Convergence in energy-lowering (disordered) stochastic spin systems

We consider stochastic processes, S-t = (S-x(t): x is an element of Z(d)) is an element of L-0(Zd) with L-0 finite, in which spin flips (i.e., changes of S-x(t)) 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.