Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
arXiv:0711.4006 · doi:10.1088/1751-8113/41/32/324009
Abstract
We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N>>1 the free energy of the system and overlap function are calculated exactly using the replica trick and Parisi's hierarchical ansatz. In the thermodynamic limit, we recover the most general version of the Derrida's Generalized Random Energy Model (GREM). The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. If the latter consists of K discrete values, the system is characterized by a K-step Replica Symmetry Breaking solution. We argue that our construction is in fact valid in any finite spatial dimensions . We discuss implications of our results for the singularity spectrum describing multifractality of the associated Boltzmann-Gibbs measure. Finally we discuss several generalisations and open problems, the dynamics in such a landscape and the construction of a Generalized Multifractal Random Walk.
25 pages, published version with a few misprints corrected
References in corpus (6)
- Anderson Transitions
- Amorphous-amorphous transition and the two-step replica symmetry breaking phase
- Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
- On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces
- Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition
- Mean field theory of spin glasses: statics and dynamics