Zeros of Gaussian Weyl–Heisenberg Functions and Hyperuniformity of Charge

We study Gaussian random functions on the complex plane whose stochastics are invariant under the Weyl–Heisenberg group (twisted stationarity). The theory is modeled on translation invariant Gaussian entire functions, but allows for non-analytic examples, in which case winding numbers can be either positive or negative. We calculate the first intensity of zero sets of such functions, both when considered as points on the plane, or as charges according to their phase winding. In the latter case, charges are shown to be in a certain average equilibrium independently of the particular covariance structure (universal screening). We investigate the corresponding fluctuations, and show that in many cases they are suppressed at large scales (hyperuniformity). This means that universal screening is empirically observable at large scales. We also derive an asymptotic expression for the charge variance. As a main application, we obtain statistics for the zero sets of the short-time Fourier transform of complex white noise with general windows, and also prove the following uncertainty principle: the expected number of zeros per unit area is minimized, among all window functions, exactly by generalized Gaussians. Further applications include poly-entire functions such as covariant derivatives of Gaussian entire functions.


Gaussian Weyl-Heisenberg Functions
We study zero sets of Gaussian circularly symmetric random functions on the plane F : C → C whose covariance kernel is given by twisted convolution: (1.1) Here, H : C → C is a function called twisted kernel. Gaussianity means that for each z 1 , . . . , z n ∈ C, (F(z 1 ), . . . , F(z n )) is a normally distributed complex random vector. Circularity means that F ∼ e iθ F, for all θ ∈ R, and implies that F has vanishing expectation and pseudo-covariance, i.e., E [F(z)] = 0, E [F(z)F(w)] = 0, for all z, w ∈ C. Hence, the stochastics of F are completely encoded in the twisted kernel (1.1). While the covariance structure (1.1) without the complex exponential factor would mean that F is stationary, the presence of the oscillatory factor means that F is twisted stationary: In other words, the stochastics of F are invariant under twisted shifts: Hence, G is a Gaussian entire function on the plane with correlation kernel given by the Bargmann-Fock kernel [35], and its zero set is well-studied [24]. In terms of G, the twisted stationarity property of F (1.3) is an instance of the projective invariance property [29], and, indeed, reflects the invariance of the stochastics of G under Bargmann-Fock shifts: (1.4) Example 1.2 (The short-time Fourier transform of complex white noise) Given a window function g ∈ S(R), the short-time Fourier transform of a function f : R → C is (1.5) The short-time Fourier transform is a windowed Fourier transform, and the value V g f (x, y) represents the influence of the frequency y near x. As localizing window g, one often chooses Gaussian, or, more generally, Hermite functions, as these optimize several measures related to Heisenberg's uncertainty principle.
In signal processing, the short-time Fourier transform is often used to analyze functions (called signals) contaminated with random noise N . The corresponding zero sets play an important role in many modern algorithms, for example in the dynamics of certain nonlinear procedures to sharpen spectrograms [16,Chap. 12] or in the design of filtering masks, where landmarks are chosen guided by the statistics of zero sets [15].
Of particular interest are the zeros of the short-time Fourier transform of complex white Gaussian noise V g N [16,Chap. 15]. While many applications demand the use of different window functions g-see, e.g., [16,Sect. 10.2]-zero-statistics for the STFT are currently only understood for Gaussian windows [5,6], as these facilitate the application of the theory of Gaussian entire functions (Example 1.1). One main motivation for this article is to obtain zero-statistics for general windows g, including for example Hermite functions. (Some related numerics can be found in [16,Chap. 15].) With an adequate distributional interpretation, the STFT of complex white Gaussian noise with respect to a Schwartz window function g defines a smooth circularly symmetric Gaussian function on the plane. Twisted stationarity is revealed by the transformation F(x + iy) := e −i xy · V g N x/ √ π, −y/ √ π , (1.6) which, as shown in Sect. 6.1, indeed yields a GWHF. The twisted stationarity of F reflects the invariance of the stochastics of complex white noise under time-frequency shifts Basic questions about zero sets of short-time Fourier transforms also underlie problems about the spanning properties of the time-frequency shifts of a given function (Gabor systems) [23,26], or about Berezin's quantization [22]. The study of random counterparts provides a first form of average case analysis for such problems.

Example 1.3 (Derivatives of GEF)
The covariant derivative of an entire function G : and it is distinguished among other differential operators of order 1 because it commutes with the Bargmann-Fock shifts (1.4). As a consequence, if G is a Gaussian entire function, as in Example 1.1, the stochastics of∂ * G are also invariant under Bargmann-Fock shifts, and the transformation yields a GWHF. The corresponding twisted kernel is computed in Sect. 6.5. Zeros of covariant derivatives are instrumental in the description of vanishing orders of analytic functions [11,13]. They are also important in the study of weighted magnitudes of analytic functions G. For example, the amplitude A(z) = e − 1 2 |z| 2 |G(z)| of an entire function G satisfies Thus, the critical points of the amplitude of a Gaussian entire function G are exactly the zeros of the GWHF (1.7)-see also [12,14]. The squared amplitude A 2 (z) is also of interest, as it corresponds after normalization to the spectrogram of complex white noise with a Gaussian window (i.e., the squared absolute value of the STFT (1.5)) [5]; see also [6,Corollary 2.8].
Example 1.4 (Gaussian poly-entire functions) Iterated covariant derivatives of an analytic function G 0 , are not themselves analytic, but satisfy a higher order Cauchy-Riemann condition ∂ q G = 0, (1.10) known as poly-analyticity [4]. In Vasilevski's parlance [33], (1.9) is a true or pure poly-entire function, while the more general solution to (1.10), with G 0 , . . . , G q−1 entire, is a fully poly-entire function. Random Gaussian poly-entire functions of either pure of full type are defined by (1.9) and (1.11), letting G 0 , . . . , G q−1 be independent Gaussian entire functions.
Poly-entire functions are important in statistical physics, in the analysis of high energy systems of particles [2], and we expect their random analogs to also be useful in that field.

