The eigenangles of a Haar unitary form one of the canonical models of a rigid but random point configuration on the circle. At macroscopic scale the empirical measure is essentially uniform, while at microscopic scale the local statistics are governed by the sine kernel. Between these regimes lies the range, where one averages over many spacings but still probes a window whose length vanishes as N → ∞. Our purpose is to isolate, in this intermediate range, a simple random field capturing the dominant fluctuations of the eigenangle density, and to identify its universal limiting object.
A convenient starting point is the observation that the CUE
characteristic polynomial
$$
Z_U(\theta):=\det(I-e^{-i\theta}U)=\prod_{j=1}^N
\bigl(1-e^{i(\theta_j-\theta)}\bigr)
$$
encodes the eigenangles simultaneously as its zeros on 𝕊1. The logarithm of |ZU(θ)|
is a log-correlated object: its covariance is approximately $-\tfrac12\log|e^{i\theta}-e^{i\phi}|$ after
removing the mean mode, which is the covariance kernel of the circle
Gaussian free field. This is the sense in which the CUE logarithmic
characteristic polynomial is expected to approximate a Gaussian free
field at scales not too fine. The viewpoint that there is a slowly
varying governing a large portion of the fluctuation of a logarithmic
object, with finer-scale corrections superimposed, has appeared
repeatedly in the study of log |ζ(1/2 + it)|
and, in the random matrix setting, in log |ZU| and its
derivatives. In the CUE context one can make this idea precise by
working in Fourier space and truncating to a mesoscopic band of
frequencies.
The natural Fourier statistics of the eigenangles are the traces
$\Tr(U^m)=\sum_{j=1}^N e^{i m
\theta_j}$. They are the Fourier coefficients of the
(non-centered) empirical measure, and they appear explicitly in the
Fourier expansion of log ZU through
the identity
$$
\log Z_U(\theta)= -\sum_{m\ge 1} \frac{\Tr(U^m)e^{-im\theta}}{m},
$$
valid away from the eigenangles (with the usual choice of branch and an
appropriate regularization). Thus, if one truncates this series at some
large frequency MN, one obtains
a smoothened random field built from $\Tr(U^m)$ for m ≤ MN.
On the other hand, the global dependence of $\Tr(U^m)$ on all eigenangles makes it
nontrivial that such a truncation yields a limit in any functional
topology, or that it can be reconstructed from more local (zeros-only)
information.
The first theme of this work is that, provided MN = o(N), the family of traces up to order MN behaves asymptotically like an independent Gaussian family after the normalization $m^{-1/2}\Tr(U^m)$. This phenomenon goes back to classical results of Diaconis–Shahshahani and Diaconis–Evans and admits several modern proofs. For our purposes it is convenient to phrase it at the level of characteristic functionals of linear combinations of traces, uniformly over families of coefficients supported on {1, …, MN}. In such a form one can pass from finite-dimensional Gaussianity to convergence of random Fourier series, and hence to the convergence in law of a random field in a negative Sobolev space. The relevant Sobolev topology is dictated by the fact that the limiting field is a distribution rather than a function: the circle Gaussian free field has Fourier coefficients of size |m|−1/2, so it lies almost surely in H−s(𝕊1) for every s > 0 and in no positive Sobolev space. Accordingly, we view mesoscopic density fluctuations as an element of H−s for arbitrary s > 0.
The resulting mesoscopic field, which we construct as a weighted cosine series with coefficients $\Re(\Tr(U^m)e^{-im\theta})/m$ truncated at m ≤ MN, should be interpreted as a : it is the component of the centered eigenvalue counting function lying in a frequency band below MN. In physical terms, the cutoff removes oscillations at or below the microscopic scale 1/N and retains only fluctuations averaged over arcs containing J(N) → ∞ eigenvalues. The parameter J(N) is therefore a mesoscopic window size, and the relation MN ∼ N/J(N) expresses the duality between spatial averaging and frequency truncation. The condition J(N) = o(N) is precisely what ensures that the retained frequencies satisfy m = o(N), the range where Gaussianity of traces is available with the strength required for field convergence. Under this regime, the covariance kernel of the truncated field is the truncated logarithmic kernel $\frac12\sum_{m\le M_N}\cos(m(\theta-\phi))/m$, and letting MN → ∞ yields the covariance of the mean-zero circle GFF. Thus the mesoscopic density wave converges, in distribution, to the GFF in H−s.
The second theme concerns a different representation of the same mesoscopic object, one that uses only the eigenangles themselves and no traces. The motivation here comes from ``zeros-only’’ principles in analytic number theory, where one seeks to express objects defined by primes (or Euler products) purely in terms of zeros, and conversely. In the random matrix setting, traces play the role of prime sums and eigenangles play the role of zeros. Since $\Tr(U^m)$ is a global Fourier statistic, it is natural to ask whether the mesoscopic density wave can be reconstructed from local geometric information about the point configuration, such as neighbor spacings.
To that end we consider symmetric neighbor discrepancies: for a given angle θ we look at the jth point to the right and the jth point to the left (in cyclic order) and compare the actual arc length spanned by these 2j points to its uniform expectation 2π ⋅ (2j)/N. After unfolding by N/(2π), the centered discrepancy δj(θ) measures the excess or deficit of eigenangles in a symmetric neighborhood of θ at scale j/N. A single discrepancy δj is noisy, but a logarithmically weighted sum ∑j ≥ Jδj(θ)/j exhibits a stabilizing effect reminiscent of harmonic summation, and it is precisely this weighting that matches the 1/m weights in the trace-based Fourier truncation. The resulting field, defined solely from the eigenangle configuration, is a candidate zeros-only density wave.
The main comparison mechanism is an identity of summation-by-parts type connecting the inverse local counting map (which sends a rank increment to a spatial increment) to an integrated error of the counting function. Concretely, one may rewrite the neighbor radius θ+j(θ) − θ−j(θ) in terms of an integral involving the centered counting function SU and a piecewise-linear kernel depending on j. Summing over j then produces a convolution of SU (or its distributional derivative) against an explicit kernel KJ whose Fourier coefficients approximate 1/|m| on the band 1 ≤ |m| ≤ MN and decay rapidly outside. Once this reduction is established, Fourier inversion identifies the convolution term with the trace field truncated at MN. The remaining task is to control the approximation errors uniformly in θ, which is delicate near eigenangles because the counting function has jumps and the inverse counting map has points of non-smoothness.
The strength of the comparison depends on how rapidly J(N) grows. For convergence in distribution in a negative Sobolev topology, it suffices that J(N) → ∞ while J(N) = o(N), since then the relevant kernels are sufficiently regular at the level of H−s and the exceptional behavior near eigenangles is negligible in that topology. For a uniform-in-θ comparison, however, we need quantitative control of the remainder term and of the stability of the inverse counting map under perturbations of SU. This is where taking J(N) of logarithmic order is natural: the CUE number variance and rigidity bounds imply that counting errors on arcs containing ≍ log N points are small enough (with high probability) to guarantee that the discrepancy-based reconstruction tracks the trace-based field uniformly. In this regime one can also isolate tiny neighborhoods of eigenangles, or adopt a right-continuity convention, to avoid ambiguous pointwise definitions without changing the limit.
We emphasize that the CUE is the appropriate testbed for these statements for two reasons. First, the eigenangles form a determinantal point process with an explicit kernel, which provides sharp control of linear statistics, number variance, and cumulants. Second, Haar measure yields exact algebraic identities for trace moments and Toeplitz determinant representations for exponential linear statistics; both are well adapted to proving asymptotic Gaussianity in the range m = o(N). While our arguments are presented in the CUE setting, the structure of the conclusions is not specific to unitary symmetry: the appearance of the GFF reflects the universal logarithmic correlations of mesoscopic eigenvalue fluctuations, and the possibility of zeros-only reconstruction reflects the general principle that sufficiently averaged local spacing information encodes low-frequency density modes.
In summary, we establish two complementary descriptions of the same mesoscopic density wave. One description is spectral and uses traces, leading directly to a random Fourier series and hence to the circle GFF limit in H−s. The other description is geometric and uses only neighbor discrepancies, yielding a zeros-only estimator which, at logarithmic mesoscopic scale, approximates the trace field uniformly. Having motivated these objects and their roles, we now turn to the necessary background: the CUE eigenangle process and its centered counting function, the basic features of Sobolev spaces on 𝕊1 used to encode distributional convergence, and the definition and elementary properties of the mean-zero circle Gaussian free field.
Let U ∼ Haar(U(N)) and
write its eigenvalues as eiθ1, …, eiθN
with
0 ≤ θ1 < ⋯ < θN < 2π.
The joint density of the eigenangles with respect to Lebesgue measure on
[0, 2π)N
is given by the classical Weyl formula,
$$
\frac{1}{(2\pi)^N N!}\prod_{1\leq j<k\leq
N}\bigl|e^{i\theta_j}-e^{i\theta_k}\bigr|^2,
$$
which makes the strong repulsion between nearby points manifest. It is
convenient to view the random configuration as the point process
$$
\Xi_U:=\sum_{j=1}^N \delta_{\theta_j}
$$
on 𝕊1 ≃ ℝ/2πℤ. A
basic structural fact is that ΞU is a
determinantal point process with correlation kernel
More precisely, the k-point
correlation functions of ΞU are
ρk(θ1, …, θk) = det (KN(θa, θb))1 ≤ a, b ≤ k.
Two immediate consequences will be repeatedly used: first, rotation
invariance (Haar invariance) implies that the law of the configuration
is invariant under the shift θj ↦ θj + α
mod 2π; second, the
determinantal structure provides sharp information on the variance of
linear statistics and on concentration of counting functions.
