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For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (1-U))$ are real-valued random variables. (For the imaginary part, it is necessary to choose a branch of the logarithm. Integrating $\frac{d}{dt} \log (\det (1-t U))$ for $t$ from $0$ to $1$ provides a good choice.)

Keating and Snaith in Random matrix theory and $\zeta(1/2+it)$ exactly calculated the cumulants of the real and imaginary parts of $\log(\det (1-U))$, giving exact formulas and obtaining large $N$ limits. In particular, for the $n$th cumulant of the real part, $n>2$, they obtained a value of

$$(-1)^n \frac{2^{n-1}-1} {2^{n-1} } \zeta(n-1) \Gamma(n) + O (N^{2-n})$$

and for the $n$th cumulant of the real part, $n>2$ even, they obtained a value of $$\frac{ (-1)^{\frac{n}{2}+ 1}}{ 2^{n-1}} \zeta(n-1) \Gamma(n) + O (N^{2-n}).$$

Their proof used the Selberg integral, and had applications to the study of the Riemann zeta function.

For $\lambda_1,\dots, \lambda_N$ the eigenvalues of $U$, we of course have $\log (\det(1-U))= \sum_{i=1}^N \log (1-\lambda_i)$. Using this, we can express the $n$th cumulant as an expectation as a sum over $k$-tuples of eigenvalues for $k\leq n$. Since the eigenvalues of a Haar-random unitary matrix are a determinantal point process, any such expectation can be expressed as an integral against determinants of $k\times k$ matrices, $k\leq n$, with entries given by the kernel $K_N (\theta, \theta') = \sum_{j=0}^{N-1} e^{ i j (\theta-\theta')}$. In fact, in the case of cumulants specifically, this expression simplifies somewhat, and I believe you end up with just the terms in the Leibniz formula arising from cyclic permutations.

My question is:

Does there exist an alternate proof of the Keating-Snaith formulas (maybe with a less precise error term) via the determinantal point process description of the eigenvalues of random matrices?

It is straigtforward to get the second moment/cumulant formula this way, so the question is primarily about the higher cumulants.

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    $\begingroup$ There are some papers of Soshnikov which are in this direction, though they don't answer your question as it seems like it would take some real book-keeping and maybe a new idea or two to get the calculation Keating-Snaith have for cumulants from his work. See arxiv.org/abs/math/9908063. See Lemma 2 for U(n) and (2.7) for an identity that can be applied to more general determinantal point processes. Note that $f(t) = \log(1-e^{it})$ is not in the class (1.4) he needs in this paper, but there may be some analytic ways to get around (1.4) if that restriction is the only sticking point. $\endgroup$ Oct 7, 2023 at 10:12
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    $\begingroup$ @BradRodgers This is excellent! I think with the arguments in the paper together with one additional idea I can get the $n$th cumulant as equal to some integral independent of $N$ plus $o(1)$. There's still the matter of evaluating the integral but that's actually not the most important part for me. $\endgroup$
    – Will Sawin
    Oct 7, 2023 at 18:51
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    $\begingroup$ @BradRodgers In fact, if you post this as an answer I will accept. $\endgroup$
    – Will Sawin
    Oct 9, 2023 at 0:26
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    $\begingroup$ Glad it is a useful reference! I'll repost as an answer. $\endgroup$ Oct 9, 2023 at 4:25

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Reposting a comment as an answer: There are some papers of Soshnikov which are in this direction. See for instance Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities (https://arxiv.org/abs/math/9908063). Lemma 2 there has a limiting identity for $U(n)$ and (2.7) has an identity that can be applied to more general determinantal point processes. Note that $f(t)=\log(1−e^{it})$ is not in the class (1.4) he needs in this paper, but there may be some analytic ways to get around (1.4) if that restriction is the only sticking point.

It seems it is not the most important point for the application you have in mind, but it is still an interesting question to see if these sort of identities can directly reproduce the closed formulas Keating and Snaith obtained for limits of cumulants. This seems non-trivial to me!

Another paper of Soshnikov that uses this circle of ideas (applied to a more general setting but with cruder bounds) is Gaussian limit for determinantal random point fields (https://arxiv.org/abs/math/0006037).

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This alternative proof of the Keating-Snaith formula avoids the Selberg integral and may be what you after: The characteristic polynomial of a random unitary matrix: a probabilistic approach, by Paul Bourgade, Chris Hughes, Ashkan Nikeghbali, and Marc Yor (2007).

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    $\begingroup$ Thanks, this is very interesting, but I don't think it's quite what I want. The goal is not just to avoid the Selberg integral but to use specifically the determinantal characterization (to give clues for how to prove similar theorems about other determinantal point process). This seems to use very heavily the description via the uniform measure on $N \times N$ matrices, considering the individual matrix entries. $\endgroup$
    – Will Sawin
    Oct 6, 2023 at 18:18

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