Central Limit Theorem for Excited Random Walk in the Recurrent Regime

Let Y = Z× (Z/LZ), where L > 1 is an integer, G = {−e1, e1,−e2, e2} where ej are coordinate vectors. We denote the coordinates of points y ∈ Y by (x(y), s(y)). Consider a cookie environment on Y, that is, for each y ∈ Y, j ∈ N, there is a probability distribution ω(y, j, e) on G. Consider an excited random walk Yn = (Xn, Sn) that is P(Yn+1 − Yn = e|Y1, . . . , Yn) = ω(Yn, ln, e) where ln is the number of visits to Yn by time n. (We denote by P and E the quenched probability and expectation with fixed ω and by P and E the annealed probability and expectation.) Yn is called (multi-)excited random walk (ERW). We make the following assumptions: (A) δ(y, j) := ω(y, j, e1)− ω(y, j,−e1) ≥ 0, (B) There exists κ > 0 such that ω(y, j, e) ≥ κ, (C) ω is stationary with respect to G-shifts and ergodic. (D) Let δ(y) = ∑∞ j=1 δ(y, j) then