At the critical line, local weak limits of Potts and random cluster measures on locally tree-like expander graphs are exactly mixtures of the free and wired tree Gibbs measures, and any mixture weight is realizable.
Ferromagnetic Ising measures on large locally tree-like graphs
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Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs
At the critical line, local weak limits of Potts and random cluster measures on locally tree-like expander graphs are exactly mixtures of the free and wired tree Gibbs measures, and any mixture weight is realizable.