Standing Assumptions
The positive semi-definiteness of the covariance kernel of a GWHF F reads as follows: 1 As a consequence, H (−z) = H (z) and H (0) ≥ 0. To avoid trivial cases, we assume that H (0) = 0, since, otherwise, F would be almost surely constant. We furthermore impose the normalization We also assume that 14) which means that no two samples F(z), F(w) with z = w are deterministically correlated, as (1.14) amounts to the invertibility of the joint covariance matrix . We also assume a certain regularity of the twisted kernel: H is C 2 in the real sense, (1.15) and denote the corresponding derivatives with supraindices; e.g., H (1,1) Finally, we will always assume that F has C 2 paths in the real sense: Almost every realization of F is a C 2 (R 2 ) function. (1.16) This is the case, for example, if H ∈ C 6 (R 2 ) in the real sense [19,Theorem 5], but also other weaker assumptions suffice (see [

Zero Sets
We are mainly interested in the zero set of a GWHF F, encoded in the random measure (1.17) where δ z denotes the Dirac measure at z. This measure properly encodes the zero set of F, because, as we prove in Proposition 3.2 below, under the standing assumptions the zeros of F are almost surely simple and non-degenerate (i.e., as a map on R 2 , the differential matrix of F is invertible). Our first result describes the first point intensity of zero sets of GWHF. Theorem 1.6 (First intensity of zero sets) Let F be a GWHF with twisted kernel H satisfying the standing assumptions. Then Z F is a stationary random measure with first intensity: Concretely, for every Borel set E ⊆ C: In addition, H ≥ 0, and therefore ρ 1 ≥ 1/π.
In many important cases, the twisted kernel H is radial, and the expression for the first point intensity can be simplified.

Corollary 1.7
Let F be as in Theorem 1.6. Assume further that H (z) = P |z| 2 , where P : R → R is C 2 (R). Then P (0) ≤ −1/2 and the first point intensity of the zero set of F is .
We mention some applications of Theorem 1.6; these are further developed in Sect. 6. In the context of Examples 1.3 and 1.4 we obtain the following.

Theorem 1.8
The first intensity of the zero set of a true-type poly-entire function as in (1.9) is 1 π q − 1 2 + 1 4q−2 , while that of a full-type one as in (1.11) is 1 2π q + 1 q .
The base case q = 1 is well-known as it corresponds to a Gaussian entire function (Example 1.1), and follows from more general results [24,Sect. 2], while the case q = 2 is implicit in [12,14], since, by (1.8), it corresponds to the number of critical points of the weighted magnitude of a GEF. For large q we see that a true-type poly-entire function has on average ≈ q π zeros per unit area, while one of full-type has ≈ q 2π zeros per unit area. As a second application, we consider the short-time Fourier transform (Example 1.2), and obtain the following. Theorem 1.9 (First intensity of zeros of STFT of complex white noise) Let g : R → C be a Schwartz function normalized by ||g|| 2 = 1, and consider the following uncertainty constants: Then the zero set of V g N , i.e., the STFT of complex white noise with window g, has first intensity: Concretely, for every Borel set E ⊆ C: The constants c 1 , . . . , c 5 are real. When g is real-valued, the expression for ρ 1,g further simplifies because c 4 = c 5 = 0.
If we interpret |g(t)| 2 as a probability density on R, the uncertainty constants c 1 and c 2 correspond to the expected value and expected spread around the origin. The constants c 3 and c 4 have a similar meaning with respect to the Fourier transform of g. The constant c 5 is more subtle to interpret, as it involves correlations between g and its Fourier transform.
We spell out the particular case of Theorem 1.9 for Hermite windows: Numerical simulations related to the zeros of the STFT of white noise with h 0 and h 1 as windows can be found in [16,Chap. 15]-see also Fig. 1. Note that the expression in Corollary 1.10 is minimal for r = 0 (Gaussian case). We will prove that this is in fact an instance of a general phenomenon. Theorem 1.11 (Uncertainty principle for the zeros of the STFT of white noise) Under the assumptions of Theorem 1.9, the minimal value of ρ 1,g is 1, and it is attained exactly when g is a generalized Gaussian; that is, To compare, we note that, in terms of the uncertainty constants (1.22), Heisenberg's uncertainty inequality reads: where ||g|| 2 = 1, and is saturated by (translated and linearly modulated) Gaussian functions, see, e.g., [17,Corollary 1.35]. Generalized Gaussians (1.25) are sometimes called squeezed states and minimize a refined version of (1.26) that involves the constant c 5 in (1.22), known as the Robertson-Schrödinger uncertainty relations [32]. The proof of Theorem 1.11 exploits the invariance of squeezed states under the canonical transformations of the time-frequency plane (Weyl operators and metaplectic rotations [17]).

