The Random Weierstrass Zeta Function I: Existence, Uniqueness, Fluctuations

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.


Introduction and overview
Let Λ be a stationary random point process in R d , d ≥ 2, and let n Λ = λ∈Λ δ λ be its counting measure.We take the probability space of Λ to be (Ω, F, P), where Ω is the space of locally finite configurations in R d and F is the σ-algebra generated by the events Stationarity of Λ amounts to invariance of P under translations, i.e. under the maps T x : Ω → Ω, where T x Λ = Λ − x.Denote by c Λ the (first) intensity of Λ, i.e., assume that E[n Λ ] = c Λ m, where m is the Lebesgue measure, and consider the following question: Question 1.For which stationary point processes Λ does there exist a stationary random vector field V Λ with div V Λ = n Λ − c Λ m in the sense of distributions?
Probably, the first instance of such a field is due to Chandrasekhar, who noted in [6, Ch.IV] that, for the Poisson point process Λ in R 3 , the stationary vector field V Λ can be defined by the regularized series where the summation is over λ ∈ Λ (here and elsewhere, we skip Λ under the summation sign), and κ = 4π/3 is the volume of the unit ball in R 3 .Chatterjee-Peled-Peres-Romik [7, Proposition 1] gave the rigorous proof of this for the Poisson point process in R d with d ≥ 3.
On the other hand, such a stationary field (with a very mild regularity) does not exist for the planar Poisson process.This follows from Theorem 5.1 below but, probably, is not news for experts.Well-studied examples of stationary planar point processes possessing stationary vector fields are the limiting Ginibre ensemble and the zero set of the Gaussian Entire Function [17,22].For the limiting Ginibre ensemble this also follows from Theorem 5.1 and, likely, this is known to experts.For the zero set of the Gaussian Entire Function F (z), the field V Λ written in complex coordinates is nothing but (F ′ /F )(z) − z which is the complex gradient of the stationary potential log |F (z)|2 − |z| 2 .Plausibly, the stationary field exists for two-and three-dimensional Coulomb-type charged systems studied by physicists and mathematicians; see the survey papers [21,26,27,20] and the references therein.
Since the higher dimensional version of the question does not bring any essentially new difficulties comparing with the planar case1 , we will concentrate on the latter.In this case, the question becomes equivalent to the following one: Question 2. For which stationary planar point processes Λ does there exist a random meromorphic function f Λ with poles exactly at Λ, all simple with unit residue, such that the random function f Λ (z) − πc Λ z is stationary?
In this paper we will provide an answer to Question 2 for point processes with a finite second moment, i.e., E[n Λ (B) 2 ] < ∞ for any bounded Borel set B ⊂ C. Such stationary point processes admit a spectral measure ρ Λ ; see Section 2 below for the details and examples.In Section 4.2, we will construct a random analogue ζ Λ of the Weierstrass zeta function from the theory of elliptic functions.The function ζ Λ is meromorphic with poles exactly at Λ, all simple with unit residue, but in general it is not stationary.One of our findings is Theorem 5.1 below.In the simplifying case when the point process Λ is ergodic, it states that the following three conditions are equivalent: (i) The spectral condition (ii) The sum λ∈Λ, 1≤|λ|≤R λ −1 converges in L 2 (Ω, P) as R → ∞.
Moreover, any solution f Λ to the problem in Question 2 with some very mild regularity (e.g., E[|f Λ (0)|] < ∞) coincides with ζ Λ −Ψ up to a (deterministic) constant, so the field in (iii) is essentially unique; see Theorem 5.4 and Remark 5.6.We also note that correcting by the random constant in condition (iii) is in fact necessary, and the natural choice of Ψ is given in Lemma 3.3.
The spectral condition (i) can be thought of as a sum rule for the two-point measure of Λ, cf.Remark 5.3 below.
Curiously, if we do not impose any regularity on f Λ , the answer to Question 2 is always positive.To show this one can use Weiss' construction [33] or a modification of the Krylov-Bogoliubov averaging construction for invariant measures [5].However, when the spectral condition (i) fails, f Λ is necessarily quite "exotic" with wild growth at infinity and very heavy tails, cf.[5].
As an application of the ideas developed here, we study in [28] the variance of line integrals of f Λ along dilated rectifiable curves R Γ in the large-R limit.When Γ encloses a Jordan domain Ω, this coincides with the "charge fluctuation" in R Ω, which is a classical quantity in mathematical physics.See Remark 6.1 for a further discussion.
Before we proceed, a few words about definitions are in order.By a stationary random vector field we mean a measurable map Λ → V Λ of Ω into the space of (Borel) measurable functions on C taking values in the extended complex plane C, such that for all z ∈ C, Equivalently, V Λ is a stationary random vector field if it takes the form V Λ (z) = F (T z Λ) for some measurable function F : Ω → C.

