On new examples of ballistic random walks in random environment. (English) Zbl 1017.60104

The author considers the following model, the random walk in random environment, RWRE. For any \(k\in\mathbb Z^d\), there is a random vector \(p_k=(p_k(e))_{|e|=1}\) of probabilities summing up to one. The family \(p=(p_k)_{k\in\mathbb Z^d}\) is the random environment. It is assumed to be i.i.d. Given this family, a random walker \(X=(X_n)_{n\in\mathbb N}\) on \(\mathbb Z^d\) jumps, when present at site \(k\), to the neighbor \(k+e\) with probability \(p_k(e)\). Hence, conditioned on the (highly disordered) environment \(p\) (this setting is called the quenched setting), \(X\) is a spatially inhomogeneous Markov chain. The Markov property is lost under the so-called annealed law, which averages over the walk and the environment. The one-dimensional case has been studied intensively since the mid-seventies and is basically well understood. However, the higher-dimensional case (on which a huge research activity has been started at the end of the nineties) is still largely open and is a source of many heuristical errors.
The main purpose of the present paper are counterexamples to the wrong intuition that, if the distribution of the environment is close to the Dirac measure on the uniform distribution (i.e., if the RWRE is close to the simple random walk), the asymptotic behavior of the RWRE is similar to the one of simple random walk. Indeed, roughly speaking, if, for sufficently small \(\varepsilon>0\), the environment is such that, almost surely, all probabilities \(p_k(e)\) lie within the \(\varepsilon\)-neigborhood of \(\frac 1{2d}\), but the expected local drift in the direction of the first coordinate is not smaller than \(\varepsilon^{5/2}\) in \(d=3\) and not smaller than \(\varepsilon^{3}\) in \(d\geq 4\), then a certain condition is satisfied that was called \((\text{T}')\) in an earlier paper of the author. As a consequence, as was shown in the earlier paper, the RWRE satisfies a law of large numbers with some non-degenerate speed (and also a central limit theorem and certain large-deviation estimates) and hence exhibits a behavior that is drastically different from the behavior of the simple random walk. This result in particular provides examples of ballistic random walks in random environments which do not satisfy Kalikov’s condition. An important tool in the work is a characterization of \((\text{T}')\) in terms of a direct inspection of the environment in a finite, large box. This criterion is called the effective criterion and was explored in earlier work of the author.


60K37 Processes in random environments
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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