Charged Zeros
We now look into weighting each zero z of a GWHF F with a charge ±1, according to whether F preserves or reverses orientation around z. More precisely, we inspect the differential matrix D F of F considered as F : R 2 → R 2 and define When zeros are interpreted as phase-singularities, charges correspond to the strength of their vorticity [9]. Charges also appear naturally in the study of critical points of random functions, as one investigates the signature of corresponding Hessian matrices-see [12,Sect. 3.1] for an extended discussion.
We encode charged zeros into the random measure: Our next result shows that the corresponding first intensity is independent of the twisted kernel H .
As mentioned after (1.17), in the situation of Theorem 1.12 the zeros of F are almost surely non-degenerate, and consequently κ z = ±1. The fact that the intensity of charged zeros is constant is non-trivial, and shown in Sect. 4. Charges are straightforward to interpret in Examples 1.1, 1.3, and 1.4 , as the transformation G(z) → F(z) = e −|z| 2 /2 G(z) preserves the sign of the Jacobian at a zero-see Sect. 6.7. In the case of Gaussian entire functions, all zeros are positively charged due to the conformality of analytic functions, and, indeed, the first intensities prescribed by Theorems 1.6 and 1.12 coincide. On the other hand, Theorem 1.8 shows that a higher order poly-entire function has a large number of expected zeros per unit area, while, according to Theorem 1.12, most of the corresponding charges cancel. Charges are, in expectation, in a certain equilibrium around the universal density 1/π.
While zeros of first-order true poly-entire functions correspond to critical points of weighted magnitudes of Gaussian entire functions (Example 1.3), their charges summarize the signatures of the corresponding Hessian matrices-see [12,Sect. 3.1] or Sect. 6.8. In fact, as we show in Sect. 6.8, Theorem 1.12 can be used to rederive a particular case of [12,Corollary 5].
For the STFT of white noise (Example 1.2) the quantities related to charge are and Theorem 1.12 gives the following.

Fluctuation of Aggregated Charge
While a general GWHF can have many expected zeros (Theorem 1.6), the corresponding expected charges almost balance out (Theorem 1.12), adding up to the universal density 1/π. We now look into the stochastic fluctuation of charge when aggregated inside large observation sets, and the extent to which equilibrium is observed at large scales. A point process is called hyperuniform if the variance of the number of particles within an observation disk of radius R is asymptotically smaller than the corresponding expected number of points [30]. Such fluctuations are also called non-extensive [20], and are anomalously small in comparison to those in ordinary fluids and amorphous solids. Originally introduced in material science, hyperuniformity provides a unified framework to classify crystals and quasicrystals. The notion was subsequently developed into an abstract statistical notion, and found applications in a broad range of topics in physics, number theory, and biology [31]. In particular, hyperuniformity can be formulated, even quantitatively, for charged point processes such as (1.27), and certain classical results can be recast in this light. For example, fluctuations of charged Coulomb systems within observation disks or radius R, if non-extensive, are known to be dominated by the observation perimeter O(R) [27,28].
Our last result shows that the fluctuations of the aggregated charge of zeros of GWHF with radial twisted kernels are non-extensive, and moreover provides an asymptotic expression for the variance of charge. Theorem 1.14 (Hyperuniformity of charge) Let F be a GWHF with twisted kernel H satisfying the standing assumptions. Assume further that H (z) = P |z| 2 , where P : Then the charged measure of zeros Z κ F satisfies the following: there exists a constant C = C H > 0 such that for all z ∈ C, uniformly on z.
The hypothesis of Theorem 1.14 is satisfied in Example 1.1 (Gaussian entire functions, where all charges are positive and more refined results exist [24,Sect. 3.5]), and in poly-entire contexts (Examples 1.3, 1.4), as well as for the short-time Fourier transform of white noise (Example 1.2) with Hermite windows-see Sect. 6.7. The case of order one pure poly-entire functions may be interesting in relation to the classification of critical points of Gaussian entire functions-see Sect. 6.8.
In the context of Theorem 1.14, whenever the one-point function of the zero set of F is large, most of the positively charged zeros tend to be surrounded by negatively charged ones, a phenomenon that in the stationary setting is called (almost perfect) screening [10,34]. In the twisted stationary setting, this phenomenon is universally valid, independently of the particular kernel H . The significance of hyperuniformity thus concerns the empirical observability of the ensemble average claimed in Theorem 1.12: universal screening is observed with growing probability at all sufficiently large scales. For example, by Markov's inequality, Theorem 1.14 implies that which is consistent with the experiment in Fig. 1, where the prescribed equilibrium is observable already in one realization of a GWHF.

Organization
Our main tool is direct computation with Kac-Rice formulae and exploitation of the invariance relation (1.2). Section 2 introduces background results and required adaptations to our setting. In Sect. 4, we prove all results related to first intensities (Theorems 1.6, 1.12 and Corollary 1.7). In Sect. 5, we study second order statistics of charged zeros and prove Theorem 1.14. Section 6 develops applications to Examples 1.1, 1.3, 1.4, and 1.2, including proofs of Theorems 1.8, 1.9, Corollary 1.10, Theorem 1.11, and Corollary 1.13. The shorttime Fourier transform plays a prominent role, as time-frequency techniques are also brought to bear on the other examples. Section 8 contains auxiliary results, including a lengthy calculation, for which we also provide a Python worksheet at https://github.com/gkoliander/gwhf. Section 7 offers conclusions, a discussion on open problems, and perspectives on future work.