Related work.
There is a certain resemblance between the questions studied here and several well-studied problems.Among these is the classical question about the growth of the variance of sums and integrals of stationary processes which boil down to the existence (better to say, non-existence) of stationary primitives of stationary processes.This was studied by Robinson [24], Leonov [19], and Ibragimov-Linnik [16, Ch XVIII, §2, 3] for stationary processes on Z and on R, and by Davidovich [10] for stationary processes on the existence of a stationary primitive for a stationary point process on R. The relevant ergodic theoretic result is Schmidt's coboundary theorem [25,Theorem 11.8].
In physics papers, Lebowitz-Martin [18] and Alastuey-Jancovici [2] among other things computed the spectral measure and the reduced covariance measure for the field and potential of two-and three-dimensional Coulomb-type systems.
Questions pertaining to existence and uniqueness of stationary solutions to stochastic PDE (see, for instance, Vergara-Allard-Desassis [31]) also belong to this circle of problems.
Wide-sense stationary point processes.The main tool in the proofs of the most of our results will be the spectral theory of generalized point processes, developed in the 1950ies by Itô, Gelfand, and Yaglom.The proofs will not use the translation-invariance of the distribution of the point process in its full strength, but rely only on the translationinvariance of the mean and of the correlations of the point process.For this reason, with some obvious modifications, the corresponding results remain valid for wide-sense stationary point processes Λ.
Organization.The article is organized as follows.In Section 2, we discuss the notion of the spectral measure and some surrounding preliminaries.Here we mention in some detail the main examples we kept in mind during this work.In Section 3 we analyze the convergence properties of reciprocal sums over Λ, e.g.1≤|λ−z|≤R 1 λ j for j = 1, 2, as well as their behavior under translations of the center z of summation.These sums play a central role in the construction of the random Weierstrass function, which is carried out in Section 4. This overall scheme works for any point process, but in general the field obtained only has stationary increments.In Section 5 we discuss the existence, uniqueness and covariance structure of the invariant field V Λ under the above-mentioned spectral condition.In Section 6 we conclude with a discussion of the existence and covariance structure of random potentials, that is, solutions to the equation ∆Π Notation.We use the following notation frequently.
• C, R; the complex plane and the real line • m; the Lebesgue measure on C • f ; the Fourier transform, with the normalization • D, S; the class of compactly supported C ∞ -smooth functions and the class of Schwartz functions, respectively • E, Cov, Var; the expectation, covariance and variance (with respect to an underlying probability space (Ω, F, P)) • F inv ; the sigma-algebra of translation-invariant events • T a ; translation by a ∈ C, acting on functions by T a f (z) = f (z + a) and on sets by • ρ Λ ; the spectral measure of the point process Λ • κ Λ , τ Λ ; the truncated and reduced truncated covariance measures for Λ, respectively.
Oftentimes, we will treat sums and series where the summation variable ranges over a point process Λ.When this is clear from the context, we will abuse notation slightly and simply write We use the standard Landau O-notation and the symbol interchangeably.For limiting procedures involving an auxiliary parameter a, we use the notation to indicate that the implicit constant may depend on a.
2 The second-order structure of stationary point processes

The spectral measure
Let Λ be a stationary point process in C with a finite second moment, that is, we assume that E[n Λ (B) 2 ] < ∞ for any bounded Borel set B. The spectral measure of Λ is a nonnegative locally finite measure ρ Λ on C such that the "Parseval formula" holds: where ϕ, ψ ∈ D, n Λ (ϕ) denotes the linear statistic and ϕ, ψ are the Fourier transforms, i.e., Existence of the spectral measure follows from a version of the Bochner theorem, see