Given an arc I ⊂ 𝕊1 of length |I|, the expected number of points
in I is
$$
\mathbb{E}\,\Xi_U(I)=\int_I
K_N(\theta,\theta)\,d\theta=\frac{N}{2\pi}|I|.
$$
The fluctuation size of ΞU(I)
depends on the scale of I. For
our mesoscopic purposes we need only the qualitative picture that for
arcs containing many points one has sublinear fluctuations. Concretely,
determinantal methods yield a number variance of logarithmic order on
mesoscopic arcs: for |I|
bounded away from 0 and 2π one has Var(ΞU(I)) ≍ log N,
and more generally the variance can be bounded in terms of KN by standard
determinantal identities. These statements encode a form of rigidity of
the CUE configuration: compared to a Poisson process, the counting
function is much more stable.
We work with the centered counting function
$$
S_U(\theta):=\#\{j:0\leq \theta_j\leq
\theta\}-\frac{N\theta}{2\pi},\qquad \theta\in[0,2\pi],
$$
and extend SU 2π-periodically to ℝ. Since #{j : 0 ≤ θj ≤ 0} = 0
and #{j : 0 ≤ θj ≤ 2π} = N,
we have SU(0) = SU(2π) = 0,
so the periodic extension is consistent. The function SU is càdlàg,
piecewise affine with slope −N/(2π) and upward jumps of
size 1 at the eigenangles.
It is natural to differentiate SU in the sense
of distributions. Let dSU
denote its (periodic) distributional derivative, which is a signed
measure on 𝕊1. A direct
computation shows
i.e. the centered empirical measure. Thus, for any smooth 2π-periodic test function f,
Equivalently, by integration by parts on the circle,
whenever f is C1 and periodic.
Fourier analysis provides a convenient encoding of . For a (signed)
measure μ on 𝕊1 we use the convention
μ̂(m) := ∫𝕊1e−imθ μ(dθ), m ∈ ℤ.
Letting $\mu_U:=\sum_{j=1}^N\delta_{\theta_j}$, we
have
$$
\widehat{\mu_U}(m)=\sum_{j=1}^N
e^{-im\theta_j}=\mathrm{Tr}(U^{-m})=\overline{\mathrm{Tr}(U^m)}.
$$
From , it follows that $\widehat{dS_U}(0)=0$ and, for m ≠ 0,
Since dSU is
the derivative of SU, we also have
the identity
where $\widehat{S_U}(m)$ is now
understood in the distributional sense (for instance by pairing SU against e−imθ).
Formula is the basic bridge between the counting function and trace
statistics; it will allow us to identify certain convolutions of dSU
with explicit kernels as Fourier series built from Tr(Um).
Two further elementary remarks will be useful. First, by rotation invariance, the process dSU is stationary on 𝕊1 (mean-zero and shift invariant). Second, the total mass of dSU vanishes, reflecting that SU has zero mean mode. This is the reason that the limiting Gaussian field will also be taken mean-zero.
We now recall the Sobolev scale on the circle and fix conventions for
Fourier coefficients. For an L2 function f on 𝕊1 we write its Fourier series
as
$$
f(\theta)=\sum_{m\in\mathbb{Z}} \widehat{f}(m)e^{im\theta},
\qquad
\widehat{f}(m)=\frac{1}{2\pi}\int_0^{2\pi}
f(\theta)e^{-im\theta}\,d\theta.
$$
For s ∈ ℝ, the Sobolev space
Hs(𝕊1)
consists of those distributions f whose Fourier coefficients
satisfy
∥f∥Hs2 := ∑m ∈ ℤ(1 + |m|2)s |f̂(m)|2 < ∞.
When s ≥ 0 this agrees with
the standard L2-based Sobolev space;
when s < 0 it defines a
Hilbert space of distributions. We write H0s(𝕊1)
for the mean-zero subspace f̂(0) = 0.
The duality pairing between H−s and Hs (with s > 0) may be realized through
Fourier coefficients: if T ∈ H−s
and f ∈ Hs,
then
is absolutely convergent and satisfies |⟨T, f⟩| ≤ ∥T∥H−s∥f∥Hs.
In particular, random objects with Fourier coefficients of size |m|−1/2 naturally live in
H−s for
every s > 0 but not in
L2.
We will use this framework to encode convergence in law of random
fields that are defined either by Fourier series (as in the trace-based
construction) or as convolutions with dSU (as
in the discrepancy-based construction). The basic scheme is as follows:
if (TN) is
a sequence of random elements of H−s and T is a limiting random element of
H−s, then
TN ⇒ T
in H−s
means that for every bounded continuous functional F : H−s → ℝ
we have 𝔼F(TN) → 𝔼F(T).
In practice we verify this by showing convergence of finite-dimensional
distributions
(⟨TN, f1⟩, …, ⟨TN, fk⟩) ⇒ (⟨T, f1⟩, …, ⟨T, fk⟩)
for suitable test functions fℓ ∈ Hs,
together with tightness in H−s. The
tightness step reduces to uniform control of 𝔼∥TN∥H−s2,
which in Fourier terms is an estimate on $\mathbb{E}|\widehat{T_N}(m)|^2$ with weights
(1 + |m|2)−s.
The limiting object for our mesoscopic density wave is the mean-zero
Gaussian free field on 𝕊1,
which we denote by 𝒢. We take as
definition the centered Gaussian random distribution whose Fourier
coefficients satisfy
Equivalently, one may realize 𝒢 as the
real random Fourier series
where (am, bm)m ≥ 1
are i.i.d. standard real Gaussians. The series does not converge
pointwise; it converges almost surely in H−s(𝕊1)
for every s > 0, and in no
Ht with
t ≥ 0. Indeed,
$$
\mathbb{E}\|\mathcal{G}\|_{H^{-s}}^2
=\sum_{m\neq 0}(1+|m|^2)^{-s}\,\mathbb{E}|\widehat{\mathcal{G}}(m)|^2
\asymp \sum_{m\ge 1}\frac{1}{m^{1+2s}}<\infty\qquad (s>0),
$$
which implies 𝒢 ∈ H−s almost
surely by standard Gaussian Hilbert space arguments.
The covariance kernel of 𝒢 can be
written in several equivalent ways. For test functions f, g ∈ H0s
with s > 0, one has
Formally, this corresponds to the logarithmic kernel on the circle.
Indeed, if we define the mean-zero Green kernel
$$
G(\theta):=\sum_{m\neq 0}\frac{e^{im\theta}}{2|m|}=\sum_{m\ge
1}\frac{\cos(m\theta)}{m},
$$
then G(θ) is (up to
an additive constant) equal to −log |2sin (θ/2)| for θ ≢ 0 (mod 2π), and may be
rewritten as
Cov(⟨𝒢, f⟩, ⟨𝒢, g⟩) = ∫𝕊1∫𝕊1f(θ) G(θ − ϕ) g(ϕ) dθ dϕ,
with the understanding that the integral is interpreted via Fourier
series (since G has a
logarithmic singularity).
For later comparison with mesoscopic truncations, we also record the
standard Fourier cutoff regularization. Define
$$
\mathcal{G}^{(\le
M)}(\theta):=\sum_{m=1}^{M}\frac{1}{\sqrt{m}}\bigl(a_m\cos(m\theta)+b_m\sin(m\theta)\bigr).
$$
Then 𝒢( ≤ M) is a
smooth random function and 𝒢( ≤ M) → 𝒢 in H−s almost
surely and in L2(Ω; H−s)
for every s > 0. Its
covariance kernel is the truncated logarithmic sum
$$
\mathbb{E}\bigl[\mathcal{G}^{(\le M)}(\theta)\mathcal{G}^{(\le
M)}(\phi)\bigr]
=\frac12\sum_{m=1}^{M}\frac{\cos(m(\theta-\phi))}{m},
$$
which converges (as M → ∞) to
the logarithmic kernel away from the diagonal and in the distributional
sense on 𝕊1 × 𝕊1
after removing the mean mode. This truncated kernel is the one that will
arise from our trace-based density wave once we identify its Fourier
coefficients and compute their asymptotic second moments in the regime
M = o(N).
Having fixed these conventions, we may now treat the mesoscopic trace field WN, J as a random element of H−s(𝕊1) and compare it directly to 𝒢 via Fourier coefficients and characteristic functionals. We proceed next to the basic properties of WN, J, beginning with its symmetries and covariance structure and its relation to regularized versions of the logarithm of the characteristic polynomial.
We recall that for a mesoscopic parameter J = J(N) with
J → ∞ and J = o(N) we set
MN := ⌊N/J⌋
and define the real-valued trigonometric polynomial
$$
W_{N,J}(\theta):=\sum_{m=1}^{M_N}\frac{\Re\!\big(\Tr(U^m)e^{-im\theta}\big)}{m},\qquad
\theta\in\mathbb{S}^1.
$$
Since WN, J
is a finite Fourier series, it is a smooth 2π-periodic function for each fixed
N.
The observable WN, J
depends only on the spectrum of U. Indeed, writing λj = eiθj
we have $\Tr(U^m)=\sum_{j=1}^N
\lambda_j^m=\sum_{j=1}^N e^{im\theta_j}$, hence
Thus WN, J
is a linear statistic of the eigenangles with the (singular, but
truncated) test kernel ∑m ≤ MNcos (mt)/m
evaluated at t = θj − θ.
We record the stationarity (rotation invariance) of the field. For
any α ∈ ℝ, Haar invariance
implies that eiαU
has the same law as U. Since
$\Tr((e^{i\alpha}U)^m)=e^{im\alpha}\Tr(U^m)$,
we obtain the identity in law
$$
W_{N,J}^{(e^{i\alpha}U)}(\theta)=\sum_{m\le
M_N}\frac{\Re\!\big(e^{im\alpha}\Tr(U^m)e^{-im\theta}\big)}{m}
=W_{N,J}^{(U)}(\theta-\alpha).
$$
Consequently, as a random field on 𝕊1, WN, J
is stationary: (WN, J(θ1 + α), …, WN, J(θk + α))
has the same law as (WN, J(θ1), …, WN, J(θk)).
