Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

Details Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

2002-03-22

arxiv.org/abs/math/0203234v1

Mathematics

Probability

11 pages

Scientific paper

We consider stochastic processes, S^t \equiv (S_x^t : x \in Z^d), with each S_x^t taking values in some fixed finite set, 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.

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