Multifractality and Freezing Phenomena in Random Energy Landscapes: an Introduction

The Boltzmann-Gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet nontrivial example of disorder-induced multifractal measures. We introduce and discuss two analytically tractable models for such potentials. The first model uses multiplicative cascades and is equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by B. Derrida and H. Spohn. Second model is the infinite-dimensional version of the problem in Euclidean space which can be solved by employing the replica trick. In particular, the latter version allows one to identify the freezing of multifractality exponents with a mechanism of the replica symmetry breaking (RSB) and to elucidate its physical meaning. The corresponding 1-step RSB solution turns out to be {\it marginally stable} everywhere in the low-temperature phase. In the end we put the model in a more general context by relating to the Gaussian Free Field and briefly discussing recent developments and extensions. The appendices provide a detailed exposition of the replica analysis of the model discussed in the lectures.

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