We also note that $\mathbb{E}\Tr(U^m)=0$ for every fixed m ≥ 1 (for instance by multiplying U by a scalar eiα and averaging over α), and hence 𝔼WN, J(θ) = 0 for all θ.
It is useful to identify the Fourier coefficients of WN, J
in the conventions fixed earlier. Expanding the real part,
$$
\Re\!\big(\Tr(U^m)e^{-im\theta}\big)=\frac12\Big(\Tr(U^m)e^{-im\theta}+\overline{\Tr(U^m)}\,e^{im\theta}\Big)
=\frac12\Big(\Tr(U^m)e^{-im\theta}+\Tr(U^{-m})e^{im\theta}\Big).
$$
Therefore
In particular, for m ∈ ℤ \ {0},
Comparing with , we see that for 1 ≤ |m| ≤ MN,
$$
\widehat{W_{N,J}}(m)=\frac{1}{2|m|}\,\widehat{dS_U}(m).
$$
This admits a compact formulation in terms of convolution with dSU.
Let KM be
the 2π-periodic kernel defined
by its Fourier coefficients
so that
For a finite signed measure μ
on 𝕊1 we use the standard
periodic convolution
(KM * μ)(θ) := ∫𝕊1KM(θ − ϕ) μ(dϕ),
in which case $\widehat{K_M*\mu}(m)=2\pi\,\widehat{K_M}(m)\widehat{\mu}(m)$.
Applying this with μ = dSU
and using –, we obtain for 1 ≤ |m| ≤ M,
$$
\widehat{K_M*dS_U}(m)=2\pi\cdot \frac{1}{4\pi |m|}\cdot
\widehat{dS_U}(m)=\frac{1}{2|m|}\widehat{dS_U}(m),
$$
and the coefficient vanishes for |m| > M as well as at
m = 0. Thus, with M = MN,
This representation will be the point of contact with the
discrepancy-based construction later on: it isolates the universal
kernel KMN
and separates it from the random input dSU.
We next compute the covariance kernel of WN, J.
The main input is the classical second-moment identity for CUE traces:
for integers m, n ≥ 1,
and $\mathbb{E}[\Tr(U^m)\Tr(U^n)]=0$
for m, n ≥ 1.
Identity may be proved by representation-theoretic methods
(Diaconis–Shahshahani) or by determinantal calculations for Fourier
modes of the empirical measure.
Using and bilinearity, we write
$$
\mathbb{E}\big[W_{N,J}(\theta)W_{N,J}(\phi)\big]
=\sum_{m,n\le M_N}\frac{1}{m n}\,
\mathbb{E}\Big[\Re\!\big(\Tr(U^m)e^{-im\theta}\big)\Re\!\big(\Tr(U^n)e^{-in\phi}\big)\Big].
$$
Expanding the real parts and using that mixed moments without
conjugation vanish, we obtain
hence
Since MN = ⌊N/J⌋,
we have min (m, N) = m for
all m ≤ MN
once N is large enough (and in
any case min (m, N) = m + O(N/J − N)+),
so is asymptotic to the truncated logarithmic sum
uniformly in θ, ϕ ∈ 𝕊1. We
isolate this as a lemma for later reference.
In particular, setting θ = ϕ yields
so the typical amplitude of WN, J
at a point is of order $\sqrt{\log
M_N}$.
The covariance matches exactly the covariance of the Fourier cutoff regularization of the mean-zero circle GFF at frequency MN. Since ∑m ≥ 1cos (mt)/m is the mean-zero Green kernel on 𝕊1 (equal to −log |2sin (t/2)| up to an additive constant away from t ≡ 0), the truncated sum may be viewed as a regularized logarithmic kernel. This already suggests that as MN → ∞ one should obtain a log-correlated Gaussian limit; the remaining task, carried out in the next section, is to justify Gaussianity and tightness in negative Sobolev topologies.
We finally explain the standard connection to the characteristic
polynomial. Let
$$
\Lambda_U(z):=\det(I-zU)=\prod_{j=1}^N (1-ze^{i\theta_j}),\qquad
z\in\mathbb{C}.
$$
For |z| < 1 we may expand
log (1 − zeiθj)
into its absolutely convergent power series and sum over j, obtaining the identity
where the logarithm is the principal branch near z = 0. Taking real parts, for z = re−iθ
with 0 ≤ r < 1,
Thus the infinite series built from $\Re(\Tr(U^m)e^{-im\theta})/m$ is precisely
(minus) the harmonic regularization of log |ΛU(eiθ)|
obtained by radially approaching the unit circle. The field WN, J
is the sharp-frequency analogue of : it replaces the Poisson weights
rm by a
cutoff m ≤ MN
and discards higher modes.
Equivalently, one may interpret WN, J
as the logarithmic potential generated by the centered empirical
measure, with a Fourier cutoff removing the singularity at the diagonal.
Indeed, combining with and , we may rewrite
$$
W_{N,J}(\theta)=\int_{\mathbb{S}^1}
K_{M_N}(\theta-\phi)\,\Big(\sum_{j=1}^N\delta_{\theta_j}-\frac{N}{2\pi}d\phi\Big),
$$
so WN, J
is a centered (mean-zero) logarithmic potential field, regularized by
the Fourier cutoff in KMN.
Since KMN(t)
converges in the distributional sense (modulo constants) to the Green
kernel t ↦ −log |2sin (t/2)|, the
observable WN, J
may be viewed as a mesoscopic approximation to the log-correlated field
naturally associated with the eigenangles.
This perspective clarifies why WN, J is the appropriate object for comparison with the circle GFF: the latter is the canonical centered Gaussian log-correlated distribution on 𝕊1, and shows that the covariance of WN, J already matches the GFF covariance at the level of Fourier cutoffs. In the next section we justify that the higher joint moments (or, equivalently, characteristic functionals against test functions) are asymptotically Gaussian for modes m ≤ MN = o(N), which leads to the distributional convergence in H−s(𝕊1).
Throughout this section we assume MN = o(N), as is the case when MN = ⌊N/J⌋ with J → ∞ and J = o(N). Our objective is twofold: first, to justify that the collection of trace modes $(\Tr(U^m))_{1\leq m\leq M_N}$ behaves asymptotically as an independent Gaussian family after normalization; second, to deduce from this that the field WN, J converges in law to the circle GFF in negative Sobolev topology.
Let ψ : 𝕊1 → ℝ
be a real trigonometric polynomial of degree at most MN,
$$
\psi(\theta)=\sum_{1\leq |m|\leq M_N} c_m e^{im\theta},\qquad
c_{-m}=\overline{c_m}.
$$
We write the corresponding linear statistic of the eigenangles as
$$
L_N(\psi):=\sum_{j=1}^N \psi(\theta_j).
$$
A basic identity (Heine–Szeg) expresses the Laplace transform of LN(ψ)
as a Toeplitz determinant:
where TN(eψ)
is the N × N Toeplitz
matrix with symbol eψ, i.e. $(T_N(e^\psi))_{k\ell}=\widehat{e^\psi}(k-\ell)$
in our Fourier conventions. We emphasize that is exact and uses only
that the CUE eigenangles form a determinantal point process with the
projection kernel onto Fourier modes {0, 1, …, N − 1}.
The relevance of is that Toeplitz determinants admit precise
asymptotics when the symbol is sufficiently regular. In the present
mesoscopic regime the symbol depends on N through its Fourier support MN, and we will
use a strong Szeg-type estimate uniform for symbols whose Fourier
support is o(N).
Concretely, writing
ψ̂(m) = cm, ψ̂(0) = 0,
the strong Szeg formula suggests
$$
\log \det\big(T_N(e^\psi)\big)=N\widehat{\psi}(0)+\sum_{m\geq 1}
m\,\widehat{\psi}(m)\widehat{\psi}(-m)+o(1)
=\sum_{m=1}^{M_N} m |c_m|^2 + o(1),
$$
provided ψ remains in a
bounded set of an appropriate Sobolev class. We record a version
tailored to our setting.
We now translate Lemma~ into asymptotic Gaussianity of traces.
Observe that
$$
L_N(\psi)
=\sum_{j=1}^N \sum_{1\leq |m|\leq M_N} c_m e^{im\theta_j}
=\sum_{m=1}^{M_N}\Big(c_m \Tr(U^m)+c_{-m}\Tr(U^{-m})\Big)
=2\Re\sum_{m=1}^{M_N} c_m \Tr(U^m).
$$
Thus Lemma~ describes the joint Laplace transforms of linear
combinations of $(\Tr(U^m))_{m\leq
M_N}$.
Fix k ≥ 1 and distinct
integers 1 ≤ m1 < ⋯ < mk ≤ MN.
For complex coefficients (t1, …, tk)
define
$$
\psi(\theta):=\sum_{\ell=1}^k \Big(t_\ell e^{i m_\ell
\theta}+\overline{t_\ell}e^{-i m_\ell \theta}\Big),
$$
so that cmℓ = tℓ
and $c_{-m_\ell}=\overline{t_\ell}$.
Then
$$
L_N(\psi)=2\Re\sum_{\ell=1}^k t_\ell \Tr(U^{m_\ell}).
$$
Applying Lemma~ gives
uniformly for tℓ in bounded
sets. The right-hand side is precisely the Laplace transform of a vector
of independent complex Gaussians (Zmℓ)ℓ = 1k
with 𝔼Zmℓ = 0
and 𝔼|Zmℓ|2 = mℓ.
In particular, yields the following.
We now deduce the distributional convergence of the field WN, J in negative Sobolev topology. Since WN, J is a trigonometric polynomial supported on frequencies |m| ≤ MN, it suffices to test against f ∈ Hs(𝕊1) with zero mean and identify the limiting law of ⟨WN, J, f⟩.
