transient. Finally, we consider any finitely additive measure (supported on the group generators) and we prove that the classification of the state space depends only on the continuous part of .     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.     

Local limits and harmonic functions for nonisotropic random walks on free groups

Summary Nearest neighbour random walks on the homogeneous tree representing a free group withs generators (2s) are investigated. By use of generating functions and their analytic properties a local limit theorem is derived. A study of the harmonic functions corresponding to the random walk leads to properties that characterize ther-harmonic function connected with the local limits.     

Gaussian fluctuations for random walks in random mixing environments

We consider a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions. Continuing our previous work [2] for the law of large numbers, we prove here that the fluctuations are Gaussian when the environment is Gibbsian satisfying the “strong mixing condition” of Dobrushin and Shlosman and the mixing rate is large enough to balance moments of some random times depending on the path. Under appropriate assumptions the annealed Central Limit Theorem (CLT) applies in both nonnestling and nestling cases, and trivially in the case of finite-dependent environments with “strong enough bias”. Our proof makes use of the asymptotic regeneration scheme introduced in [2]. When the environment is only weakly mixing, we can only prove that if the fluctuations are diffusive then they are necessarily Gaussian. Partially supported by CNRS, UMR 7599 “Probabilités et Modèles aléatoires”. Partially supported by NSF grant number DMS-0302230.     

Persistent random walks in random environment

Summary Weak convergence of a class of functionals of PRWRE is proved. As a consequence CLT is obtained for the normed trajectory.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (Grant No. 476/82).     

Semi-classical limit for random walks

Let be a discrete symmetric random walk on a compact Lie group with step distribution and let be the associated transition operator on . The irreducibles of the left regular representation of on are finite dimensional invariant subspaces for and the spectrum of is the union of the sub-spectra on the irreducibles, which consist of real eigenvalues . Our main result is an asymptotic expansion for the spectral measures

along rays of representations in a positive Weyl chamber , i.e. for sequences of representations , with . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on (for which is essentially a direct sum of Harper operators).

     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Non parametric estimation for random walks in random environment

We investigate the problem of estimating the cumulative distribution function (c.d.f.) $F$ of a distribution $\nu$ from the observation of one trajectory of the random walk in i.i.d. random environment with distribution $\nu$ on $\mathbb{Z}$. We first estimate the moments of $\nu$, then combine these moment estimators to obtain a collection of estimators ${\left({\stackrel{?}{F}}_n^M\right)}_{M\ge 1}$ of $F$, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of $F$ and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.     

Quenched moderate deviations principle for random walk in random environment

We derive a quenched moderate deviations principle for the one-dimensional nearest random walk in random environment, where the environment is assumed to be stationary and ergodic. The approach is based on hitting time decomposition.     

Delayed random walks

A delayed random walk $\text{{S}{}^1{}_{\mathrm{n}}\text{, n ≥ 0}}$ is defined here as a partial sum process of independent random variables in which the first N summands (N optional) are distributed F¹,…,F_N, respectively, while all remaining summands are distributed F₀, where {F_k, _k ≥ 0} is a sequence of proper distribution functions on the real line. Delayed random walks arise naturally in the study of certain generalized single server queues. This paper examines optional times of the process such as $\text{π = inf {n: n ≥ 1 and S}{}^1{}_{\mathrm{n}}\text{≥ 0}}$. Conditions insuring the finiteness of E {π} and E {π²} are obtained, generating functions calculated, and illustrative examples given. The bivariate functions $\text{E{r}{}^{\mathrm{\pi }}\text{explsqbitS}{}^1{}_{\mathrm{\pi }}\text{rsqb}}$ and $\text{E {}\text{∑}\text{n=0}\text{π?1}\text{explsqbitS}{}^1{}_{\mathrm{n}}\text{rsqb}}$ are studied for the case where N ≡ 1.