Infinite Volume Limit and Spontaneous Replica Symmetry Breaking in Mean Field Spin Glass Models

Abstract

Mean field spin glasses [1] were introduced around 30 years ago as an approximation of realistic models for disordered magnetic alloys. In what follows, we make particular reference to the well known Sherrington-Kirkpatrick (SK) model [2], but most of the results we present can be proven in greater generality. The Hamiltonian of the SK model in a magnetic fieldhfor a given configuration of theNIsing spinsσ _i = ±1 i = 1 … Nis defined as

$$ {H_N}(\sigma ,h;J) = - \frac{1}{{\sqrt N }}\sum\limits_{1 < i < j \leqslant N} {{J_{ij}}{\sigma _i}{o_j} - h\sum\limits_{i = 1}^N {{\sigma _i},} } $$

(1)

where the couplings J_ijare quenched i.i.d. Gaussian random variables with zero mean. The model can be generalized in many ways, for instance allowing for more general disorder distributions, or introducing interactions among p-ples of spins, withp >2 (p-spin model [3]). The main object of interest is the infinite volume quenched free energy density, defined as

$$ {H_N}(\sigma ,h;J) = - \frac{1}{{\sqrt N }}\mathop \sum \limits_{1 \leqslant i < j \leqslant N} {J_{ij}}{\sigma _i}{\sigma _j} - h\sum\limits_{i = 1}^N {{\sigma _i}} , $$

(1)

HereZ _N is the disorder dependent partition function, at inverse temperature ß, andEdenotes average with respect to the quenched disorderJ.Due to their mean field character, these models are exactly solvable, at least in the framework of Parisi theory of replica symmetry breaking [1]. This predicts a solution for the quenched free energy, which we denote byF ^Parisi (ß, h)expressed as thesupremumof a suitable trial functional over the space of functional order parameters [1]. From the rigorous point of view, the situation is much more delicate, and even the problem of proving that the limit in (2) exists remained open for a very long time.