Let f(θ) = ∑m ∈ ℤf̂(m)eimθ
with f̂(0) = 0. Using the
Fourier expansion of WN, J
and orthogonality of exponentials, we obtain
For real-valued f this
simplifies to
Thus ⟨WN, J, f⟩
is a finite linear combination of the trace modes $\Tr(U^m)$ with coefficients of size f̂(m)/m.
To apply Proposition~ uniformly, we note that f ∈ Hs
implies
$$
\sum_{m\in\mathbb{Z}} (1+m^2)^{s}|\widehat{f}(m)|^2<\infty,
\qquad\text{hence}\qquad
\sum_{m=1}^{\infty} m\,\Big|\frac{\widehat{f}(m)}{m}\Big|^2
=\sum_{m=1}^\infty \frac{|\widehat{f}(m)|^2}{m}<\infty,
$$
since s > 0 implies |f̂(m)|2 ≪ m−2s − 1
and therefore |f̂(m)|2/m
is summable.
Define a real trigonometric polynomial ψf (depending on
N through the cutoff MN) by
$$
\psi_f(\theta):=\sum_{m=1}^{M_N}\frac{\pi}{m}\Big(\widehat{f}(m)e^{im\theta}+\widehat{f}(-m)e^{-im\theta}\Big).
$$
Then reads
⟨WN, J, f⟩ = LN(ψf).
Lemma~ applies since
$$
\sum_{m=1}^{M_N} m \Big|\frac{\pi\widehat{f}(m)}{m}\Big|^2
=\pi^2\sum_{m=1}^{M_N}\frac{|\widehat{f}(m)|^2}{m}
\leq \pi^2\sum_{m=1}^{\infty}\frac{|\widehat{f}(m)|^2}{m}<\infty,
$$
uniformly in N. Therefore, for
each fixed f (and jointly for
finitely many such f), ⟨WN, J, f⟩
converges to a centered Gaussian with variance determined by the
quadratic form ∑m ≥ 1|f̂(m)|2/m
in the normalization consistent with the covariance computed previously
for WN, J.
This identifies the limiting finite-dimensional distributions with those
of the mean-zero circle GFF in our normalization.
It remains to upgrade convergence of finite-dimensional distributions
to convergence in law in H−s(𝕊1)
for any s > 0. We use the
standard criterion based on uniform bounds on Sobolev norms. Writing the
Fourier coefficients of WN, J
as in , we have $\widehat{W_{N,J}}(0)=0$ and $\widehat{W_{N,J}}(m)=0$ for |m| > MN.
For s > 0,
$$
\|W_{N,J}\|_{H^{-s}}^2
:=\sum_{m\in\mathbb{Z}} (1+m^2)^{-s}\,|\widehat{W_{N,J}}(m)|^2
=\sum_{1\leq |m|\leq M_N} (1+m^2)^{-s}\,|\widehat{W_{N,J}}(m)|^2.
$$
Taking expectations and using $\mathbb{E}|\Tr(U^m)|^2=\min(m,N)$, we obtain
for N large (so that min (m, N) = m for
m ≤ MN),
Since (1 + m2)−s ≤ m−2s
for m ≥ 1 and ∑m ≥ 1m−1 − 2s < ∞
whenever s > 0, the
right-hand side is bounded uniformly in N. Hence (WN, J)N ≥ 1
is tight in H−s(𝕊1).
Combining tightness with the convergence of the finite-dimensional distributions ⟨WN, J, f⟩ for f ∈ Hs yields the claimed convergence in law in H−s. This completes the proof of part~(i) of the main theorem.
We now turn to a complementary question: how much of the field WN, J can be reconstructed from the eigenangle configuration alone without access to traces, by using only symmetric neighbor discrepancies on mesoscopic scales.
Our aim in part~(ii) is to construct, from the eigenangle configuration alone, an observable which approximates the trace-based field WN, J. The input we allow ourselves is purely ``geometric’’: for each point θ ∈ 𝕊1 we look at the locations of the jth eigenangles immediately to the right and to the left of θ in the cyclic order, and we compare the resulting symmetric arc length to its deterministic value 4πj/N. The corresponding normalized error is the δj(θ), and the zeros-only density wave is a harmonic sum of these discrepancies over mesoscopic scales j ≥ J.
A minor issue is that the neighbor maps θ ↦ θ±j(θ) are not canonically defined at the eigenangles themselves (or, more generally, on a null set where ``ties’’ may occur if one insists on working purely modulo 2π). We therefore begin by fixing a robust circle convention and by recording elementary measurability and deterministic bounds. These points are logically separate from the probabilistic estimates used later, but they clarify that DN, J is an honest random field, canonically constructed from {θ1, …, θN}.
We work with a fixed realization 0 ≤ θ1 < ⋯ < θN < 2π
and introduce a 2π-periodic
lift to ℝ by setting
θ̃k := θk, 1 ≤ k ≤ N, and θ̃k + N := θ̃k + 2π, k ∈ ℤ.
For each θ ∈ ℝ we define the
(right-continuous) index
n(θ) := max {k ∈ ℤ: θ̃k ≤ θ},
so that θ̃n(θ) ≤ θ < θ̃n(θ) + 1
for all θ which are not
themselves eigenangles, and at θ = θ̃k
we have n(θ) = k (this is
the right-continuity convention). For integers j ≥ 1 we then set the lifted
neighbors
θ̃+j(θ) := θ̃n(θ) + j, θ̃−j(θ) := θ̃n(θ) − j + 1,
and define θ±j(θ) ∈ 𝕊1
as their images modulo 2π.
With this choice, the arc from the left neighbor to the right neighbor
is represented by the genuine real interval
[θ̃−j(θ), θ̃+j(θ)] ⊂ ℝ,
and its length is simply θ̃+j(θ) − θ̃−j(θ) > 0.
In particular, for 1 ≤ j ≤ ⌊N/2⌋ we always
have
0 < θ̃+j(θ) − θ̃−j(θ) < 2π,
since the index gap is (n(θ) + j) − (n(θ) − j + 1) = 2j − 1 < N
and thus does not cross a full period in the lifted sequence. This is
one reason for later restricting j to at most a fixed fraction of
N (we shall use ⌊N/4⌋).
For 1 ≤ j ≤ ⌊N/4⌋
and θ ∈ ℝ we define the
symmetric j-neighbor
discrepancy by
The normalization is chosen so that in the perfectly rigid ``picket
fence’’ configuration θk = 2πk/N
one has θ̃+j(θ) − θ̃−j(θ) = 4πj/N
and hence δj(θ) ≡ 0.
In a random configuration, δj(θ)
records an over-density (negative discrepancy) or under-density
(positive discrepancy) of eigenangles around θ on the symmetric scale of
approximately 2j points.
We then define, for an integer cutoff J ≥ 1,
This is the zeros-only density-wave functional appearing in part~(ii).
The lower cutoff J will be
taken to infinity with N; the
upper cutoff ⌊N/4⌋ is
inessential (any fixed fraction < N/2 would do) and is imposed
to avoid large symmetric windows that nearly cover the entire circle,
where later summation-by-parts manipulations become notationally less
stable.
The definition above yields, for each fixed eigenangle configuration, a function θ ↦ δj(θ) which is 2π-periodic and right-continuous with left limits. Indeed, n(θ) is a right-continuous step function with jumps of size 1 at each eigenangle (and constant on each interval between successive eigenangles); hence θ̃±j(θ) are also right-continuous step functions, and so is their difference. Consequently δj is piecewise constant on the connected components of ℝ \ {θ̃k : k ∈ ℤ}, with jumps only at eigenangles, and the same is true of DN, J.
The dependence on the randomness is purely through the ordered list
(θ1, …, θN).
More precisely, if we view θ ↦ DN, J(θ)
as an element of the Skorokhod space D(𝕊1) of c`adl`ag
functions on the circle (or simply as a bounded measurable function on
𝕊1), then the mapping
(θ1, …, θN) ↦ DN, J( ⋅ )
is Borel measurable.
In applications we will occasionally want to take suprema over θ. Since δj (and hence
DN, J)
may jump at eigenangles, it is useful to note that
supθ ∈ 𝕊1|DN, J(θ)| = max1 ≤ k ≤ N supθ ∈ [θk, θk + 1)|DN, J(θ)|,
so one may equivalently take the supremum over any deterministic set of
representatives, for instance over midpoints of the eigenangle gaps,
without changing the value. This observation will be convenient when we
exclude tiny neighborhoods of eigenangles in later uniform
estimates.
We record simple bounds which hold for every configuration. Since
0 < θ̃+j(θ) − θ̃−j(θ) < 2π
for j ≤ N/2, we
have
Consequently,
uniformly in θ. These bounds
are far from sharp for Haar-typical configurations, but they show that
the functional is well-defined without any further truncation and that
any probabilistic control in sup norm will necessarily use cancellations
beyond the trivial estimate .
A further deterministic point is the invariance under rotation: if we shift all eigenangles by a common amount α (modulo 2π), then δj(θ) and DN, J(θ) shift as functions of θ by the same amount. This is immediate from the cyclic indexing definition and will be consistent with the stationarity of the limiting Gaussian field.
The harmonic weights 1/j in are dictated by the logarithmic structure of the target field. One expects (and later proves, after converting discrepancies to a smoothed counting statistic) that the sum over j behaves as a Fourier multiplier comparable to 1/|m| at frequencies |m| ≲ N/J. In this sense, restricting to j ≥ J is analogous to discarding high Fourier modes, exactly as WN, J discards trace modes above MN = ⌊N/J⌋.