The reduced covariance measure
The spectral measure of a point process is the Fourier transform of "the reduced covariance measure" κ Λ , which is a signed measure on C such that see Daley and Vere-Jones [8, Ch. 8] (their notation is slightly different from the one we use here).Recalling that where c Λ is the first intensity of the point process Λ (i.e., the mean number of points of Λ per unit area) and ν Λ is the reduced two-point measure of Λ, we get that where τ Λ = ν Λ − c 2 Λ m is the (reduced) truncated two-point measure of Λ.Note that in the physics literature it is tacitly assumed that the measures ν Λ and τ Λ have densities, called the two-point function and truncated two-point function, respectively.
Similarly, the reduced covariance measure is defined for random stationary processes and random stationary measures in C.
The total variation of any reduced covariance measure κ Λ (and therefore of the reduced truncated measure τ Λ ) is also translation-bounded [8,Ch. 8].
It is worth mentioning that for many point processes the tails of the measure τ Λ decay very fast, which means that on high frequencies the spectral measure is close to the Lebesgue measure c Λ m.On low frequencies the behavior of the spectral measure is governed by the Stillinger-Lovett sum rules, which control the zeroth and the second moments of κ Λ [21].

The conditional intensity of Λ
We denote by F inv ⊂ F the sigma-algebra of translation-invariant events in F. The random variable Hence, the random variable c Λ does not degenerate to the deterministic intensity c Λ if and only if n Λ (R D) asymptotically has the variance of the maximal possible order R 4 , i.e., "hyper-fluctuates", and this in turn is equivalent to ρ Λ ({0}) > 0. Note that if the point process Λ is ergodic, then the sigma-algebra F inv contains only events of probability 0 or 1, and therefore, c Λ is constant.
The simplest example of a point process with a spectral measure having an atom at the origin is a random mixture of two independent Poisson processes having different intensities (such processes are called Cox point processes).For a more general construction, take Λ to be any ergodic point process with spectral measure ρ Λ , and denote by L a positive non-degenerate random variable with Var[L] < ∞.Then Λ ′ = L − 1 2 Λ is a point process with finite second moment.In view of (2.4), we get that and hence (2.5) gives that ρ

Examples
Here, we will make a short stop to present several examples of stationary two-dimensional point processes, which we kept in mind starting this work.For all these examples, the spectral measure can be computed without much effort.
The Poisson point process.In this case, the two-point function identically equals c 2 Λ (c Λ is the intensity of the Poisson process), the truncated two-point function vanishes, , and (2.1) is nothing but the classical Parseval-Plancherel formula.
The limiting Ginibre ensemble.This is the large N limit of the eigenvalues of the Ginibre ensemble of N ×N random matrices with independent standard complex Gaussian entries.
One important feature of the limiting ensemble is its determinantality, which means that its k-point functions can be expressed in terms of the determinants see, for instance, [17,Section 4.3.7].This immediately yields the simple expression −π −2 e −π|z| 2 for the truncated two-point function and that c Λ = π −1 , which, in turn, yields that the spectral measure is absolutely continuous with the density π −1 (1 − e −π|ξ| 2 ).
Zeroes of the Gaussian Entire Function.The Gaussian Entire Function (GEF, for short) is defined by the random Taylor series with independent standard complex Gaussian coefficients ζ n .The most basic facts about GEFs and their zeroes can be found in [17,22].The two-point function and the spectral measure of the zero point process were explicitly computed by Forrester-Honner [11] and Nazarov-Sodin [23].The intensity is given by c Λ = π −1 , the truncated two-point function equals h(|z|), where , while the spectral measure is absolutely continuous with the density "Stationarized" random Gaussian perturbation of the lattice.This is a stationary point process defined as Λ a = ν + ζ a ν + U ν∈Z 2 , where ζ a ν are independent complex-valued Gaussian random variables with the variance a > 0, and U is uniformly distributed on [0, 1] 2 and is independent of all ζ a ν s.In this case, the spectral measure is also not difficult to compute (see, for instance, Yakir [35, §3]).It is a sum of an absolutely continuous measure, which is similar to the one of the limiting Ginibre ensemble, and a discrete measure with atoms at Z 2 \ {0}, Moreover, the reduced covariance measure κ Λ a is given by This can be obtained by a direct computation on the spatial side, or by taking the inverse Fourier transform of ρ Λ a .In the limit as a → 0, we obtain the randomly shifted lattice Λ = ν + U ν∈Z 2 with the purely atomic spectral measure ρ Λ = ν∈Z 2 \{0} δ ν .
3 Reciprocal sums over stationary point processes

