Pre-freezing of multifractal exponents in random energy models with a logarithmically correlated potential

Boltzmann-Gibbs measures generated from logarithmically correlated random potentials are multifractal. We investigate the abrupt change ('pre-freezing') of multifractality exponents extracted from the averaged moments of the measure—the so-called inverse participation ratios. The pre-freezing can be identified with termination of the disorder-averaged multifractality spectrum. The naive replica limit employed to study a one-dimensional variant of the model is shown to break down at the pre-freezing point. Further insights are possible when employing zero-dimensional and infinite-dimensional versions of the problem. In particular, the latter version allows one to identify the pattern of the replica symmetry breaking responsible for the pre-freezing phenomenon.