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Reviewing some of the literature on random matrices I have seen several studies and results on characteristic polynomials of random matrices, usually of fixed size/degree $N$. Zeros then are either on the circle for e.g. unitary or orthogonal matrices (CUE, COE) or on $\mathbb R\subseteq\mathbb C$ for hermitian matrices (GUE, GOE,...). These random polynomials were used by Keating and Snaith to make influential conjectures on moments of $\zeta$ at the last turn of millenium.

Now I have seen that recently probabilists have defined and studied random analytic functions which are the infinite $N$ limit of the above characteristic polynomials. They use a single random function, $\xi_\infty(s)$, with slight modifications for different gaussian or circular ensembles.

I have a question about this construction: How is it supposed to be related to the Riemann $\zeta$? In particular $\zeta$ has additional trivial zeros at $2\mathbb Z^-$ which are not present in $e^{s(C-\pi i)}\xi_\infty(s)$ the limit random characteristic function for GUE, with zeros in a real interval. This function includes a scaling by $\sqrt N$ to keep eigenvalues of the same size as $N\rightarrow\infty$. To refine my question I could say: What hypothetically happens when passing from the "purely" random function $\xi_\infty$ to the "phenomenologically" random function $\zeta$ that creates trivial zeros? Is it possible to create L-functions by some derandomization process applied to random operators?

I could imagine that Wigner's own path from deterministic atoms to random matrices yield relevant hints. A model for the present situation would be: take a chaotic (here ergodic) process and parametrize an operator with it, appropriately, like for random Schrödinger operators modeling electrons in disordered lattices. The appearance of zeros in $\zeta$ also reminds me some phenomena for partition functions of statistical mechanical systems, like appearance of phase transitions in infinite systems which are not possible in finite systems. Also the Yang-Lee "formalism" for extracting knowledge of phase transitions from the location of zeros of partition functions.

Any comment, fleeting thought... is welcome.

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    $\begingroup$ You just add the extra zeroes, coming from the missing local factor at $\infty$. I don't see why there should be anything else to it. $\endgroup$ – Will Sawin Dec 10 '17 at 22:01
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    $\begingroup$ Random matrix theory is not modeling the zeros of $\zeta(s)$; it's modeling the zeros of $\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ $\endgroup$ – Stopple Dec 10 '17 at 22:36
  • $\begingroup$ See also this question: mathoverflow.net/questions/277030/… $\endgroup$ – Stopple Dec 10 '17 at 22:43
  • $\begingroup$ Thanks. I realize that I was assuming inconsciously that the natural definitions are really the most basic ones, the characteristic polynomial $\det(sI-M)$ on one side and really $\zeta(s)$, without the $\Gamma$ factor cancelling trivial zeros. I feel there could be some mysterious way of passing from one side to the other explaining the appearance of extraneous zeros in a natural way, in a construction from random matrices. I find it hard to explain, but if both sides are inextricably linked... Sorry for the vagueness. I'll think about it. $\endgroup$ – plm Dec 10 '17 at 23:39
  • $\begingroup$ Random matrices are completely unrelated to the trivial zeros. Likewise the conjecture that there exists a chaotic system which, when quantized, provides the zeta zeros only applies to the nontrivial ones. $\endgroup$ – Marcel Dec 10 '17 at 23:43

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