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**A law of large numbers for random walks in random environment.**
*(English)*
Zbl 0965.60100

Let \((X_n)_{n\in\mathbb N_0}\) be a random walk in random environment on \(\mathbb Z^d\) with arbitrary \(d\in\mathbb N\). More precisely, there is a random environment consisting of an i.i.d. collection \((\omega(x,e))_{e\in\mathbb Z^d,|e|=1}\), \(x\in\mathbb Z^d\), of random \(2d\)-vectors whose components are positive and sum up to one. Given the environment, the walker, given that he/she is at \(X_n=x\) at a time \(n\in\mathbb N_0\), jumps to the neighbor \(x+e\) with probability \(\omega(x,e)\). Conditioned on the environment, the process \((X_n)_n\) is Markov, but this is not true under the ‘annealed\' law, where one also averages over the environment. While the one-dimensional setting is quite well understood, many seemingly simple questions are completely open in the multi-dimensional setting.

In the present paper, it is assumed that the values of the environment are bounded away from zero, which is a kind of ellipticity assumption. Furthermore, Kalikow’s condition is assumed, which roughly requires the following: For a certain given vector \(l\in\mathbb R^d\), for certain auxiliary Markov chains (whose transition probabilities are transformed with the expected number of hits up to leaving a given set \(U\subset\mathbb R^d\) containing zero) the expectation of \(\langle l,(X_1-X_0)\rangle\) (where \(X_1-X_0\) is the first step of the Markov chain) is bounded away from zero, uniformly in the set \(U\) and in the starting point.

Kalikow’s condition appeared first in 1981 and is difficult to check directly, but for a number of cases considered in the literature, it is known to hold. Under these assumptions, the authors derive a strong law of large numbers (with deterministic, non-degenerate velocity) for the endpoint of the walker under the annealed law. The main tool of the proof is a certain renewal structure for the multi-dimensional walk. This property is relatively easy to find and to employ in one dimension and proved very useful there, but it is rather complicated to apply in the present multi-dimensional setting. By Kalikow’s condition, one has a property like ‘transience in direction \(l\)\' which is helpful.

In the present paper, it is assumed that the values of the environment are bounded away from zero, which is a kind of ellipticity assumption. Furthermore, Kalikow’s condition is assumed, which roughly requires the following: For a certain given vector \(l\in\mathbb R^d\), for certain auxiliary Markov chains (whose transition probabilities are transformed with the expected number of hits up to leaving a given set \(U\subset\mathbb R^d\) containing zero) the expectation of \(\langle l,(X_1-X_0)\rangle\) (where \(X_1-X_0\) is the first step of the Markov chain) is bounded away from zero, uniformly in the set \(U\) and in the starting point.

Kalikow’s condition appeared first in 1981 and is difficult to check directly, but for a number of cases considered in the literature, it is known to hold. Under these assumptions, the authors derive a strong law of large numbers (with deterministic, non-degenerate velocity) for the endpoint of the walker under the annealed law. The main tool of the proof is a certain renewal structure for the multi-dimensional walk. This property is relatively easy to find and to employ in one dimension and proved very useful there, but it is rather complicated to apply in the present multi-dimensional setting. By Kalikow’s condition, one has a property like ‘transience in direction \(l\)\' which is helpful.

Reviewer: W.König (Berlin)

### MSC:

60K40 | Other physical applications of random processes |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |

### Keywords:

random walk in random environment; law of large numbers; Kalikow’s condition; renewal structure
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\textit{A.-S. Sznitman} and \textit{M. Zerner}, Ann. Probab. 27, No. 4, 1851--1869 (1999; Zbl 0965.60100)

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