2 research outputs found
Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
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.Comment: 25 pages, published version with a few misprints correcte