Notation
We use t for real variables and z, w for complex variables. We always use the notation The real and imaginary parts of z ∈ C are otherwise denoted (z) and (z), respectively. The differential of the (Lebesgue) area measure on the plane will be denoted for short d A, while the measure of a set E is |E|. The derivatives of a function F : C → C interpreted as F : R 2 → R 2 are denoted by F (1,0) (real coordinate) and F (0,1) (imaginary coordinate). Higher derivatives are denoted by F (k, ) . Vectors (z 1 , . . . , z n ) ∈ C n are identified with column matrices (z 1 , . . . , z n ) ∈ C n×1 ; (z 1 , . . . , z n ) t denotes transposition, while (z 1 , . . . , z n ) * denotes transposition followed by coordinatewise conjugation. We let denote the matrix with the property: The Jacobian of F : C → C at z ∈ C is the determinant of its differential matrix D F considered as F : R 2 → R 2 : The following observations will be used repeatedly: (2.2)

Gaussian Vectors and Intensities
By a Gaussian vector we always mean a circularly symmetric complex Gaussian random vector, i.e., a random vector X on C n such that ( (X ), (X )) is normally distributed, has zero mean, and vanishing pseudo-covariance: A complex Gaussian vector X on C n is thus determined by its covariance matrix If Cov[X ] is non-singular, then X is absolutely continuous and has probability density Gaussian vectors are not a priori assumed to have non-singular covariances. The zero vector, for example, is a singular Gaussian vector.
If (X , Y ) is a Gaussian vector on C n+m and h : C n → R is a function, the conditional expectation E h(X ) Y = 0 is defined by Gaussian regression. Informally, this involves finding a linear combination of X , Y which is uncorrelated to Y . The following remark makes this intuition precise.
Assume further that C is nonsingular. Let Z be a circularly symmetric Gaussian random vector in C n with covariance Then, for any locally bounded h : is the Gaussian regression version of the conditional expectation [3, Eq. (1.5)].
Whenever it exists, the first intensity or one-point intensity of a random signed measure μ on C is a measurable function ρ : Second order intensities are defined in the article as needed. Objects related to charged zeros are denoted with a superscript κ. Background on random Gaussian functions can be found in [1,3].

Kac-Rice Formulae
The formulae that describe the statistics of the level sets of Gaussian functions are generically known as Kac-Rice formulae. The following result is quoted from [3, Theorem 6.2]-with the notation Jac f :

Proposition 2.2 [Expected number of roots] Let U ⊂ R d be open, Z
: U → R d a Gaussian random field, and u ∈ R d . Assume that: For each t ∈ U , Z(t) has a non-degenerate distribution (i.e., its covariance it positivedefinite), Then for every Borel set E ⊂ U :

Covariance Structure of First Derivatives
As a first step in the investigation of a GWHF F, we describe the stochastics of the Gaussian vector at a given point z ∈ C. We start by calculating the covariances The following lemma will help us simplify further calculations.
Proof The conclusion follows directly from (1.12).
We now specialize the previous calculations at z = w and see that the covariance matrix of (3.1) is We will be mainly interested in conditional expectations of the form According to Remark 2.1, these are E h(Z ) where Z ∈ C 2 is a circularly symmetric Gaussian random vector with covariance matrix: As we can see, the covariance matrix (3.4) and thus the expectation (3.3) do not depend on the specific given point z. The following result formalizes these observations.
where Z = (Z 1 Z 2 ) ∈ C 2 is a circularly symmetric Gaussian random variable with covariance matrix given by in (3.4).
The zeros of F are almost surely simple and thus isolated. More precisely, Proof For (a) note that the twisted stationarity condition (1.2) implies that e i (·w) F(·−w) and F(·) are circularly symmetric complex Gaussian fields with the same covariance structure. The claim about absolute values follows immediately since Part (b) follows from discussion above, as Jac For (c), we apply [3, Proposition 6.5]. The required hypotheses are that F be C 2 almost surely, as we assume, and that the probability density of F(z) be bounded near 0, uniformly in z. This last requirement is satisfied because F(z) is a circularly symmetric complex Gaussian vector with variance V ar[F(z)] = H (0) = 1.

Kac-Rice Formulae for GWHF
The second preparatory step is to check that various Kac-Rice formulae are applicable to GWHF and obtain corresponding intensities for zeros.

