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?
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.