This chapter discusses the metastable states in glassy systems. Glassy systems often have a “complex” landscape, which is responsible for the non-trivial behaviour they exhibit. The notion of complex landscape, despite being evocative, remains rather vague and requires a quantitative characterization. Metastable states must have finite lifetime and the definition of state has to include a reference to time. For example, it seems reasonable to consider a metastable region as a state when the equilibration time in that region is much smaller than the nucleation time required driving the system to the stable phase. Spin Glasses are magnetic systems where the mutual interactions between the spins can be either ferromagnetic or anti-ferromagnetic. This is usually modeled assuming quenched random variable to represent the interactions. These models are characterized by the presence of a very large number of metastable states. The standard way to investigate the individual and global features of metastable states is to compute the number density of TAP solutions, or, rather, the corresponding complexity.
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