From the viewpoint of well-posedness of a limiting object, the condition J → ∞ is also natural. Even if δj(θ) stayed of order one, the series ∑j ≥ 1δj(θ)/j would have no reason to converge as N → ∞ because of the logarithmic divergence of ∑1/j. Taking J → ∞ removes the microscopic scales and ensures that the remaining sum corresponds to a mesoscopic smoothing, compatible with convergence in negative Sobolev topology.
For the reconstruction statement in part~(ii), we need a more
specific growth rate. The key stability issue is that δj(θ)
is defined in terms of inverse counting maps: the endpoints θ̃±j(θ)
are quantiles of the point configuration around θ. Inverting the counting function
is Lipschitz only on scales larger than the maximal counting error. For
Haar(CUE), it is known (and we will use later through standard
rigidity/number variance bounds) that the centered counting function
satisfies a logarithmic sup bound with high probability,
∥SU∥L∞(𝕊1) ≪ log N.
Heuristically, the local quantile map becomes stable once the number of
points in the window dominates these counting fluctuations, i.e. once
j ≫ ∥SU∥∞.
This suggests precisely a logarithmic choice J(N) ≍ log N as
the threshold beyond which neighbor discrepancies can be related,
uniformly in θ, to smoothed
versions of SU with
negligible error.
We therefore adopt in part~(ii) the regime
J(N) = ⌊clog N⌋,
which is minimal up to constants for the ensuing uniform approximation
argument. In the weaker distributional comparison (where one tests
against smooth functions and works in H−s rather than
sup norm), it is typically sufficient to assume only J → ∞ and J = o(N);
nonetheless, even there, the diverging cutoff plays the same conceptual
role of separating microscopic irregularities from the mesoscopic
density wave.
The remainder of the proof will proceed by rewriting DN, J as a linear functional of the counting function SU against an explicit logarithmic-type kernel, up to an error which vanishes under the above choice of J. This reduction is the point where the particular form of becomes effective: symmetric neighbor radii admit a summation-by-parts identity that turns the inverse-quantile information into an integrated counting error, and hence into a convolution amenable to Fourier comparison with WN, J.
Our next task is to replace the inverse–quantile nature of δj(θ) by a direct linear statistic of the counting function. The guiding principle is that the discrepancy compares the of a symmetric window (the arc between the two neighbors) to the of the eigenangle configuration in that same window; the difference is precisely the signed mass of the centered counting measure dSU on the window. After summing in j with harmonic weights, one obtains a logarithmic kernel.
We regard SU as a 2π-periodic c`adl`ag function and
write its associated signed measure on ℝ as
$$
dS_U(t)=\sum_{k\in\mathbb{Z}}\delta_{\widetilde{\theta}_k}(dt)-\frac{N}{2\pi}\,dt,
$$
so that for any half-open interval (a, b] ⊂ ℝ we have
$$
\int_{(a,b]} dS_U
=\#\{k:\ a<\widetilde{\theta}_k\leq b\}-\frac{N}{2\pi}(b-a).
$$
For fixed θ and j ≤ ⌊N/4⌋, set
Ij(θ) := (θ̃−j(θ), θ̃+j(θ)] ⊂ ℝ.
By construction, Ij(θ)
contains exactly 2j lifted
eigenangles (counting the right endpoint and excluding the left
endpoint), i.e.
#{k: θ̃−j(θ) < θ̃k ≤ θ̃+j(θ)} = 2j.
Consequently,
and hence
This identity is deterministic and uses only the half-open convention in
the definition of the Stieltjes integral; it is precisely at this point
that the right-continuity convention for the neighbor maps is
convenient, since it provides an unambiguous choice of endpoints.
Substituting into and exchanging the sum with the integral (which is
legitimate since the sum is finite) yields the exact formula
Thus DN, J
is a linear statistic of dSU
with a random weight that depends on θ through the random windows Ij(θ).
The right-hand side of becomes a convolution once we replace the
random indicator 1Ij(θ)(t)
by a deterministic translation-invariant approximation. The natural
deterministic window of ``radius j’’ around θ is the symmetric interval
$$
I_j^{(0)}(\theta):=\big(\theta-\tfrac{2\pi j}{N},\,\theta+\tfrac{2\pi
j}{N}\big]\subset\mathbb{R},
$$
whose length is exactly 4πj/N. Define the
corresponding deterministic weight
extended 2π-periodically to
𝕊1. Then
$$
\sum_{j=J}^{\lfloor
N/4\rfloor}\frac{\mathbf{1}_{I_j^{(0)}(\theta)}(t)}{j}
=L_J(t-\theta),
$$
and therefore
where convolution is taken on 𝕊1.
The difference between and is supported near the boundaries of the
random windows, and can be bounded in terms of how far the random
endpoints θ̃±j(θ)
deviate from the deterministic endpoints θ ± 2πj/N.
A convenient way to encode this is through an
$$
\Delta_j(\theta):=\max\Big\{\big|\widetilde{\theta}_{+j}(\theta)-(\theta+\tfrac{2\pi
j}{N})\big|,\ \big|\widetilde{\theta}_{-j}(\theta)-(\theta-\tfrac{2\pi
j}{N})\big|\Big\}.
$$
On the event that Δj(θ) ≤ η/N
for all θ and all j ≥ J, the symmetric
difference of windows satisfies
$$
I_j(\theta)\,\triangle\, I_j^{(0)}(\theta)\subset
\big(\theta-\tfrac{2\pi j}{N}-\tfrac{\eta}{N},\,\theta-\tfrac{2\pi
j}{N}+\tfrac{\eta}{N}\big]
\ \cup\
\big(\theta+\tfrac{2\pi j}{N}-\tfrac{\eta}{N},\,\theta+\tfrac{2\pi
j}{N}+\tfrac{\eta}{N}\big],
$$
and hence, by ,
The right-hand side involves only the total variation of dSU on
a union of boundary layers of thickness O(η/N). In
particular, if we have a rigidity-type control on the number of
eigenangles in arcs of length O(η/N), then the
contribution of these layers is o(1) after summing the harmonic
weights provided J is at least
logarithmic. We will implement this probabilistically in the uniform
comparison step, but for the present reduction it is enough to isolate
the corresponding remainder term.
The weight LJ defined in is
explicitly computable in Fourier series. For m ≠ 0, let
$$
\widehat{L}_J(m):=\frac{1}{2\pi}\int_{-\pi}^{\pi}L_J(u)e^{-imu}\,du.
$$
Since the summands in are even, L̂J(m)
is real and L̂J(−m) = L̂J(m).
A direct calculation gives
$$
\widehat{L}_J(m)
=\frac{1}{2\pi}\sum_{j=J}^{\lfloor N/4\rfloor}\frac{1}{j}\int_{-2\pi
j/N}^{2\pi j/N} e^{-imu}\,du
=\sum_{j=J}^{\lfloor N/4\rfloor}\frac{1}{j}\cdot \frac{\sin(2\pi m
j/N)}{\pi m}.
$$
The sum in j is a truncated
harmonic exponential sum, and it is at this point that the mesoscopic
scale MN = ⌊N/J⌋
appears: for |m| ≪ N/J, the
phase increment 2πm/N is small
compared to the lower limit 1/J in the harmonic sum, so one
expects L̂J(m)
to behave like 1/|m| up to an
error of order 1/J; for |m| ≫ N/J,
oscillation forces additional decay.
Rather than keeping LJ itself, it is
conceptually cleaner to replace it by a kernel with Fourier multiplier
11 ≤ |m| ≤ MN/|m|,
which is the multiplier relevant for WN, J.
Accordingly, define the 2π-periodic kernel
so that its Fourier coefficients satisfy
$$
\widehat{K}_J(m)=\frac{\mathbf{1}_{1\leq |m|\leq M_N}}{2|m|},\qquad
m\in\mathbb{Z}\setminus\{0\},
$$
with K̂J(0) = 0 (the
constant term is irrelevant for convolution against dSU,
whose total mass is 0). In particular,
KJ is a
truncated logarithmic kernel: as MN → ∞ one has
the classical identity
$$
\sum_{m=1}^{M}\frac{\cos(mu)}{m}
=-\log\big|2\sin(u/2)\big|+O(1)
\quad\text{for }u\not\equiv 0\ (\mathrm{mod}\ 2\pi),
$$
with an error uniform on compact subsets of 𝕊1 \ {0}.
We now record the reduction in a form tailored to later comparison with WN, J.
Lemma~ isolates two independent mechanisms behind the approximation . The first term in is a : it reflects that the raw harmonic windowing in j produces a kernel LJ whose Fourier multiplier is only approximately 1|m| ≤ MN/|m|. This contribution is purely analytic and is already small once J → ∞, provided we control ∥SU∥L∞.
The second term in is the genuinely ``zeros-only’’ difficulty: it measures the cost of replacing the random inverse-counting windows Ij(θ) by deterministic windows centered at θ. This term is small precisely when the inverse counting map is stable, i.e. when the endpoint fluctuations θ̃±j(θ) − (θ ± 2πj/N) are negligible compared to the scale 2πj/N. For CUE, the typical size of the counting error is logarithmic in N, so stability is ensured uniformly once j ≥ J with J a sufficiently large constant multiple of log N. In the next section we will combine this stability with standard rigidity/number-variance bounds to show that the second term in vanishes in probability uniformly in θ (with a harmless exclusion of tiny neighborhoods of eigenangles if one wishes to avoid pointwise issues at jumps). This will complete the conversion of DN, J into the convolution statistic KJ * dSU, at which point Fourier inversion will identify it with the trace field WN, J.
We now verify that the convolution statistic appearing in Lemma~ is the trace field WN, J. This is a deterministic Fourier identity which uses only the definition of the Stieltjes measure dSU and the fact that the eigenangles encode the traces of powers of U.
Recall that, as a signed 2π-periodic measure,
$$
dS_U(t)=\sum_{k=1}^N\delta_{\theta_k}(dt)-\frac{N}{2\pi}\,dt,
$$
and in particular, for each integer m ≠ 0,
(We used that the Lebesgue term has vanishing mth Fourier coefficient for m ≠ 0.)
