The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately) $$ g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u). $$ The same pair correlation function plays a (conjectured) role in the theory of the Riemann zeta function. See for example https://en.wikipedia.org/wiki/Pair_correlation_conjecture
Question: Is there any simple way of constructing an increasing sequence of real numbers whose pair correlations distribution is as given above? I mean as a kind of (more or less simple) stochastic process. The result should be an infinite sequence.
I know that this is related to the theory of determinantal point processes, but do not really understand how to do it. Maybe someone can help.