On an explicit construction of 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 $N\to\infty$ the free energy of the system in the thermodynamic limit coincides with the most general version of 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 $N\ge1$.