Combining Lemma~ and Lemma~, we obtain the exact decomposition
with rN, J
satisfying the structural bound . Thus the proof of the uniform
reconstruction statement in (ii) reduces to showing that supθ|rN, J(θ)| → 0
in probability for J = ⌊clog N⌋.
Fix c > 0 and set J = ⌊clog N⌋ and
MN = ⌊N/J⌋.
We work with the decomposition
rN, J(θ) = rN, Jker(θ) + rN, Jinv(θ),
where
$$
r^{\mathrm{ker}}_{N,J}(\theta):=\big((K_J-L_J)*dS_U\big)(\theta),
\qquad
r^{\mathrm{inv}}_{N,J}(\theta):=
-\int_{\mathbb{R}}
\Bigg(\sum_{j=J}^{\lfloor
N/4\rfloor}\frac{\mathbf{1}_{I_j(\theta)}(t)-\mathbf{1}_{I_j^{(0)}(\theta)}(t)}{j}\Bigg)\,dS_U(t),
$$
so that is simply the crude bound |rN, Jinv(θ)| ≤ (⋯)
obtained by total variation. For the argument, we will not rely on this
total-variation estimate; instead we treat rN, Jinv(θ)
as a linear statistic of the determinantal process with a random test
function and control it through standard rigidity/number-variance
input.
The field rN, Jker
is a trigonometric polynomial with frequencies ≪ MN, and we
may express it directly in Fourier modes. Writing the Fourier series of
RJ := KJ − LJ
as RJ(u) = ∑m ≠ 0R̂J(m)eimu,
we have
For m = o(N)
we have $\mathbb{E}|\Tr(U^m)|^2=m$
(indeed min (m, N) = m),
and the Fourier analysis preceding Lemma~ yields the pointwise
estimate
with additional decay for |m| ≫ MN
(coming from the oscillation in L̂J(m)).
Restricting to |m| ≤ MN
(the only range relevant for the dominant part of the field) and using
orthogonality, we obtain the L2 bound
Since J ∼ clog N, this
gives log MN/J2 → 0,
hence rN, Jker(θ) → 0
in L2, uniformly in
θ.
To upgrade this to a supremum over θ, we use the fact that rN, Jker
is a trigonometric polynomial of degree O(MN);
hence it is Lipschitz with
$$
\| (r^{\mathrm{ker}}_{N,J})'\|_{L^\infty}
\ll M_N \sum_{m=1}^{M_N} |\widehat{R}_J(m)|\,|\Tr(U^m)|.
$$
By Cauchy–Schwarz and , the right-hand side has bounded second moment of
order o(1) ⋅ MN;
consequently, a standard discretization argument (evaluate on a grid of
mesh MN−2
and use the Lipschitz bound to pass from the grid to the continuum)
yields
We omit routine details, since any polynomial mesh finer than MN−1
suffices once one has an o(1)-control of the L2 size of the
coefficients in together with Bernstein inequalities for trigonometric
polynomials.
The inverse-neighbor maps θ ↦ θ±j(θ)
are well-defined Lebesgue-a.e. and extended by right-continuity. To
avoid any pointwise ambiguity at discontinuities of SU, it is
convenient to first prove uniform control on the set
$$
\Omega_\varepsilon:=\Big\{\theta\in\mathbb{S}^1:\
\dist(\theta,\{\theta_1,\dots,\theta_N\})\geq \varepsilon/N\Big\},
$$
and then pass to all θ using
right-continuity and the fact that 𝕊1 \ Ωε
has Lebesgue measure O(ε) and consists of N tiny arcs. In the present setting
one may take, for example, ε = N−9.
On Ωε, the
counting function is locally constant and the neighbor endpoints are
stable under small perturbations of θ. The key quantitative input is a
rigidity/number-variance bound for CUE: for any arc A ⊂ 𝕊1 of length |A|, the centered count ∫AdSU
is subgaussian with variance ≍ log (N|A|+2), uniformly
in the location of A. In
particular, for j ≥ J
the arc length 4πj/N is at least
of order (log N)/N,
so the typical counting error on such arcs is $O(\sqrt{\log j})$, and the relative error
$O(\sqrt{\log j}/j)$ tends to 0 as soon as j → ∞.
Using this (together with a union bound on a grid in θ and j and a monotonicity argument to
pass from the grid to the continuum), one proves that for every fixed
α > 0 there exists Cα < ∞ such
that, with probability at least 1 − N−α,
and in particular Δj(θ) = o(j/N)
uniformly for j ≥ J
and θ ∈ Ωε.
Fix θ ∈ Ωε
and define the random test function
$$
g_\theta(t):=\sum_{j=J}^{\lfloor
N/4\rfloor}\frac{\mathbf{1}_{I_j(\theta)}(t)-\mathbf{1}_{I_j^{(0)}(\theta)}(t)}{j},
$$
so that rN, Jinv(θ) = −∫gθ(t) dSU(t).
On the event , each term 1Ij(θ) − 1Ij(0)(θ)
is supported on a symmetric difference consisting of two short arcs
whose total length is O(Δj(θ)),
with Δj(θ)
much smaller than the underlying scale j/N. Moreover, since the
harmonic weights are summable, gθ is uniformly
bounded:
$$
\|g_\theta\|_{L^\infty}\leq \sum_{j=J}^{\lfloor
N/4\rfloor}\frac{2}{j}\ll \log(N/J)\ll \log N.
$$
The relevant feature is that gθ has on low
modes once j ≥ J,
because each summand is a small translation error of a fixed-shape
window. More precisely, one checks from the explicit Fourier transform
of an interval indicator that for each m ≠ 0,
$$
|\widehat{g_\theta}(m)|
\ll \sum_{j=J}^{\lfloor N/4\rfloor}\frac{1}{j}\cdot
|m|\,\Delta_j(\theta)\cdot \frac{1}{|m|}
\ll \sum_{j=J}^{\lfloor N/4\rfloor}\frac{\Delta_j(\theta)}{j},
$$
and therefore, using ,
$$
|\widehat{g_\theta}(m)|
\ll \frac{1}{N}\sum_{j=J}^{\infty}\frac{\sqrt{\log j}}{j^{3/2}}
\ll \frac{\sqrt{\log J}}{N\sqrt{J}}.
$$
Inserting this into the determinantal variance formula for CUE linear
statistics (in the form $\Var(\int h\,dS_U)\ll
\sum_{m\neq 0}\min(|m|,N)\,|\widehat{h}(m)|^2$ for piecewise
H1/2 functions
h) yields
$$
\sup_{\theta\in\Omega_\varepsilon}\Var\big(r^{\mathrm{inv}}_{N,J}(\theta)\big)
\ll \sum_{m=1}^{N} m\,|\widehat{g_\theta}(m)|^2
\ll N^2 \cdot \Big(\frac{\log J}{N^2 J}\Big)
\ll \frac{\log J}{J}
\to 0,
$$
since J → ∞. Hence, for each
fixed θ ∈ Ωε,
we have rN, Jinv(θ) → 0
in L2, and in
particular in probability.
To obtain a bound over θ ∈ Ωε,
we again discretize. The map θ ↦ rN, Jinv(θ)
is piecewise constant between eigenangles (because the windows Ij(θ)
change only when θ crosses an
eigenangle), while the deterministic windows Ij(0)(θ)
vary continuously in θ. It
follows that θ ↦ rN, Jinv(θ)
has controlled oscillation on Ωε on scales
≫ 1/N. Choosing a grid {θℓ} of mesh
N−2 contained in
Ωε and
applying a union bound together with the L2 estimate above
yields
(The passage from the grid to the continuum uses that on Ωε the neighbor
maps are locally constant and the deterministic part is Lipschitz, so no
large fluctuations can occur between nearby grid points.)
Combining and gives
$$
\sup_{\theta\in\Omega_\varepsilon}|r_{N,J}(\theta)|\xrightarrow{\mathbb{P}}0.
$$
Finally, if we take the supremum over all θ ∈ 𝕊1, we may either (a)
adopt the right-continuous conventions for θ ↦ θ±j(θ)
and note that the exceptional set consists only of the eigenangles
themselves, or (b) keep the supremum over Ωε as stated. In
either interpretation, recalling we obtain
$$
\sup_{\theta\in\mathbb{S}^1}|D_{N,J}(\theta)-W_{N,J}(\theta)|
=\sup_{\theta\in\mathbb{S}^1}|r_{N,J}(\theta)|
\xrightarrow{\mathbb{P}}0,
$$
which is precisely the uniform reconstruction statement in (ii).
The results proved above are asymptotic and distributional, and the comparison statement in~(ii) is of a uniform, probabilistic nature. It is therefore useful to record a small collection of numerical experiments that (a) illustrate the fields WN, J and DN, J for moderate N, (b) visually corroborate the reconstruction phenomenon DN, J ≈ WN, J when J ≍ log N, and (c) check the covariance structure and approximate Gaussianity predicted by the GFF limit in~(i). We emphasize that this section is purely computational and plays no role in the proofs; it is best read as a diagnostic guide for implementation choices and for interpreting the observables.
A standard method to sample U ∼ Haar(U(N)) is
to draw a complex Ginibre matrix G with i.i.d. entries gab ∼ 𝒩(0, 1) + i𝒩(0, 1),
perform a QR decomposition G = QR with Q unitary and R upper-triangular, and then
normalize the phases by setting
$$
U := Q \cdot
\mathrm{diag}\Big(\frac{R_{11}}{|R_{11}|},\dots,\frac{R_{NN}}{|R_{NN}|}\Big)^{-1}.
$$
We then compute the eigenvalues of U (or directly its eigenangles) and
sort them as 0 ≤ θ1 < ⋯ < θN < 2π.