Convergence of three series
Recall the notation (Ω, F, P) for the probability space on which the point process Λ is defined.To define the random Weierstrass zeta function, we need three lemmas.Lemma 3.1.Let Λ be a stationary point process in C having a finite second moment.
Then almost surely and in L 2 (Ω, P), The behavior of these two sums as R → ∞ will be important for us.
Lemma 3.2.Let Λ be a stationary point process in C having a finite second moment.
Then there exists a random variable The convergence of Ψ 1 (R) in L 2 (Ω, P) requires an additional property of the spectral measure ρ Λ of Λ.
Lemma 3.3.Let Λ be a stationary point process in C with spectral measure ρ Λ , and assume that Then there exists a random variable It is worth mentioning that the conditional convergence of the series Λ λ −ℓ with ℓ = 1, 2, when Λ is the limiting Ginibre process, or the zero process of GEF appear as auxiliary results in Ghosh-Peres [14, Sections 8 and 10].

Proof of the three lemmas
The Bessel function of order ν ∈ Z + is given by We will frequently use two basic properties of the Bessel function, namely the asymptotic formulas and sup For the proof of both facts see, for instance, [32,Ch.7].
To show that Ψ 3 (∞) ∈ L 2 (Ω, P), we use the Parseval identity (2.1) to move to the spectral side, which gives The integrand above is bounded uniformly in R. Thus, we need to check how fast it decays as |ξ| becomes large.For this, we use the asymptotic formula (3.3) for the Bessel function and see that, for |ξ| ≥ 1, and the function on the RHS is dρ Λ -integrable.Hence, we can apply the dominated convergence theorem and deduce that Ψ 3 (R) converge in L 2 (Ω, P) as well.
For any R ≥ 1, and by the Parseval formula (2.1), applied with we have We may rewrite the inner integral on the RHS as where χ = arg ξ, and where in the last equality we used that (J Thus, By the near-origin asymptotics (3.2) of the Bessel functions, J 1 (x)/x is bounded near the origin, and, together with the asymptotic formula (3.3) and the translation-boundedness of ρ Λ , we get lim That is, Ψ 2 (R) is a Cauchy sequence in L 2 (Ω, P).

Proof of Lemma 3.3
We start by computing and so, for any R ′ > R, the Parseval formula (2.1) gives that we can use the Cauchy-Green formula to obtain and plugging this into the above formula for Since J 0 is bounded, we can use (3.4) and the asymptotic formula (3.3) for J 0 to get lim This completes the proof.

Translation properties of reciprocal sums
The random variables Ψ ℓ (R) are defined by summation over annuli centered at the origin.
It will be essential to understand the effect of translating the center of summation in these sums.This amounts to understanding the summation over the lunar domains formed as the symmetric difference of two large disks with different centers.
Lemma 3.4.Let Λ be a stationary point process in C with finite second moment, with Although we will need this in the paper, we remark that the convergence is locally uniform in u, v, z.
In the case when Λ is the Poisson point process, Lemma 3.4 was proved by Chatterjee, Peled, Peres and Romik in [7, Lemma 8] where they obtain the analogous result for d ≥ 3, but the same proof works also for d = 2.
Proof.By stationarity of Λ, it suffices to prove the lemma for z = 0. We will assume that R is large enough so that both u and v are contained inside RD.By the Cauchy-Green formula it holds that {|x−u|≤R} dm(x) x = π ū, and thus Introduce the notation Clearly, , since the law of the point process Λ, conditioned on Y , has the intensity c Λ .Thus, Plugging in the definition of X and Y yields that where, Since the integrand is bounded by some constant C = C(u, v) > 0 (independent of R), we can bound I 1 as To bound the second integral I 2 , we use the Cauchy-Green formula to compute the inner integral: Plugging back the above in the definition of I 2 gives us By the standard stationary phase bound (see Proposition A.1 in the Appendix), for any . Hence, we see that Plugging back the bounds on I 1 and I 2 into (3.5),we get that which gives the lemma.
4 Fields and potentials with stationary increments

The Weierstrass zeta function
There is an evident analogy with the classical Weierstrass zeta function from the theory of elliptic functions.Suppose for a moment that Λ is a non-degenerate lattice in C, then In our context, it is more convenient to use a different normalization, which goes back to Eisenstein.Letting and noting that, for each R, Let c Λ be the inverse area of the fundamental domain of Λ.Then, it is not difficult to show (see Taylor [30,Appendix]) that in this case we have the limiting relation It is worth to mention that in [30] Taylor computed the Fourier expansions of the

