Subexponential tail asymptotics for a random walk with randomly placed one-way nodes
Large deviations for a special case of one-dimensional random walk in random environment are considered. The model is defined as follows. Given an i.i.d.\ sequence \((\omega_x)_{x\in\mathbb Z}\) with values in \([0,1]^{\mathbb Z}\) (the random environment), the random walk \((X_n)_{n\in\mathbb N_0}\) on \(\mathbb Z\) jumps, when present at \(x\in\mathbb Z\), to \(x+1\) with probability \(\omega_x\) and to \(x-1\) with probability \(1-\omega_x\). The paper considers the special case where \(\omega_x\) takes values \(1\) or \(p\) only, where \(p\in(0,1/2]\) is a parameter. Hence, there is a delicate competition in the environment between one-way nodes (where the walker is always pushed to the right) and sites pointing the walker to the left on an average. In this case, Alili has shown that an almost sure law of large numbers holds, i.e., \(\lim_{n\to\infty}X_n/n=v_\alpha\) exists with probability one, for almost all realisations of the environment. The first result of the present paper identifies the speed \(v_\alpha\) in terms of expectations involving the smallest index \(x\in\mathbb N\) such that \(\omega_x=1\), i.e., in terms of the length of the node-free zone right to the origin. Henceforth, it is assumed that the drift \(v_\alpha\) is positive. The main results concern the decay of the probabilities \(P(X_n/n\leq v)\) and \(P_\omega(X_n/n\leq v)\) for \(v\in(0,v_\alpha)\), where \(P\) averages over the environment and the walk, and \(P_\omega\) over the random walk only, with the environment fixed. These probabilities turn out to decay subexponentially fast, and the rate is related to the almost sure behavior of the length of the longest node-free interval in the environment between the origin and the site \(n\), as \(n\to\infty\). In particular, there are cases in which the rate is non-deterministic. The proofs of the results are adaptations and extensions of material from earlier joint work of the author and \textit{O. Zeitouni} [in: Random walks. Bolyai Soc. Math. Stud. 9, 127-165 (1999; Zbl 0953.60009)].
scientific article; zbMATH DE number 1751670