On an explicit construction of Parisi landscapes in finite dimensional Euclidean space,  with some applications

 

Abstract:

We construct an 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 the
free energy of the system in the thermodynamic limit coincides
with the most general version of the Derrida's Generalized Random
Energy Model. The low-temperature behaviour depends
essentially on the spectrum of length scales involved in the
construction of the landscape. We argue that our construction is
in fact valid in any finite spatial dimensions, which therefore provides

an explicit realization of a Parisi landscape. We show how this

can be used to construct generalized multifractal time series, where

the whole multifractal spectrum depends on the "epoch" (ie the log of

the time lag).