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I have read several times by now that the couplings in the SYK model are drawn randomly from a gaussian distribution. I was wondering what exactly is meant by that. To elaborate, when I compute an observable in the SYK model I start in some way from the Lagrangian. This already includes the couplings, so how do I know they have "previously" been determined randomly? What is the difference to just fixing the couplings all to be let's say 0.4, because I like that particular number and they just randomly turned out to all be 0.4?

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The SYK model is always considered in the large-N limit (N = number of Majorana modes), and the probability that the distribution of couplings in a particular realization differs appreciably from the probability distribution you used to specify the model goes to zero as N -> infinity. This is the Law of Large Numbers at work.

Your question does make sense for other disordered models in which we don't take the large-N limit. Then we indeed have to distinguish between the disorder-averaged properties of the model, averaged over all disorder realizations with a given probability distribution, and the properties of a particular disorder realization which might be atypical.

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  • $\begingroup$ So should I average the hypothetical observable I calculated over all realizations of SYK models with the appropriate weight for the given realization of the couplings? $\endgroup$ – QuantumStudent Dec 10 '18 at 14:32
  • $\begingroup$ @QuantumStudent Like I said, in the large N limit it won't matter. "Almost all" (i.e. all except a set of measure zero) realizations will give the same expectation value. $\endgroup$ – Dominic Else Dec 10 '18 at 14:34

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