On statistical mechanics of a single particle in high-dimensional random landscapes

Yan V. Fyodorov

School of Mathematical Sciences, University of Nottingham, UK

Mon, Dec. 17th 2007, 14:15

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

I am going to discuss recent results of the replica
approach to statistical mechanics of a single classical particle
placed in a random Gaussian landscape. The particular attention will
be paid to the case of landscapes with logarithmically growing
correlations and to its recent multiscale generalisations. Those
landscapes give rise to a rich multifractal spatial structure of the
associated Boltzmann-Gibbs measure. In the limit of large spatial
dimension the free energy of the model is shown to reproduce exactly
the most general version of Derrida's Generalized Random Energy
Model. If time allows, I will briefly mention also the case of a
random landscape constructed locally by adding many squared
Gaussian-distributed terms, as well as related results on counting
stationary points of random Gaussian surfaces.