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Rigorous solution of the Gardner problem. (English) Zbl 1034.82042
Summary: We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between \(N\) spins which solve the problem of storing a given set of \(p\) random patterns. The Gardner formula for this volume in the limit \(N\), \(p\to\infty\), \(p/N\to\alpha\) is proven for all values of \(\alpha\). Besides, we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics.

MSC:
82C32 Neural nets applied to problems in time-dependent statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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