Random energy models: broken replica symmetry and activated dynamics

International audience ; The random energy model (REM) and the generalized random energy model (GREM) are simple spin glass models which play an important role in the theory of spin glasses. The connection with more complex spin glass models can be made using the pspin generalisation of the SK model, which, in the large p limit, gives precisely the REM. The REM and GREM allow us to illustrate and to test in a simple framework several central ideas of the theory both from an equilibrium and from a dynamical point of view. They were also used in several contexts such as the problem of protein folding or errorcorrecting codes. This chapter presents the basic ideas and some recent developments for two aspects: (1) The random energy models and replica symmetry breaking by B. Derrida and P. Mottishaw (2) Aging in the activated dynamics of the REM and the p-spin SK models by V. Gayrard To appear as a contribution (specifically, Chapter 31) to the edited volume ``Spin Glass Theory \& Far Beyond - Replica Symmetry Breaking after 40 Years'', World Scientific.