Lemma 3.3 Let F be a GWHF with twisted kernel H satisfying the standing assumptions.
Then the first intensities of the uncharged and charged zero sets are independent of z and given by That is, the random measures (1.17) and (1.27) satisfy In addition, we define the semi-charged two-point intensity τ κ 2 : C → R by Then τ κ 2 is well-defined and serves as density for the following semi-charged factorial moment: Proof We first apply Proposition 2.2 to obtain: The required regularity hypotheses are verified by Proposition 3.2. Since F(z) is a Gaussian circularly symmetric complex random variable with zero mean and variance H (0) = 1, p F(z) (0) = 1 π , and (3.5) follows. The independence of z follows from Property (b) in Proposition 3.2.
Similarly, Proposition 2.3 gives: for all compact E ⊆ C and bounded and continuous ϕ : R → R. Formally applying this formula to the non-continuous function ϕ(x) = sgn(x) yields (3.6). To justify such application, fix a compact set E ⊆ C and let ϕ n : R → [−1, 1] be continuous and such that ϕ n (x) = sgn(x), for |x| > 1/n. First note that sgn(Jac F(z)) (3.10) almost surely, as convergence can only fail when F(z) = 0 and Jac F(z) = 0, and this is a zero probability event according to Property (c) in Proposition 3.2. To show that (3.10) also holds in expectation, we estimate note that E #{z ∈ E, F(z) = 0} = ρ 1 |E| < ∞, and invoke the Dominated Convergence Theorem. We now inspect the right-hand side of (3.9). By Property (b) in Proposition 3.2, where Z = (Z 1 Z 2 ) ∈ C 2 is a circularly symmetric Gaussian random variable with covariance matrix given by (3.4).
we can again invoke the Dominated Convergence Theorem to conclude that the right-hand side in (3.11) converges to the right-hand side of (3.12). Summarizing, we have that which yields (3.6). For (3.7), we first note that sgn(Jac F(z)) · sgn(Jac F(w)).
Let δ > 0 and consider the random Gaussian field on C 2 given bỹ We apply Proposition 2.3 toF and use a regularization argument as before to learn that, for any Borel setẼ ⊆ C 2 , where To apply the weighted Kac-Rice formula it is important that the covariance matrix ofF(z, w) be non-singular, as granted by (1.14). Proposition 2.3 thus gives (3.15), where λ is the value of the probability density ofF(z, w) at 0, which is indeed given by (3.16). We let E ⊆ C be compact, chooseẼ ⊆ E × E in (3.15) according to the signs of Jac F(z) and Jac F(w), let δ → 0, and use the monotone convergence theorem to deduce (3.8), albeit with a function depending on (z, w) in lieu of τ κ 2 . It thus remains to show that E Jac F(z) Jac F(w) F(z) = F(w) = 0 depends only on the difference z −w. To this end, note that the twisted stationarity condition (1.2) means that and F are circularly symmetric complex Gaussian fields with the same covariance. Thus, Let us calculate the right-hand side of the previous equation. Writing ζ = a +ib, we compute Similar equations hold of course for w in lieu of z. We note that the event is precisely the event {F(z − ζ ) = F(w − ζ ) = 0}, and that under this event, and similarly for w in lieu of z. As a consequence, under (3.18), Jac F ζ (z) = Jac F(z − ζ ) and Jac F ζ (w) = Jac F(w − ζ ). Plugging these observations into (3.17) we conclude that That is, the number E Jac F(z) · Jac F(w) F(z) = F(w) = 0 depends only on z − w.

First Intensities
We can now derive the main results on first intensities.
Proof of Theorem 1.6 By Kac-Rice's formula (3.5), By Property (b) in Proposition 3.2, where Z = (Z 1 Z 2 ) ∈ C 2 is a circularly symmetric Gaussian random variable with covariance matrix given by in (3.4). Let us assume initially that is non-singular and denote H := det( ), so that H > 0. Hence, We use the following formula [9,eq. (4.32)] where the last integral is to be understood as a principal value. By Lemma 8.1, Using Lemma 3.1, we note that the off-diagonal element γ in satisfies and, hence, This shows that λ, μ ∈ (0, +∞); we may assume that λ ≥ μ. A direct calculation further shows that Using (4.2) we write Hence, with the aid of (4.3), we compute Finally, if is singular, we let Z τ := Z + τ X with X an independent standard circularly symmetric random vector on C 2 and τ ∈ (0, 1). Then Z τ has covariance τ = + τ I and the calculation above shows that Furthermore, by continuity, Since Z 1 , Z 2 , X 1 , X 2 are normal, they are square integrable with respect to the underlying probability, and thus the right hand side of (4.4) is integrable. Hence, by dominated convergence, This completes the proof.
As an application of Theorem 1.6, we derive a simplified expression for radial twisted kernels.

and (1.18) simplifies to (1.21).
Finally, we derive the first intensity of charged zeros.
Proof of Theorem 1. 12 We proceed along the lines of the proof of Theorem 1.6. This time we use Kac-Rice's formula (3.6), which gives that ρ κ 1 is the following constant: where Z = (Z 1 Z 2 ) ∈ C 2 is a circularly symmetric Gaussian random variable with covariance matrix given by in (3.4). Thus, where γ is the covariance E Z 1 Z 2 in (3.4).

Sufficient Conditions for Hyperuniformity
The following lemma gives sufficient conditions for the hyperuniformity of the charged zero set (1.27). These are formulated in terms of the first intensity of the uncharged zero set ρ 1which is constant by Theorem 1.12-and the semi-charged two-point intensity defined in Lemma 3.3, cf. (3.7).

