We give a series of combinatorial results that can be obtained from any two collections (both indexed by Z×N) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions. © 2011 Elsevier Inc.
Holmes, M., & Salisbury, T. S. (2012). A combinatorial result with applications to self-interacting random walks. Journal of Combinatorial Theory. Series A, 119(2), 460–475. https://doi.org/10.1016/j.jcta.2011.10.004