  Boltzmann-Gibbs measures generated by 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.
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.
