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.
Best Answer
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).