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Abstract
This is an advanced course in statistical mechanics with the purpose of introducing the main ideas and tools to describe the low temperature phase of disordered systems. One of its goals is to explain the deep physical meaning of the replica-symmetry-breaking formalism and how it allows us to appropriately describe systems with a low temperature phase characterized by a very rugged free-energy landscape (multiple equilibria). Assuming a basic knowledge of phase transitions in magnetic systems, e.g., Curie-Weiss model, we will present a detailed study (calculation of the free energy) of two prototypical models of disordered systems: the Hopfield model (neural network) and the Sherrington-Kirkpatrick model (disordered mean-field Ising model), the latter exhibiting a low temperature phase with the characteristic fractal free-energy landscape described by the famous full replica-symmetry breaking ansatz from Parisi.