Lemma 5.1 Let F be a GWHF with twisted kernel H satisfying the standing assumptions. Suppose that
Then there exists C > 0 such that for all z 0 ∈ C, Proof We let z 0 ∈ C and use (3.8) and Theorem 1.12 to compute In terms of the function ϕ(z) := 1 π 2 ρ 1 − 1 ρ 1 τ κ 2 (z), the expression for the variance reads The last expression measures the average deviation within the disk B R (z 0 ) between the indicator function of that disk and its convolution with ϕ. Precise estimates are given in Lemma 8.3-whose proof is deferred to Sect. 8. The hypotheses of Lemma 8.3 are met due to (5.1) and (5.2), and we readily obtain (5.3) and (5.4).

Computations for Radial Twisted Kernels
For radial twisted kernels, the following proposition provides an expression for the integrals in (5.2). We use the notation of Lemma 5.1.

Following Remark 2.1, this conditional expectation is E h(Z )
where Z ∈ C 4 is a circularly symmetric Gaussian random variable with covariance matrix: Here, A = 2,3;2,3 (z) 2,3;2,3 (z, w) 2,3;2,3 (z, w) * 2,3;2,3 (w) , where i, j;k,l is the submatrix of containing the rows i and j and columns k and l, and Thus, with the convention that P, P , and P are understood to be evaluated at |z − w| 2 . By Lemma 3.3, With this information we can calculate explicitly E to obtain Step 2 (Conclusions) We first verify (5.6) and (5.7); this then implies that the integrals in (5.8) and (5.9) are absolutely convergent. To this end, we use (5.17) and estimate and, similarly, while lim s→∞ I (s) = 0 by (5.5). Hence, For (5.9), integration by parts gives A direct calculation shows that . (5.18) Hence, by (5.5), as claimed in (5.9).

Proof of Theorem 1.14
We now derive the main result on hyperuniformity of charge. We invoke Lemma 5.1. Condition (5.1) is satisfied as shown in Proposition 5.2, (5.7), while (5.2) is seen to hold by comparing the explicit expressions given in Corollary 1.7 and Proposition 5.2. The asymptotic value of the variance in (5.4) is computed in (5.9).

The Short Time Fourier Transform of White Noise
Let g : R → C be a Schwartz function. As a first step towards the definition of the short-time Fourier transform of white noise, we consider its distributional formulation. For a Schwartz function f : R → C, we write (1.5) as where (x, y)g denotes the time-frequency shift 2 We define the STFT of a distribution f ∈ S (R) by (6.1), using the distributional interpretation of the L 2 inner product ·, · , and note that this defines a smooth function on R 2 . The adjoint short-time Fourier transform V * g : S(R 2 ) → S(R), provides the following concrete description of the distributional STFT: See [21,Chap. 11] for more background on the STFT of distributions. Let N be complex white noise on R, that is, where W 1 and W 2 are independent copies of the Wiener process (Brownian motion with almost surely continuous paths), and the derivative is taken in the distributional sense. The short-time Fourier transform of complex white noise is the random function: see [5,6] for other definitions and a comprehensive discussion on their equivalence. Then V g N is Gaussian because, as a consequence of (6.2), for any Schwartz function ϕ ∈ S(R 2 ), V g N , ϕ = N , V * g ϕ is normally distributed. In addition, V g N is circularly symmetric, as, for any θ ∈ R, e iθ · V g N = V g e iθ · N ∼ V g N . One readily verifies that 3) The following lemma relates the STFT of white noise and GWHFs.
Then F is a GWHF with twisted kernel and the standing assumptions are satisfied. In addition, the zero set of V g N has a first intensity ρ 1,g related to that of the zero set of F by Proof F is Gaussian and circularly symmetric because V g N is. Using (6.3), we inspect the covariance of F: We now verify the standing assumptions. Since g is Schwartz, H is C ∞ , and (1.15) and (1.16) hold. The normalization condition (1.13) is indeed satisfied since H (0) = V g g(0) = ||g|| 2 2 = 1. To check the non-degeneracy condition (1.14) note first that, by Cauchy-Schwarz, If equality holds for some z = x + iy, then there exists λ ∈ C such that This implies, Since g ∈ L 2 (R) \ {0}, we must have x = 0. Hence, which implies y = 0, since g ≡ 0. Hence z = 0. Finally, since F(z) = 0 if and only if V g N (z/ √ π) = 0, (6.4) follows.

Calculation of the First Intensity
We now apply our results to the short-time Fourier transform of complex white noise.
Proof of Theorem 1. 9 We consider the functions F and H as in Lemma 6.1, and the first intensities of their zero sets, ρ 1 and ρ 1,g , related by (6.4). We calculate where we used that, by Lemma 3.1, H (1,1) (0) ∈ R. Note also that c 4 ∈ R by Lemma 3.1, while, clearly, c 1 , c 2 , c 3 , c 5 ∈ R. Thus, (1.19) is given by and Theorem 1.6, together with (6.4), yield as claimed. Finally, if g is real valued, integration by parts gives showing that c 4 = 0, while clearly c 5 = 0.

