Random walks with jumps in random environments (examples of cycle and weight representations) (Q2769671)

Let \(E\) be a denumerable set of states. A cycle is a -- up to cyclic permutations -- totally ordered finite subset of \(E\). For a cycle \(c\) and states \(i,j\in E\), let \(J_c(i)\) and \(J_c(i,j)\) denote the indicator functions on the events that \(i\) respectively the pair \((i,j)\) appears in the cycle \(c\). Given a set \({C}\) of cycles and a map \({C}\ni c\mapsto w_c\in(0,\infty)\) of cycle weights, one can define a stochastic matrix \(P\) by putting \(P_{ij}= \sum_{c\in{{C}}}w_c J_c(i,j)/\sum_{c\in{{C}}}J_c(i)\). We say that \(P\) is given in terms of a cycle and weight representation. It is known that recurrent and reversible Markov chains admit a cycle and weight representation. NEWLINENEWLINENEWLINEThe present paper considers two special Markov chains on \(\mathbb N\) for which also a cycle and weight representation exists. In these examples, the walker makes jumps of size \(+2\) and \(-1\) (respectively \(-2\) and \(+1\)) with probabilities that depend on the present position only (not on the time). The cycle and weight representation for the first of these two Markov chains turns out to be unique, while for the second chain the author only provides a sufficient criterion for uniqueness. Furthermore, a criterion for transience, null recurrence and positive recurrence is given in terms of the convergence of certain series, which are recursively defined in terms of the transition probabilities. NEWLINENEWLINENEWLINEAs a second main result, a random walk in random environment on \(\mathbb Z\) is considered, where the walker makes steps \(+2\) or \(-1\), and the sequence of transition probabilities is assumed to be ergodic. It is shown that, for almost all realizations of the environment, this walk admits a unique cycle and weight representation. Furthermore, necessary and sufficient criteria are given for the almost sure recurrence, and for divergence towards \(\infty\) or \(-\infty\). The criteria are in terms of the expectation of the logarithm of certain infinite continued fractions defined in terms of the transition probabilities.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].