Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential

TL;DR

This paper studies the abrupt change in multifractal exponents, called pre-freezing, in logarithmically correlated random energy models, revealing the role of replica symmetry breaking in this phenomenon.

Contribution

It introduces the concept of pre-freezing of multifractal exponents and analyzes its origin using different dimensional variants of the model, highlighting the breakdown of naive replica limits.

Findings

Pre-freezing corresponds to termination of the multifractality spectrum.

Naive replica limit fails at the pre-freezing point.

Infinite-dimensional models reveal the pattern of replica symmetry breaking.

Abstract

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.