In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a discrete time process on Z defined by parameters (p1,..., pM) ∈ [0, 1]M for some positive integer M, where the walker upon the ith visit to z ∈ Z moves to z + 1 with probability pi (mod M), and moves to z - 1 with probability 1 - pi (mod M). We give an explicit formula in terms of the parameters (p1,..., pM) which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that 1/M Σmi=1 pi = 1/2 all behaviors are possible, and may depend on the order of the pi. Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.
Kozma, G., Orenshtein, T., Shinkar, I. (2016). Excited random walk with periodic cookies. ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 52(3), 1023-1049 [10.1214/15-AIHP669].
Excited random walk with periodic cookies
Orenshtein T.
;
2016
Abstract
In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a discrete time process on Z defined by parameters (p1,..., pM) ∈ [0, 1]M for some positive integer M, where the walker upon the ith visit to z ∈ Z moves to z + 1 with probability pi (mod M), and moves to z - 1 with probability 1 - pi (mod M). We give an explicit formula in terms of the parameters (p1,..., pM) which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that 1/M Σmi=1 pi = 1/2 all behaviors are possible, and may depend on the order of the pi. Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.
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