Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finitedimensional Euclidean spaces
Abstract
We construct an Ndimensional 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 → ∞ 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 lowtemperature 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 Kstep replica symmetry breaking solution. We argue that our construction is in fact valid in any finite spatial dimensions N >= 1. We discuss the implications of our results for the singularity spectrum describing multifractality of the associated BoltzmannGibbs measure. Finally we discuss several generalizations and open problems, such as the dynamics in such a landscape and the construction of a generalized multifractal random walk.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2008
 DOI:
 10.1088/17518113/41/32/324009
 arXiv:
 arXiv:0711.4006
 Bibcode:
 2008JPhA...41F4009F
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics
 EPrint:
 25 pages, published version with a few misprints corrected