Slowdown and neutral pockets for a random walk in random environment
type: Item
instance of: scholarly article
title: Slowdown and neutral pockets for a random walk in random environment
zbMATH Open document ID: 0947.60095
publication date: 2000-03-30T00:00:00Z
review text: A random walk in random environment on \(\mathbb Z^d\) is studied in the case where the drifts are either neutral or `point to the right'. The model is defined as follows. Let \(\omega=(\omega_x)_{x\in\mathbb Z^d}\) denote an i.i.d.\ sequence of random variables \(\omega_x=(\omega_x(e)) _{e\sim 0}\) (with marginal distribution \(\mu\)), taking values in the set of probability measures on the set of all the \(2d\) neighbors of \(x\in\mathbb Z^d\). Given \(\omega\), we define a random walk \((X_n)_{n\in\mathbb N_0}\) on \(\mathbb Z^d\), which steps from \(x\in\mathbb Z^d\) to a neighbor \(x+e\) of \(x\) with probability \(\omega_x(e)\). The walk starts from \(X_0=y\) under \(P_{\omega,y}\). Hence, \(\omega\) serves as a random environment for the random walk \((X_n)_{n\in\mathbb N_0}\), and the \(\omega_x\)'s are the local transition probabilities. The consideration of the walk under \(P_{\omega,y}\) is the so-called `quenched' setting, and under \(P_y=\mu^{\otimes \mathbb Z^d}(d\omega) P_{\omega,y}\) one has the so-called `annealed' setting. This model has been studied a lot since the mid-seventies in one dimension, but is much more difficult to handle in dimensions \(d\geq 2\). In fact, there are many interesting questions yet open (general recurrence and transience criteria are lacking, for instance), and rigorous results are available only since a few years (for example a large-deviation principle by M. Zerner). One of the main reasons for the difficulty is the lack of reversibility of the walk, which amounts to a lack of self-adjointness for related spectral properties. The author studies the case where the environment has either neutral sites or drifts pointing to the right, more precisely: For some \(\delta>0\), the probabilities \(\mu(\omega_0(e)=1/2d\), \(\forall e\sim 0)\) and \(\mu(\sum_{e\sim 0}\omega_0(e)e\cdot {\text e}_1\geq \delta)\) are both positive and sum up to one, where \({\text e}_1\) denotes the first canonical basis vector. The author shows th…
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 1389146
zbMATH Keywords: random walk in random environment in general dimension
zbMATH Keywords: drift to the right
zbMATH Keywords: quenched and annelaed probabilities
author: Alain-Sol Sznitman
reviewed by: W. König
MaRDI profile type: Publication
full work available at URL: s004400050239
OpenAlex ID: W1993346772
DOI: 10.1007/S004400050239
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1389146