The Uncertainty Principle for Zeros
In order to show that generalized Gaussian windows minimize the expected numbers of zeros of the STFT with complex white noise, we first show that the corresponding intensities are invariant under certain transformations that preserve the class of Gaussians. Lemma 6.2 Let g : R → C be a Schwartz function, and x 0 , ξ 0 , ξ 1 ∈ R. Let Then the first intensities of the zero sets of V g N and V g 1 N coincide:

Proof
We proceed in two steps, and exploit different properties of the STFT. We first assume that ξ 1 = 0 and use the so-called covariance of the STFT under time-frequency shifts: which can be verified by direct calculation or deduced from [21, Lemma 3.1.3]. Applying this formula to each realization of complex white noise f = N , we deduce that Z F g 1 and Z F g are related by a deterministic translation: We now assume that ξ 0 = x 0 = 0, so that g 1 and g are related by the unitary operator U : L 2 (R) → L 2 (R), The operator U is also an isomorphism on the spaces of Schwartz functions and tempered distributions. For a distribution f , we use the formula ]. Let N be complex white noise; then so is U N (both generalized Gaussian processes have the same stochastics). In addition, by Lemma 6.1 and Theorem 1.6, the zero sets of V g N and V g 1 U N have first intensities and these are constant. Hence, for any Borel set E ⊆ R 2 , by (6.6), as S is a linear map with determinant equal to 1. Finally, the general case without assumptions on ξ 0 , ξ 0 and x 0 follows from the discussed special cases by successively considering the effect of the time-frequency shift (x 0 , ξ 0 ) and quadratic modulation U .
We can now prove the announced uncertainty principle for zero sets.
Proof of Theorem 1. 11 We use the notation of the proof of Theorem 1.9. Recall the relation (6.4). As shown in Theorem 1.6 and its proof, ρ 1 as given by (1.18) satisfies ρ 1 ≥ 1/π and achieves the value 1/π exactly when H = 0. We now describe the functions attaining that minimum.

Hermite Windows
We now consider Hermite functions Proof of Corollary 1. 10 We write H (z) = P(|z| 2 ) with P(t) = L r (t)e −t/2 . By Lemma 6.1, H satisfies the standing assumptions. We can therefore apply Corollary 1.7. Inspecting (6.11) we obtain Using (6.4), we conclude

Derivatives of Gaussian Entire Functions
Let G 0 be a Gaussian entire function, that is, a circularly symmetric random function with correlation kernel, and consider the iterated covariant derivatives where q ∈ N. G is called a Gaussian poly-entire function of pure type. The following lemma provides an identification with a GWHF.

Lemma 6.3
Let G be a Gaussian poly-entire function of pure-type, as in (6.14). Then is a GWHF with twisted kernel satisfying the standing assumptions. Here, L n denotes the Laguerre polynomial (6.11).
Proof We consider the complex Hermite polynomials H k, j (z,z) defined by: Conjugating the last equation we obtain: We combine (6.13) and (6.14), expand e zw into series, and compute where the last equality is proved in [18,Eq. 3.19]; see also [18,Proposition 3.7] and [25,Sect. 2]. Hence, as desired. Finally, note that F is also the GWHF associated in Sect. 6.4 with the STFT with Hermite window h q−1 . Hence, the standard assumptions hold by Lemma 6.1.

Gaussian Poly-entire Functions
We now look into Gaussian poly-entire function of full type (cf. Example 1.4). These are defined as where G 0 , . . . , G q−1 are independent Gaussian entire functions, and q is called the order of G. The following lemma identifies G with a GWHF, by means of the generalized Laguerre polynomial Lemma 6.4 Let G be a Gaussian poly-entire function of full type of order q, as in (6.16). As an application, we obtain the following.
Proof of Theorem 1.8 By Lemmas 6.3 and 6.4, we can apply Corollary 1.7 with P(t) = L q−1 (t)e −t/2 or P(t) = q −1 L (1) q−1 (t)e −t/2 . In the first case (pure type), the calculation was carried out in the proof of Corollary 1.10 (where r = q − 1). For the second case (full type), we note that L (1) We thus compute, and, therefore,

Charges
We start with the following general observation.
Lemma 6.5 Let F, G : C → C be C 1 in the real sense, and z 0 ∈ C. If F(z 0 ) = 0 and G(z 0 ) = 0, then the charges of F and F · G at z 0 coincide.
Proof Using (2.2) we see that the charge of F · G at z 0 is which is also the charge of F at z 0 .
We first apply Theorem 1.12 to the short-time Fourier transform, and obtain formulas in terms of (1.28).
Proof of Corollary 1.13 By Lemma 6.1, the short-time Fourier transform of complex white noise can be identified with a GWHF by the transformation At a zero ζ = a + ib, and, consequently, Applying Theorem 1.12 with the change of variable z =ζ / √ π we obtain sgn Jac F(ζ ) as claimed.
For the STFT of white noise with a Hermite window (6.10), the twisted kernel is given in (6.12), and we can apply Theorem 1.14 with After a change of variables as in the proof of Corollary 1.13, we obtain uniformly on z 0 . Finally, we note that we can also apply Theorems 1.12 and 1.14 to poly-entire functions. Let G be a Gaussian poly-entire function of pure-type, as in (6.14). According to Lemma 6.3, the function is a GWHF. By Lemma 6.5, the charges of F and G at a zero ζ coincide: κ ζ = sgn(Jac F(ζ )) = sgn(Jac G(ζ )).
A similar argument applies to poly-entire functions of full-type (cf. Example 1.4 and Sect. 6.6). Hence, Theorem 1.12 shows that the first intensity of the charged zeros of G is 1/π. Similarly, Theorem 1.14 applies to G and concrete expressions for the asymptotic charged particle variance can be obtained with the polynomials P(r ) = e −r /2 · L q−1 (r ) pure-type (6.14) 1 q · e −r /2 · L (1) q−1 (r ) full-type (6.16) .

