Maximal Entropy Random Walks in Z: Random and non-random environments

Abstract: The Maximal Entropy Random Walk (MERW) is a natural process on a finite graph, introduced a few years ago with motivations from theoretical physics. The construction of this process relies on Perron-Frobenius theory for adjacency matrices.Generalizing to infinite graphs is rather delicate, and in this article, we treat in a fairly exhaustive manner the case of the MERW on Z with loops, for both random and nonrandom loops. Thanks to an explicit combinatorial representation of the corresponding Perron-Frobenius eigenvectors, we are able to precisely determine the asymptotic behavior of these walks. We show, in particular, that essentially all MERWs on Z with loops have positive speed.