Abstract
We extend the construction by Külske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain.
As a new phenomenon we also show in a Potts example that for a degenerate non-reversible chain this CLT approximation is not enough, and that the metastate can have less symmetry than the symmetry of the interaction and a Gaussian approximation of disorder fluctuations would suggest.
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Formentin, M., Külske, C. & Reichenbachs, A. Metastates in Mean-Field Models with Random External Fields Generated by Markov Chains. J Stat Phys 146, 314–329 (2012). https://doi.org/10.1007/s10955-011-0391-8
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DOI: https://doi.org/10.1007/s10955-011-0391-8