AGING ON PARISIS TREE

We present a detailed study of simple 'tree' models for off equilibriu m dynamics and aging in glassy systems. The simplest tree describes th e landscape of a random energy model, whereas multifurcating trees occ ur in the solution of the Sherrington-Kirkpatrick model. An important ingredient taken from these models is the exponential distribution of deep free-energies, which translate into a power-law distribution of t he residence time within metastable 'valleys'. These power law distrib utions have infinite mean in the spin-glass phase and this leads to th e aging phenomenon. To each level of the tree is associated an overlap and the exponent of the time distribution. We solve these models for a finite (but arbitrary) number of levels and show that a two-level tr ee accounts very well for many experimental observations (thermoremane nt magnetization, a.c. susceptibility, second noise spectrum....). We introduce the idea that the deepest levels of the tree correspond to e quilibrium dynamics whereas the upper levels correspond to aging. Temp erature cycling experiments suggest that the borderline between the tw o is temperature dependent. The spin-glass transition corresponds to t he temperature at which the uppermost level is put out of equilibrium but is subsequently followed by a sequence of (dynamical) phase transi tions corresponding to non equilibrium dynamics within deeper and deep er levels. We tentatively try to relate this 'tree' picture to the rea l space 'droplet' model, and speculate on how the final description of spin-glasses might look like.