The random Weierstrass zeta function
We return to the probabilistic setting, and let Λ be a stationary point process with finite second moment, and recall the quantities Ψ 1 (R) and Ψ 2 (R) defined in (3.1).Lemmas 3.1 and 3.2 in Section 3 allow us to define the random meromorphic function where Ψ 2 (∞) = lim R→∞ Ψ 2 (R).Note that, for any R > 1, Again by Lemmas 3.1 and 3.2, the last two terms on the right-hand side of (4.2) tend to zero as R → ∞, where the convergence is in L 2 (Ω, P) and is locally uniform in z.This hints that the function ζ Λ is not too far from being a stationary one.
Theorem 4.1.Let Λ be a stationary point process in C having a finite second moment.
Then the random meromorphic function ζ Λ , as defined in (4.1), has stationary increments.
That is, for any a ∈ C, the distribution of the random meromorphic function Proof.We have where .
Hence, by Lemmas 3.1 and 3.2, for any z ∈ C fixed.The above, together with the "lunar lemma" (Lemma 3.4), gives us and therefore, lim where H a is the L 2 (Ω, P)-limit This completes the proof, modulo the proof of Claim 4.2.
Claim 4.2.For R > 0, let f R,Λ be stationary random functions, and assume that there exist a random function f ∞,Λ such that for all z ∈ C. Then f ∞,Λ is stationary as well.
Proof.By the definition of stationarity, there exist random variables h R such that, for P-a.e.Λ ∈ Ω and for any z ∈ C, we have that By assumption, there exists another random variable, h ∞ , such that Moreover, by the invariance of P we have (h R − h ∞ ) • T z → 0 as well.We claim that for a.e.Λ ∈ Ω and for any z ∈ C, Indeed, wherever h ∞ is defined, Since both f ∞,Λ (z) − f R,Λ (z) and (h R − h ∞ ) • T z tend to 0 in L 2 (Ω, P) and since R > 0 was arbitrary, the right-hand side must vanish.Consequently, the representation (4.6) for f ∞,Λ follows.
As a corollary to Theorem 4.1, we observe that the distribution of the random meromorphic function is stationary (as above, the convergence is in L 2 (Ω, P) and is locally uniform in z).To obtain an equivariant representation of ℘ Λ similar to (4.4) it suffices to note that where the last equality follows from the crude bound which in turn is obtained by a simple covering argument and the triangle inequality.This yields the representation ℘ Λ (z) = P (T z Λ), where Looking ahead a little, we note that the representation (4.2) of ζ Λ suggests that existence of the L 2 (Ω, P)-limit becomes equivalent to existence of a stationary vector field V Λ (z).In its turn, it appears that existence of the limit (4.10) is easy to check on the spectral side.
If any of the three conditions (a)-(c) hold, we choose the particular normalization for the stationary random field.
We remark that in Condition (b), it is in fact sufficient to assume that Ψ 1 (R j ) is convergent in L 2 (Ω, P) along a sequence R j → ∞.Indeed, one can show that this condition directly implies (a).We will not pursue the details here.(c) ⇒ (a).First we prove that, for any ϕ ∈ D, the random variable where C ϕ is the Cauchy transform of ϕ, and Since the Cauchy transform C ϕ is a bounded on C, |λ|≤R C ϕ (λ) ϕ n Λ (RD), which implies that the first term on the right-hand side of (5.2) belongs to L 2 (Ω, P).By Lemma 3.1, we know that sup z∈spt(ϕ) which implies that the second term in the sum is in L 2 (Ω, P).The random variable Ψ 1 (R) satisfies the bound and the right-hand side has finite second moment by assumption, so we conclude that the third term on the right-hand side of (5.2) also lies in L 2 (Ω, P).Finally, Lemma 3.2 tells us that the last term on the right-hand side of (5.2) is in L 2 (Ω, P) as well, and all together we get that ζ Λ (ϕ) ∈ L 2 (Ω, P).
which implies that V Λ has a spectral measure ρ V Λ .In view of [34,Theorem 5], the identity ∂V Λ = π(n Λ − c Λ m) gives the relation Since ρ V Λ is locally finite, the result then follows by solving for ρ V Λ in (5.3).For the reader's convenience, we sketch a proof of (5.3).
Claim 5.2.Assume that V is a stationary random function such that for some random in the sense of distributions, and suppose moreover that V(ϕ) ∈ L 2 (Ω, P) for any ϕ ∈ D. Then whence, . Rewriting both sides in terms of the corresponding spectral measures (note that the spectral measure of n Λ − cm may differ from ρ Λ by at most an atom at the origin) and using that ∂ϕ(ξ) = πiξ ϕ(ξ), we get for some constant a. Next, we recall that the Fourier transforms of functions in D are dense in the Schwartz space S (see Remark 2.1).For an arbitrary compact set K ⊂ C \ {0}, we approximate its indicator function 1l K by a uniformly bounded sequence (ϕ n ) ⊂ S, converging to 1l K pointwise.Passing to the limit in (5.5), we get which gives (5.3).
With the proof of Claim 5.2 complete, we are done with the proof of Theorem 5.1.
Let us note that the possible atom at the origin of the spectral measure ρ Λ is irrelevant for the spectral condition (a).
The spectral measures of the zero process of GEFs, of the limiting Ginibre ensemble and of the stationarized random perturbation of the lattice satisfy spectral condition (a).
Hence, for these point processes, the generalized random function V Λ is stationary, while for the Poisson process it only has stationary increments.All this can be proved directly for each of these processes.Theorem 5.1 provides us with a unified reason for this phenomenon.
Remark 5.3.The following set of conditions on the reduced covariance measure κ Λ yields the spectral condition (a) in Theorem 5.1: (κ 1 ) existence of the 1st moment: Indeed, existence of the first moment of κ Λ yields that the spectral measure ρ Λ is absolutely continuous with a non-negative C 1 -smooth density h.By condition (κ 2 ), h vanishes at the origin.Since h is continuously differentiable, we conclude that h(ξ) = O(|ξ|) as ξ → 0, which yields the spectral condition (a).
The zeroth sum-rule (κ 2 ) is known to imply suppressed fluctuations of n Λ (see, for instance, [21, Section 1C]).Ghosh and Lebowitz [12] observed that the combination of (κ 2 ) with a stronger than (κ 1 ) decay of correlations yields an interesting geometric property of Λ known as number-rigidity.

