Excited random walk with periodic cookies

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 (p(1)...,pm) is an element of [0, 1](M) for some positive integer M, where the walker upon the ith visit to z is an element of Z moves to z + 1 with probability pi (mod M), and moves to z - 1 with probability 1 - p(i(mod M)). We give an explicit formula in terms of the parameters (p(1),..., p(m)) which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that 1/M Sigma(M)(i=1) p(i) = 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.