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We prove the continuity of the extremal decomposition measure of the freestate of low temperature Potts models, and more generally of ferromagneticfinite-spin models, on a regular tree, including general clock models. Thedecomposition is supported on uncountably many inhomogeneous extremal states,that we call glassy states. The method of proof provides explicit concentrationbounds on branch overlaps, which play the role of an order parameter fortypical extremals. The result extends to the counterpart of the free state(called central state) in a wide range of models which have no symmetry,allowing also the presence of sufficiently small field terms. Our work shows inparticular that the decomposition of central states into uncountably manyglassy states in finite-spin models on trees at low temperature is a genericphenomenon, and does not rely on symmetries of the Hamiltonian.

Continuity of the extremal decomposition of the free state for finite-spin models on Cayley trees