Statistical Field Theory for Neural Networks by Moritz Helias & David Dahmen

Author:Moritz Helias & David Dahmen
Language: eng
Format: epub
ISBN: 9783030464448
Publisher: Springer International Publishing

where the expectation value 〈〉x(J) is over realizations of x for one given realization of J. It is convenient to express the observable in its Fourier transform (with suitably defined and measure ) using Eq. (2.​4)

where naturally the moment-generating functional appears as . The mean observable averaged over all realizations of J can therefore be expressed as

in terms of the generating functional that is averaged over the frozen disorder, as anticipated above.

We call a quantity self-averaging, if its variability with respect to the realization of J is small compared to a given bound 𝜖. Here 𝜖 may, for example, be determined by the measurement accuracy of an experiment. With the short hand δO[x] := O[x] −〈〈O[x]〉x(J)〉J we would like to have

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