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.
Brownian particles with electrostatic repulsion on the circle: Dyson 's model for unitary random matrices revisited
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Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models
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.