By Anton Bovier, Pierre Picco

Geared toward graduates and power researchers, it is a finished creation to the mathematical points of spin glasses and neural networks. it may be helpful to mathematicians in likelihood thought and theoretical physics, and to engineers operating in theoretical desktop technology.

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**Example text**

We mention that this strict separation into two steps was not followed in [BG5]. However, it appears to be the most natural and reasonable procedure. Gentz [G1] used this strategy in her proof of the central limit theorem, but only in the regime M2 / N 1 o. To get sufficiently good estimates when a > 0, a sharper analysis is required in part (ii). To get explicit results, from now on we will work in a more restricted class of examples that includes the Hopfield model. 1) with P 2:: 2, and we will require only that the variables ~f have mean zero, variance one, and are bounded.

For prior results under stronger assumptions, see [Ge,Si,Gi]. Such results are generally proven by tedious combinatorial methods, combined with truncation techniques. 2) with K a numerical constant and valid for small E. More recently, a bound of the form exp ( was proven by the authors in [BG5], using a concentration estimate due to Talagrand. In [T4] a simplified version of that proof is given. Now we will give the simplest proof of such a result we can think of. -',v A~,v. For positive symmetric matrices it is clear that eigenvalue of A.

Also throughout this section we assume that the ~f are symmetric Bernoulli variables. 1. 2). Without restriction of generality let us consider the vicinity of m*e 1. , IIvl12 :::; E < m*. 6)). In terms of the variable v, this condition implies that both IVll2 :::; eva and Ilvll~~=~ ::; eva for some constant e (depending on p), where we have set v = (O,V2,V3,'" ,VM). 1 ) 11 But note that we consider only the case M aNP-l. rv aN rather than M rv A. Bovier and V. Gayrard 56 where we have set tf == a~f.