In every dimension d≥2 there exists a unique β_*^{(d)}>0 such that the uniform density on the sphere is the unique global minimizer of the USA free energy up to the linear-stability threshold K_# for β≤β_*, yielding a continuous transition, while for β>β_* the uniform density is not globally minimiz
Phase transitions and linear stability for the mean-field Kuramoto - Daido model, February 2026
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For 1/(n+1)-periodic interactions with Fourier decay, the phase transition in mean-field free energies on the circle is continuous at the linear stability threshold of the uniform state.
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In every dimension d≥2 there exists a unique β_*^{(d)}>0 such that the uniform density on the sphere is the unique global minimizer of the USA free energy up to the linear-stability threshold K_# for β≤β_*, yielding a continuous transition, while for β>β_* the uniform density is not globally minimiz
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For 1/(n+1)-periodic interactions with Fourier decay, the phase transition in mean-field free energies on the circle is continuous at the linear stability threshold of the uniform state.