Partial annealing and overfrustration in disordered systems

We study disordered systems with the replica method keeping the number of replicas finite and negative. This is shown to bias the distribution of samples towards overfrustrated ones. General results on the thermodynamics of such a system is presented. The physical situation described by this finite-n approach is one where the usually quenched variables evolve on long timescales, their evolution being driven by the quasi-equilibrium correlations of the thermalized variables. In the case of neural networks this amounts to a coupled dynamics of neurons (on fast timescales) and synapses (on longer timescales). The storage capacity of the Hopfield model is shown to be substantially increased by these coupled dynamics.