Quenched sub-exponential tail estimates for one-dimensional random walk in random environment
Let \(\omega=(\omega_x)_{x\in\mathbb Z}\) be an i.i.d. sequence of \([0,1]\)-valued random variables, serving as a random environment. The one-dimensional random walk in random environment, \((X_n)_{n\in\mathbb N_0}\), is defined as the Markov chain on \(\mathbb Z\) that makes a step to the right neighbor of its present location, \(x\), with probability \(\omega_x\) and to the left neighbor with probability \(1-\omega_x\). The conditional behavior of \((X_n)_{n}\) given the environment \(\omega\) is referred to as the quenched setting. Solomon (1975) proved a quenched law of large numbers and determined the value of the speed, \(v_\alpha\in[-1,1]\). Greven and den Hollander (1994) derived a quenched large-deviation principle for \(X_n/n\) with a deterministic rate function. In the case where the expectation of \((1-\omega_0)/\omega_0\) is smaller than one but its essential supremum, \(\rho_{\text{max}}\), is not smaller than one, the speed \(v_\alpha\) is positive and the rate function vanishes on the interval \([0,v_\alpha]\). Note that the condition \(\rho_{\text{max}}>1\) means that the random environment has positive drifts with positive density. The article under review derives refined estimates for the quenched probability of the event \(\{X_n<nv\}\) for \(v\in(0,v_\alpha)\), as \(n\to\infty\) in the case mentioned above. In the subcase where \(\rho_{\text{max}}>1\), the logarithm of this probability decays roughly like \(-n^{1-1/s}\) where \(s>1\) is defined by making the expectation of \([(1-\omega_0)/\omega_0]^s\) equal to one. In the subcase where \(\rho_{\text{max}}=1\), this logarithm decays like \(-n(\log n)^{-2}\) with upper and lower bounds that both vanish if \(v-v_\alpha\) vanishes. An important ingredient of the proof is the asymptotics of the annealed probability (i.e., taken over the walk and the environment) of this event, which was derived earlier by \textit{A. Dembo, Y. Peres} and \textit{O. Zeitouni} [ibid. 181, No.~3, 667-683 (1996; Zbl 0868.60058)].
scientific article; zbMATH DE number 1214087