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We consider Boltzmann-Gibbs measures associated with log-correlated Gaussianfields as potentials and study their multifractal properties which exhibitphase transitions. In particular, the pre-freezing and freezing phenomena ofthe annealed exponent, predicted by Fyodorov using a modifiedreplica-symmetry-breaking ansatz, are generalised to arbitrary dimension andverified using results from Gaussian multiplicative chaos theory.

Pre-freezing transition in Boltzmann-Gibbs measures associated with log-correlated fields