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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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