First Derivatives of GEF
We now interpret the statistics of zeros of Gaussian pure poly-entire functions of order 1, and show how they recover the well-known first order statistics of critical points of weighted magnitudes of Gaussian entire functions (cf. Examples 1.3).
Let G be a Gaussian entire function as in Example 1.1 and consider its amplitude A(z) = e − 1 2 |z| 2 |G(z)|. Then, by (1.8), the critical points of A are exactly the zeros of the GWHF F(z) = e − 1 2 |z| 2∂ * G(z). By Theorem 1.8 (with q = 2), the first intensity of the critical points of A is therefore 5/3. Second, consider a critical point z 0 of A. Then, by Proposition 3.2, with probability one, z 0 is not a zero of G, and near z 0 we can write G(z) = L(z) 2 with L analytic. Hence, As the factor L/L is smooth (in the real sense) and non-zero near z 0 , we conclude by Lemma 6.5 that the charge of F at z 0 is that is, the opposite of the sign of the determinant of the Hessian matrix of A at z 0 . Hence, κ z = 1 if z 0 is a saddle point of A, while κ z = −1 if A has a local maximum at z 0 (while local minima are excluded, as they are zeros of G [24, Sect. 8.2.2]). Thus, by Theorem 1.12, the first intensity of the quantity "saddle points − local maxima" is 1. Combining this with the first intensity of the total critical points, we conclude that the first intensity of the local maxima of A is 1/3 whereas that of the saddle points is 4/3. While the calculation of first intensities of different kinds of critical points of G is wellknown-they follow for example as the limit of more precise results for polynomial spaces in [12, Corollary 5]-the hyperuniformity of the statistics of "saddle points − local maxima" is, to the best of our knowledge, a novel consequence of Theorem 1.14.

Conclusions and Outlook
We introduced the notion of twisted stationarity for an ensemble of random functions and obtained basic statistics for their zeros. In comparison to the model case of translation invariant Gaussian entire functions, a novel element is found: GWHF may either preserve or reverse orientation around a zero, and zero statistics are thus augmented with the new attribute of charge.
While our result on hyperuniformity of charge is a first step in the exploration of repulsion between zeros of GWHF, as it shows that a universal form of screening is observed at large scales, many important questions remain open. First, Theorem 1.14 was obtained under the assumption that the twisted kernel is radial, which means that statistics are rotationally invariant. We do not know if hyperuniformity of charge holds also for non-radial twisted kernels. Second, no variance estimates were derived for uncharged zeros. We conjecture that the uncharged number variance grows like the perimeter of the observation disk. Finally, numerical experience suggests that the repulsion between zeros of the same charge is stronger than that between oppositely charged ones, but we do not yet have formal statistics justifying that claim.
The short-time Fourier transform of white noise is a case in point application of our results, because they open the door to the use of non-Gaussian windows. This new freedom has prospective applications in signal processing which we expect to develop in future work. Indeed, when analyzing a signal, one can often choose the STFT window, and the potentially rich zero statistics that we derived hold simultaneously for all such choices.

Computations with Gaussians
Lemma 8.1 Let ∈ C 2×2 be positive definite and t ∈ R. Then fix a > 0, b ∈ R, and consider both sides of (8.1) as functions of the complex variable ξ = t 2 + id. For ξ ∈ iR (i.e., t = 0) and d 2 < ac − b 2 (i.e., positive definite), (8.1) holds because it expresses the fact that the probability density of a complex Gaussian is normalized. We will show that both sides of (8.1) are analytic functions on the domain To this end, we first rewrite Here, the integrand is an analytic function in ξ . To show the analyticity of the integral, we note that for any compact subset C ⊆ A, we have ϑ C := sup ξ ∈C [ξ ] 2 < ac − b 2 . Thus, the absolute integrand satisfies Hence, the absolute integral is uniformly bounded for ξ ∈ C. Applying Morera's theorem and Fubini's theorem, we can conclude that the integral is analytic as well.
The right-hand side of (8.1) can be rewritten as and is also analytic in ξ as long as ac = (b + ξ)(b − ξ). In particular, for ξ ∈ A we have that Hence, both sides of (8.1) are analytic on A and coincide on the set A ∩ iR. By the identity theorem of analytic functions, they thus coincide on A.

Lemma 8.2
Let v be a 4-dimensional circularly symmetric complex Gaussian vector with covariance matrix . Then where per is the permanent and i, j;k,l is the submatrix of containing the rows i and j and columns k and l.

Lemma 8.3
Let ϕ : C → R be an integrable function that satisfies Then there exists a universal constant C > 0 such that for all z 0 ∈ C, In addition, letting the following holds: Proof We first note the elementary facts for some constant C > 0. For each w ∈ C, rescaling and translating yields We calculate where we used Fubini's theorem. For (8.3), we use (8.5) and estimate For (8.4), we use (8.6) to obtain as r → +∞, where we used the dominated convergence theorem, as allowed by (8.5).
Inspection of each term in the other factor in I combined with (5.5) shows that lim sup r →∞ r 4 I (r 2 ) < ∞.
Funding Open access funding provided by Austrian Science Fund (FWF).
Data Availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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