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In this paper we discuss the rigorous approach to the statistical mechanics of mean field desordered Ising model. The description of the thermodynamic behavior for the desordered systems is a formidable problem, and the mean field spin glass represents, for its complex behavior, a kind of paradigm for the study of complexity. The most advanced results in this direction have been obtained by means of the non—rigorous methods of theoretical physics, and are associated to the important discovery of spontaneous replica simmetry breaking and the Parisi ansatz. On the othe side, in order to improve the comprehension of complex systems, it is necessary to build suitable rigorous methods. We explain one of these techniques, the marginal martingale approach, and compare the "theoretical physics" results with the rigorous ones. In particular, we give a rigorous proof for the existence of a functional order parameter for the Sherringhton—Kirkpatrick model, and show how, by means of an suitable assumption, the ultrametric structure of the pure state space can be found. The tools developed for this mathematical study can be used to study in a rigorous way the neural network thermodynamics. In the next two paragraphs we compare the mean field spin glass model to the ferromagnetic and the neural network models and discuss the concept of functional order parameter for the desordered systems. Then, in the fourth paragraph the marginal martingale is defined and we give a representation theorem for it. Finally, by means of the cavity ansatz we find rigorously the Parisi expression of the ultrametric topology of the pure phases space. This work has been partially supported by MURST (Italian Ministry of University and Scientific and Technological Research) and INFN (Italian National Institute for Nuclear Physics).
keywords spin glass, marginal martingale, neural network, ultrametric topology
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