Recurrence of random walks in the Ising spins
Consider the 1/2-Ising model in \(Z^ 2\). Let \(\sigma_ j\) be the spin at the site \((j,0)\in Z^ 2\), \((j=0,\pm 1,\pm 2,...)\). Let \(\{X_ n\}^{+\infty}_{n=0}\) be a random walk with the random transition probabilities such that \[ P(X_{n+1}=j\pm 1| X_ n=j)=p_ j^{\pm}\equiv 1/2\pm \nu (\sigma_ j-\mu)/2. \] We show a case where \(E[p^+_ j]>E[p^-_ j]\), but \(\lim_{n\to \infty}X_ n=-\infty\) a.s. or \(X_ n\) is recurrent a.s.
random walk with the random transition probabilities
scientific article; zbMATH DE number 3982238