Recurrence of random walks in the Ising spins

Abstract

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.