In these notes the main theoretical concepts and techniques in the field of mean-field spin-glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spinglass model is analyzed (the p-spin spherical model) by using three different approaches. Thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition. Dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the Mode Coupling approximation, and the dynamical transition. And finally complexity, covering the mean-field (TAP) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid on the mutual consistency of the results obtained from the different methods.

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We derive the zero-temperature phase diagram of spin glass models with a generic fraction of ferromagnetic interactions on the Bethe lattice. We use the cavity method at the level of one-step replica symmetry breaking (1RSB) and we find three phases: A replica-symmetric (RS) ferromagnetic one, a magnetized spin glass one (the so-called mixed phase), and an unmagnetized spin glass one. We are able to give analytic expressions for the critical point where the RS phase becomes unstable with respect to 1RSB solutions: we also clarify the mechanism inducing such a phase transition. Finally we compare our analytical results with the outcomes of a numerical algorithm especially designed for finding ground states in an efficient way, stressing weak points in the use of such numerical tools for discovering RSB effects. Some of the analytical results are given for generic connectivity.

We study analytically the dynamics of a generalized p-spin model, starting with a thermalized initial condition. The model presents birth and death of states, hence the dynamics (even starting at equilibrium) may go out of equilibrium when the temperature is varied. We give a full description of this constrained out of equilibrium behavior and we clarify the connection to the thermodynamics by computing (sub-dominant) tap states, constrained to the starting equilibrium configuration.PACS numbers: 75.10. Nr, 64.70.Pf, 75.50.Lk Many interesting physical systems live for very long times out of equilibrium, and, in this regime, they display highly non trivial behaviors which are still to be understood (e.g. rejuvenation and memory effects in spin glasses). In general, these systems fall out of equilibrium when some external parameter is changed. For example, fragile glassforming liquids undergo a dramatic slowing down of their relaxational dynamics when the temperature is dropped below the glass transition temperature 1 . This effect is sharpened in certain mean-field models where, at a critical temperature T d , a transition occurs from an equilibrium kind of dynamics to an off-equilibrium aging one 2 . The phenomenon is ubiquitous and can be found also in very different fields: e.g. in local search algorithms for solving hard optimization problems the time-complexity may become extremely large by varying a macroscopic parameter 3 . A better understanding of the mechanisms leading to the dramatic slowing down in out of equilibrium dynamics is a subject of broad interest and wide applicability.In describing the dynamical slowing down (and possible eventual arrest) the common view suggests that at a low temperature a huge number of metastable states appears (with energies higher than the equilibrium one, E 0 ), making relaxation to equilibrium very slow, and even impossible if interactions are long ranged and metastable states lifetimes diverge in the thermodynamic limit. This picture has been verified by solving the out of equilibrium Langevin dynamics of a particularly simple mean-field model, the so-called fully connected spherical p-spin, whose Hamiltonian is 4where the N spins σ i are continuous variables subject to the spherical constraint i σ 2 i = N and the couplings are i.i.d. random variables with zero mean and variance p!/(2N p−1 ). In this model (hereafter p ≥ 3) if we consider a quench, that is if we choose an initial configuration of high energy and let the system relax at a fixed value of the temperature T < T d (T d being the dynamic transition temperature), the asymptotic dynamics remains trapped at the energy level of the highest and most numerous metastable states, the so-called threshold states. Time-translation invariance and the dynamic fluctuation-dissipation relation are violated and aging is observed in correlation and response functions 5 .These features are intriguing and experimentally relevant, since aging behaviour has been observed in many disordered systems. Nevertheless, in order t...