Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees

Christopher Shriver

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arXiv · 2011.00653

Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees

Christopher Shriver

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Abstract

We extend results of R. Holley beyond the integer lattice to a large class of groups which includes free groups. In particular we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-invariant measure converges weakly to the set of Gibbs measures when evolved under Glauber dynamics. These results are proven using a new notion of free energy density relative to a sofic approximation by homomorphisms. Any measure which minimizes free energy density is Gibbs.

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Christopher Shriver. 2020-11-02. Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees. https://arxiv.org/abs/2011.00653

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