Generalization of the cavity method for adiabatic evolution of Gibbs states

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Mean field glassy systems have a complicated energy landscape and an enormous number of different Gibbs states. In this paper, we introduce a generalization of the cavity method in order to describe the adiabatic evolution of these glassy Gibbs states as an external parameter, such as the temperature, is tuned. We give a general derivation of the method and describe in details the solution of the resulting equations for the fully connected p-spin model, the XOR-SAT problem and the anti-ferromagnetic Potts glass (or "coloring" problem). As direct results of the states following method, we present a study of very slow Monte-Carlo annealings, the demonstration of the presence of temperature chaos in these systems, and the identification of a easy/hard transition for simulated annealing in constraint optimization problems. We also discuss the relation between our approach and the Franz-Parisi potential, as well as with the reconstruction problem on trees in computer science. A mapping between the states following method and the physics on the Nishimori line is also presented.