Metastates in mean-field models with random external fields generated by Markov chains

TL;DR

This paper generalizes the concept of metastates in mean-field models to cases where the external disorder is generated by an ergodic Markov chain, revealing new phenomena in non-reversible cases.

Contribution

It extends metastate construction to Markov chain disorder, showing analogous results to i.i.d. cases and identifying new asymmetry phenomena in degenerate non-reversible chains.

Findings

Metastates can be constructed with Markov chain disorder similar to i.i.d. cases.

In non-reversible chains, metastates may exhibit less symmetry.

Degenerate non-reversible chains can lead to asymmetries not predicted by Gaussian approximations.

Abstract

We extend the construction by Kuelske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are 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 the metastate can have less symmetry than the symmetry of the interaction and a Gaussian approximation of disorder fluctuations would suggest.