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Quenched limits for transient, zero speed one-dimensional random walk in random environment. (English) Zbl 1179.60070

The authors study the nearest-neighbor one dimensional random walk in a random i.i.d. environment. The regime is such that the walk is transient but with zero speed: \(X_n\) is of order \(n^s\) with \(s<1\). In the paper the quenched limiting distributions are studied. One could expect that the quenched limiting distributions are of the same type as the annealed limiting distributions since annealed probabilities are averages of quenched probabilities: this turn out not to be the case. The authors show that under the quenched law no limit laws are possible. There exists a localized regime, that is some sequences \((n_k)\) and \((x_k)\) dependening on the environment only such that \(X_{n_k}-x_k=o(\log{n_k})^2\). Also there exists a spread out regime, that is some sequences \((t_m)\) and \((s_m)\) depending on the environment only, such that \(\log{s_m}/\log{t_m}\to s<1\) and \(P_{\omega}(X_{t_m}/s_m\leq x)\to 1/2\) for all \(x>0\) and \(\to 0\) for \(x\leq 0\).

MSC:

60K37 Processes in random environments
60F05 Central limit and other weak theorems
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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