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Slowdown estimates and central limit theorem for random walks in random environment. (English) Zbl 0976.60097

The author investigates random walks on the \(d\)-dimensional lattice in a random environment. The environment is given by an i.i.d. collection of \(2d\)-dimensional vectors which specifiy the jump probabilities at each site. He proves a central limit theorem under a condition which goes back to Kalikow and implies transience of the random walk. In a previous joint work of the author with M. P. W. Zerner [Ann. Probab. 27, No. 4, 1851-1869 (1999; Zbl 0965.60100)], it was shown that Kalikow’s condition implies in fact that the random walk has a deterministic, non-zero drift. Then, large deviation probabilities are considered. While the speedup probabilities decay exponentially, the slowdown probabilities have a subexponential decay. This is connected to the occurrence of \`\` traps\'\' in the environment. These are related to \`\` pockets of atypically low eigenvalues\'\' for Brownian motion among Poissonian obstacles. For random walks which are neutral or biased to the right, the author determines the exact rate of decay of the slowdown probabilities, thereby improving his earlier results. An important object in the proofs is the renewal time.

MSC:

60K40 Other physical applications of random processes
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82B43 Percolation

Citations:

Zbl 0965.60100
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