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Chaotic time dependence in a disordered spin system

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Stochastic Ising and voter models on d are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t→∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) – which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature – if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d≥ 2. These results are based on an analysis of the “localization” properties of Random Walks with Random Rates.

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Received: 10 August 1998

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Fontes, L., Isopi, M. & Newman, C. Chaotic time dependence in a disordered spin system. Probab Theory Relat Fields 115, 417–443 (1999).

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  • Mathematics Subject Classification (1991): 60K35, 82B44