All plots below are based on these eigenangles; the only additional
ingredients are finite sums and simple nearest-neighbor manipulations.
For the comparison in~(ii), it is numerically preferable to work
directly with eigenangles throughout, so that both WN, J
and DN, J
can be evaluated without explicitly forming high powers Um.
Although WN, J
is defined in terms of traces $\Tr(U^m)$, for moderate N and mesoscopic cutoff MN = ⌊N/J⌋
one can compute these traces via the eigenangles:
$$
\Tr(U^m)=\sum_{k=1}^N e^{i m \theta_k},\qquad 1\leq m\leq M_N.
$$
Thus, for a given grid θ ∈ {θℓgrid}ℓ = 1L ⊂ 𝕊1,
we may evaluate
$$
W_{N,J}(\theta^{\mathrm{grid}}_\ell)
=\sum_{m=1}^{M_N}\frac{1}{m}\Re\!\Big(\Big(\sum_{k=1}^N e^{i m
\theta_k}\Big)e^{-i m \theta^{\mathrm{grid}}_\ell}\Big).
$$
The naive cost of this evaluation is O(NMN + LMN)
if one first computes all traces and then evaluates the trigonometric
polynomial on the grid. In the regime relevant for~(ii), say J = ⌊clog N⌋, we
have MN ∼ N/(clog N),
so for N up to a few thousand
this is entirely feasible.
Two implementation details are worth noting.
The observable WN, J
is mean-zero and uses only m ≥ 1, so one may simply omit m = 0. Numerically, it is convenient
to store the coefficients
$$
A_m:=\frac{\Tr(U^m)}{m},\qquad 1\leq m\leq M_N,
$$
and then evaluate ℜ(∑m ≤ MNAme−imθ).
We view angles modulo 2π and
use the principal representatives in [0, 2π). When evaluating e±imθ,
the periodicity is automatic, but when comparing with DN, J(θ)
it is convenient to ensure that both are evaluated on the same grid in
[0, 2π) and plotted with
wrap-around.
The field DN, J(θ)
depends only on the ordered eigenangles and the neighbor maps θ ↦ θ±j(θ).
A practical way to evaluate DN, J
on a grid is as follows. For each θ, find the index k(θ) such that θk(θ) ≤ θ < θk(θ) + 1
(with cyclic convention at 2π). Then the jth neighbor to the right is θ+(j)(θ) = θk(θ) + j
and to the left is θ−(j)(θ) = θk(θ) − j + 1
(again cyclic), and we set
$$
\delta_j(\theta)=\frac{N}{2\pi}\big(\theta_{+j}(\theta)-\theta_{-j}(\theta)\big)-2j,
\qquad
D_{N,J}(\theta)=\sum_{j=J}^{\lfloor
N/4\rfloor}\frac{\delta_j(\theta)}{j}.
$$
In practice, one must interpret θ+j(θ) − θ−j(θ)
as the positive arc-length from θ−j(θ)
to θ+j(θ)
in cyclic order; equivalently one may lift the eigenangles to an
increasing sequence on ℝ by appending
θk + N = θk + 2π
and use this lift to compute differences without ambiguity.
Two further numerical remarks parallel the technical points in the proof.
The maps θ ↦ θ±j(θ)
are piecewise constant with jumps at eigenangles. If one evaluates on a
grid that includes eigenangles, one may adopt the right-continuous
convention used above, or else simply exclude tiny neighborhoods of
eigenangles. Numerically, we recommend either evaluating at midpoints
$\frac12(\theta_k+\theta_{k+1})$ (which
avoids discontinuities by construction), or removing points within
distance ε/N of any
θk, with
ε chosen modestly (in
experiments, any polynomially small choice behaves similarly).
The upper limit ⌊N/4⌋ is not
canonical; it is chosen to keep the symmetric window [θ−j(θ), θ+j(θ)]
away from wrapping around the circle in a degenerate way and to remain
in a mesoscopic regime where the harmonic weights ∑1/j behave as expected.
Numerically, varying the upper cutoff within a fixed fraction of N changes only the very
low-frequency component of the resulting field.
For a typical run with N between 103 and 104 and with J = ⌊clog N⌋ (say c between 1/2 and 2), one may compute both fields on a uniform grid θℓgrid = 2πℓ/L with L on the order of 103–104, or else on the eigenangle midpoints. Plotting θ ↦ WN, J(θ) and θ ↦ DN, J(θ) on the same axes typically reveals two visually indistinguishable log-correlated profiles, and plotting the difference θ ↦ DN, J(θ) − WN, J(θ) yields a comparatively flat curve.
In these experiments, the magnitude of the difference is strongly
affected by J. If J is taken too small (for instance,
bounded or growing very slowly), the discrepancy field DN, J
includes many short-range contributions that are sensitive to the
discontinuities of the inverse counting map and the local fluctuations
of gaps; the resulting error is visible at moderate N. By contrast, when J is of order log N, the harmonic sum begins
sufficiently far out that the stability mechanism (as in the proof
of~(ii)) becomes effective, and the difference is dominated by a
small-amplitude, slowly varying remainder. This is consistent with the
form of the decomposition
DN, J(θ) − WN, J(θ) = rN, J(θ),
where the principal contributions to rN, J
arise from (i) the mismatch between the ideal Fourier cutoff kernel and
the discrepancy-induced kernel, and (ii) the random perturbation of
window endpoints caused by inverting the local counting function.
From a computational perspective, one may use the empirical
quantity
ΔN, J(∞) := maxℓ|DN, J(θℓgrid) − WN, J(θℓgrid)|
as a proxy for the supremum norm in~(ii). Repeating over many draws of
U yields an empirical
distribution of ΔN, J(∞)
which decreases as N increases
when J = ⌊clog N⌋,
whereas it stabilizes or decreases much more slowly if J is kept fixed.
To check the GFF covariance structure predicted by~(i), we may
estimate the covariance function
CN, J(ϑ) := 𝔼[WN, J(ϑ) WN, J(0)]
by averaging WN, J(ϑ)WN, J(0)
over many independent Haar samples. Lemma~3.1 predicts that
$$
C_{N,J}(\vartheta)\approx
\frac12\sum_{m=1}^{M_N}\frac{\cos(m\vartheta)}{m},
$$
up to an error of order 1/J in
the regime MN = o(N).
Numerically, the agreement is typically excellent already for moderate
N provided one uses enough
samples to reduce Monte Carlo noise. Moreover, as MN grows, the
truncated sum approaches the classical logarithmic kernel
$$
\frac12\sum_{m=1}^{M}\frac{\cos(m\vartheta)}{m}
= -\frac12\log\big|2\sin(\vartheta/2)\big| +
O\Big(\frac{1}{M\,|\sin(\vartheta/2)|}\Big),
$$
valid away from ϑ ≡ 0 (mod 2π). Thus,
plotting CN, J(ϑ)
against $-\frac12\log|2\sin(\vartheta/2)|$ (after
subtracting an M-dependent
constant if desired) gives a direct visual confirmation of
log-correlation at mesoscopic cutoffs.
One may similarly test the covariance of the zeros-only field DN, J; when J = ⌊clog N⌋, its covariance function closely tracks that of WN, J, providing an additional check of the reconstruction phenomenon at the level of second moments.
A complementary diagnostic, closer to the mechanism behind
Proposition~C and Theorem~A, is to examine the distribution of
normalized traces
$$
X_m:=\frac{\Tr(U^m)}{\sqrt{m}},\qquad 1\leq m\leq M_N,
$$
for MN = o(N).
Empirically, for each fixed m
the real and imaginary parts of Xm appear
approximately $\mathcal{N}(0,\tfrac12)$, and for distinct
m the cross-correlations are
close to 0. One may quantify this by
computing the empirical covariance matrix of (Xm)m ≤ MN
over repeated draws of U, or
by producing Q–Q plots of ℜ(Xm) and ℑ(Xm) against
standard normals. While such tests are necessarily limited by sample
size and by the fact that MN is not
asymptotically negligible for very small N, they align well with the
principle that the low Fourier modes of the CUE counting measure behave
essentially as independent Gaussians up to the o(N) threshold.
These checks also illuminate the meaning of the H−s topology in~(i): the Sobolev norm effectively weights the mth Fourier coefficient by |m|−s, so that convergence is robust to the omission of high modes and to modest distortions near the cutoff.
Finally, it is helpful to verify that both observables behave correctly on simple deterministic configurations, independent of Haar randomness.
If we take θk = 2πk/N,
then SU ≡ 0, hence
(KJ * dSU) ≡ 0
for any kernel KJ with mean
zero, and therefore WN, J ≡ 0.
On the other hand, θ+j(θ) − θ−j(θ) = 4πj/N
for all θ away from the grid
points, giving δj(θ) = 0
and thus DN, J ≡ 0.
This confirms that both observables measure deviations from uniform
density rather than absolute density.
If we artificially place a macroscopic fraction of the points in a
shorter arc while leaving the remaining points sparse elsewhere, then
the symmetric neighbor window around θ inside the dense region has length
smaller than 4πj/N, yielding
δj(θ) < 0
for a range of j and hence a
negative excursion of DN, J(θ).
The trace field WN, J
exhibits the same qualitative sign and profile, consistent with the
interpretation of both fields as density waves.
In summary, these numerical illustrations serve three purposes: they provide a concrete view of the random log-correlated profiles generated by WN, J; they demonstrate that the zeros-only construction DN, J captures the same mesoscopic field when J ≍ log N; and they offer practical checks (covariance matching and trace Gaussianity) that mirror the theoretical inputs used in the proofs. The next section places these observables in a broader context, relating them to carrier-wave heuristics and to the corresponding constructions for the Riemann zeta function.
We close by interpreting the observables introduced above as canonical ``carrier wave’’ functionals, by relating the zeros-only reconstruction to Farmer’s density-wave methodology, and by indicating a few consequences and possible extensions, especially those suggested by the random-matrix–to–zeta dictionary.
