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Precise large deviation estimates for a one-dimensional random walk in a random environment. (English) Zbl 0922.60059
Suppose that the integers are assigned i.i.d random variables \(\omega_x\), taking values in \([1/2, 1[\). The sequence \((\omega_x)\) serves as an environment. This environment defines a random walk \((X_n)\), called random walk in random environment (RWRE), which, when at \(x\), moves one step to the right with probability \(\omega_x\), and one step to the left with probability \(1-\omega_x\). In a more general setup, when \(\omega_x\) can also have values between \(0\) and \(1/2\), Solomon gave a formula for the a.s. speed \(v_\alpha\) of the RWRE, i.e. of the a.s. limit of \(X_n/n\). In the setting of this paper, \(v_\alpha > 0\). For \(0< v < v_\alpha\), let \(a_n : = (1/n^{1/3})\log P[X_n/n \leq v]\), where \(P\) denotes the so-called annealed measure which averages the walk and the environment sequence. In the case where the distribution of \(\omega_x\) has an atom on \(1/2\), the authors determine the limit of \(a_n\). This proves a conjecture of Dembo, Peres and Zeitouni who had shown that \(\liminf a_n\) and \(\limsup a_n\) are strictly between \(-\infty\) and \(0\). The proof is based on a coarse graining scheme, which resembles the techniques used in the study of Brownian motion in a Poissonian potential. More precisely, it involves a coarse graining of the environment into blocks of size \(n^{1/3+\delta}\), some small \(\delta\), and classifying them as “biased” blocks (if the empirical measure of \(\omega_i\)’s in the block has a significant proportion of \(\omega_i>1/2\)) and “fair” blocks (if not). The biased blocks serve as effective barriers, in the sense that the random walk only rarely crosses such a block from right to left. Handling stretches of fair blocks is done by Chebycheff’s inequality, and most of the effort is invested in proving that long stretches of fair blocks which are shorter than the maximal stretch do not contribute much to the tail asymptotics.

MSC:
60G50 Sums of independent random variables; random walks
60F10 Large deviations
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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