Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

Abstract

We consider stochastic processes, $S^t \equiv (S_x^t :x \in \mathbb{Z}^d ) \in \mathcal{S}_0^{\mathbb{Z}^d }$ with $\mathcal{S}_0$ finite, in which spin flips (i.e., changes of S ^t _x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.