On the explicit construction of Parisi landscapes in finite dimensional Euclidean spaces

AnN-dimensional Gaussian landscape with multiscale translation-invariant logarithmic correlations has been constructed, and the statistical mechanics of a single particle in this environment has been investigated. In the limit of a high dimensional N → ∞, 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 behavior depends essentially on the spectrum of length scales involved in the construction of the landscape. The construction is argued to be valid in any finite spatial dimensions N ≥1. The text was submitted by the authors in English.