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