Recurrence of random walks in the Ising spins

Consider the 1/2-Ising model inZ ². Let σ _j be the spin at the site (j, 0)∈Z ² (j=0, ±1, ±2, ...). Let ({ X_n } _{n = 0}^{ + infty } ) be a random walk with the random transition probabilities such that $$P(X_{n + 1} = j pm 1|X_n = j) = p_j^ pm equiv 1/2 pm v(sigma _j - mu )/2$$ We show a case whereE[p _j ⁺ ≧E[p _j ^? ], but (mathop {lim }limits_{n to infty } X_n = - infty ) is recurrent a.s.