Chapter 10 - Transient random walks on dynamically oriented lattices

Abstract

This chapter emphasizes on transient random walks on dynamically oriented lattices. Random walks as a tool in mathematical physics have been widely used in classical statistical mechanics to study critical phenomena. Analogous methods in quantum statistical mechanics require the study of random walks on oriented lattices. Random walks in random and non-random environments have been intensively studied. The horizontal lines are unidirectional towards a random or deterministic direction. Depending on the orientation, the walk could be either recurrent or transient. The transience naturally arises when the orientations are all identical in infinite regions. The chapter focuses on spatially inhomogeneous or dependent distributions of the orientations. It introduces the lattices for which the distribution of the horizontal orientation is generated by a dynamical system, namely orientations and increments. For ergodic dynamical systems, a strong law of large numbers is also proved.