The definition
$$
W_{N,J}(\theta)=\sum_{m=1}^{M_N}\frac{\Re\big(\Tr(U^m)e^{-im\theta}\big)}{m}
$$
is not ad hoc: it is precisely the low-frequency part of the
log-characteristic polynomial. Indeed, on the unit circle one has the
identity (in a principal value sense)
$$
\log\big|\det(I-e^{-i\theta}U)\big|
=-\sum_{m\geq 1}\frac{\Re\big(\Tr(U^m)e^{-im\theta}\big)}{m},
$$
so that WN, J
is the truncation of −log |det (I − e−iθU)|
to Fourier modes 1 ≤ m ≤ MN.
The weight 1/m is therefore
forced by harmonic analysis and by the logarithm. From this perspective,
Theorem~A is a mesoscopic version of the well-known principle that the
log-characteristic polynomial is asymptotically a log-correlated
Gaussian field after suitable regularization: we are not approximating
the full field at microscopic resolution, but rather its
smoothed/low-frequency content, and in that regime a clean convergence
in H−s is
available.
The cutoff MN = ⌊N/J⌋ is also natural. The trace statistics $\Tr(U^m)$ are asymptotically Gaussian only up to m = o(N), and the determinantal/Szeg inputs used in Proposition~C become stable precisely when the Fourier support is o(N). Thus J → ∞ is not an auxiliary technical condition: it encodes the requirement that we stay below the ``Nyquist’’ threshold where the finite-size constraints of U(N) begin to correlate the modes.
Theorem~B provides a zeros-only proxy DN, J which, for J ≍ log N, uniformly reconstructs the same mesoscopic field as WN, J. Conceptually, the existence of such a proxy rests on two complementary facts.
The map θ ↦ θ±j(θ)
is an inverse to the local counting function, and symmetric windows
[θ−j(θ), θ+j(θ)]
encode integrated counting errors. Summation-by-parts identities then
convert harmonic weights ∑δj(θ)/j
into convolutions of SU against an
explicit kernel. This mechanism is deterministic and is the analytic
heart of Lemma~6.1.
Although inversion is discontinuous at eigenangles, it becomes stable
when we average over scales j ≥ J with J → ∞. One may view J as a regularization parameter:
small j are sensitive to
individual gaps, while large j
average over enough points to suppress the local instabilities of the
inverse counting map. The requirement J ≍ log N in the uniform
statement is then consistent with the familiar heuristic that log N is the scale at which union
bounds and rigidity estimates begin to control suprema over θ.
In this light, the field DN, J is not merely an alternative estimator of WN, J; it is an intrinsic object attached to the point process itself. It uses only nearest-neighbor geometry, and it does so in the uniquely natural way dictated by the harmonic weights that correspond to a logarithm.
Farmer’s density-wave plots for zeta are based on the idea that
fluctuations in the zeros (encoded through neighbor discrepancies) carry
a coherent long-range signal, which he interprets as a ``wave’’
modulating the local density. The observable
$$
\sum_{j\geq J}\frac{\delta(\gamma_{-j},\gamma_{j})}{j}
$$
in that context is the direct analogue of our DN, J,
and the qualitative features observed numerically (large-scale
oscillations, apparent log-correlation, and stability once J is taken moderately large) match
the behavior of DN, J
in the CUE setting.
Theorem~B may therefore be read as a random-matrix justification of the following interpretation: the density wave is the low-frequency component of the log-characteristic polynomial (or, on the zeta side, the low-frequency component of log |ζ| after regularization), and the neighbor-discrepancy functional is a zeros-only method to recover that component. In the CUE model this is an identity up to a small remainder, and the size of the remainder is controlled uniformly when J ≍ log N.
We emphasize a subtle but important point. The zeros-only reconstruction does not claim that information in the characteristic polynomial is encoded in the local gaps. Rather, it identifies a particular, canonical projection of the field—the Fourier band 1 ≤ m ≤ N/J with the logarithmic weights—as being recoverable from neighbor statistics. This is exactly the ``carrier’’ part in carrier-wave heuristics: the component that varies on macroscopic/mesoscopic scales and can be viewed as a random environment in which microscopic fluctuations live.
Once one accepts the GFF limit in Theorem~A, it is natural to ask what this implies for extremes of the underlying fields. For log-correlated fields, maxima typically exhibit a log –log log structure (with model-dependent constants), and the characteristic polynomial of CUE is conjectured to fall into this universality class. Our results do not directly address the maximum of the full log |det (I − e−iθU)|, but they do yield two consequences at the mesoscopic level.
Because DN, J
and WN, J
are uniformly close for J ≍ log N, any functional
that is continuous with respect to the supremum norm (in particular, the
maximum over θ on a fine grid,
or smoothed maxima) has the same asymptotics for both fields. Thus the
extreme-value behavior of WN, J
can, in principle, be studied using only eigenangles, without
traces.
The truncation parameter J
separates the log-correlated coarse field'' from thefine
field’’ coming from modes m > MN.
In many arguments for log-correlated maxima, one decomposes the field
into a sum of approximately independent increments across scales. Here
J provides an explicit coarse
component with a clean GFF limit. A natural next problem is to quantify
the dependence between the coarse field WN, J
and the complementary high-frequency tail, and to determine whether
DN, J
can be augmented with additional local terms to capture some of that
tail.
A plausible working conjecture is that, at least for J in a polylogarithmic range, the maxima of WN, J already exhibit the correct leading-order growth for the maxima of the fully regularized log-characteristic polynomial, with an error that can be controlled as J → ∞. Establishing such a statement would require nontrivial multiscale control (and is well beyond the scope of the present work), but Theorem~B indicates that the relevant coarse component can be accessed from zeros alone.
The motivation for a zeros-only carrier wave is that, for $\zeta(\tfrac12+it)$, the
prime side'' and thezero side’’ are connected by the
explicit formula. Heuristically, one expects a decomposition in which a
smoothed log |ζ| behaves like
a log-correlated Gaussian field whose Fourier modes correspond to prime
sums, while the zeros encode the same object through fluctuations of the
counting function of zeros on the critical line. Farmer’s methodology
suggests that one can bypass primes and recover the carrier wave
directly from zeros; Theorem~B proves the analogous statement in the CUE
model.
To make an analogous theorem for zeta precise, one would need replacements for the two main inputs used here.
Proposition~C is a statement about joint Gaussianity of $\Tr(U^m)$ for m = o(N). On the
zeta side, one would seek joint Gaussianity (or at least asymptotic
quadratic log-moments) for smoothed prime sums
$$
\sum_{p\leq T^{1/J}} \frac{p^{-it}}{\sqrt{p}}\, w(\log p),
$$
in a range corresponding to m ≤ T1/J,
with J → ∞ and J ≍ log log T playing the
role of log N. The ``recipe’’
and the ratios conjecture are designed precisely to predict such joint
moments for Euler products and logarithmic derivatives, and thus provide
a plausible substitute for the Szeg/Toeplitz machinery available in
CUE.
Lemma~6.1 is essentially deterministic and only uses that the zeros form
a simple point configuration on a circle. For zeta, the zeros lie on the
line and the relevant object is the centered zero counting function
S(t), together with
the inverse map giving the location of the jth zero to the left/right of a
point. The same summation-by-parts mechanism should convert neighbor
discrepancies into a convolution of dS against a kernel with
Fourier transform 1|m| ≤ M/|m|.
The analytic difficulty is then to control the remainder uniformly,
which in CUE is done using rigidity/number-variance bounds. On the zeta
side one would require uniform control on the fluctuations of S(t) on short intervals (at
least in probability with respect to t), at a strength comparable to what
is provided unconditionally for CUE by determinantal structure.
If these two ingredients can be established (even conditionally), the CUE theorem suggests a precise formulation of Farmer’s principle: at a cutoff J comparable to log log T, the density-wave functional built from neighbor discrepancies of zeros should approximate the low-frequency part of $\log|\zeta(\tfrac12+it)|$, uniformly in t over long ranges after excluding tiny neighborhoods of zeros.
We record a few concrete directions suggested by the present results.
The statements above are qualitative (o(1) in probability). It is natural
to ask for quantitative rates in N and the dependence on c in J = ⌊clog N⌋. In
particular, one expects a tradeoff: larger J improves the stability of
inversion but removes more low frequencies from the reconstruction.
Making this tradeoff explicit would clarify which portion of the field
is practically recoverable from zeros at a given N.
The determinantal structure of CUE is convenient but not conceptually
essential for the discrepancy-to-convolution step. One may ask whether
analogous zeros-only reconstructions hold for other circular ensembles,
for β-ensembles on the line,
or for mesoscopic statistics in general log-gas models where rigidity is
available. In such settings one expects the limiting log-correlated
field to persist (with β-dependent variance), and the
reconstruction should then identify a canonical coarse field determined
by the configuration alone.
Theorem~A identifies the H−s limit of the
coarse field. A sharper description of the joint law of (WN, J, tail),
and of how much of the tail is encoded in higher-order local statistics
of eigenangles, would bring the discussion closer to full carrier-wave
decompositions.
In sum, the picture emerging from (i)–(ii) is that there is a canonical mesoscopic carrier wave associated with the eigenangle configuration, and that it admits two equivalent representations: a trace/Fourier representation WN, J and a purely geometric neighbor-discrepancy representation DN, J. The former connects directly to the log-characteristic polynomial and the circle GFF; the latter connects directly to zeros-only methodologies and, in particular, to Farmer’s density-wave plots. The random matrix model thus supplies a setting in which the carrier-wave philosophy can be formulated as a theorem, and it suggests a concrete route by which analogous statements for ζ may be attacked under appropriate probabilistic input on zero statistics and prime sums.