Uniqueness
Theorem 5.4.Let V Λ be a generalized random function satisfying the following properties: (α) It is stationary.
(β) There exists a random constant c such that V Λ (z) + πcz is meromorphic with with poles exactly at Λ, all simple and with unit residue.
(γ) For any test function ϕ ∈ D, the random variable Then the spectral condition (a) of Theorem 5.1 holds, and the random fields V Λ and V Λ differ by a constant in L 2 (Ω, P) which is measurable with respect to F inv , the sigma-algebra of translation invariant events.
Proof.By Claim 5.2, the spectral measure ρ V of V Λ agrees with |ξ| −2 ρ Λ outside the origin, and hence the spectral condition (a) in Theorem 5.1 holds.We may therefore speak about the random function V Λ , and we have We next observe that where ν R is the current of integration ν R (f ) := {|z|=R} f (z)dz with respect to the differ- so it follows that ν R (0) = 0.As a consequence, the atom at the origin does not matter, so by repeating the same calculation backwards we arrive at Because |z|=R z dz = 2πiR 2 , the residue theorem gives us that and the same holds with V Λ replaced by V Λ and c replaced by πc Λ .But then and juxtaposing this identity with the fact that lim we get that c = c Λ , almost surely.
Let G Λ = V Λ − V Λ , so that G Λ is a stationary random entire function.If we choose the test-function ϕ to be radial with total integral 1, then where we used the mean value property of holomorphic functions to arrive at the second equality.The LHS is the difference of two random variables in L 2 (Ω, P), and the variance of each term is independent of z.Hence G Λ is a random entire function with Armed with this, we get the bound which, together with positivity, implies that the random variable is finite almost surely.The mean value property implies that for all |ζ| ≥ 1, so in view of Liouville's theorem G Λ is almost surely constant.
Finally, to see that G Λ (0) is measurable with respect to F inv , we note that for all R ≥ 1.By Wiener's ergodic theorem [3, Theorem 3], we get that lim measurable with respect to F inv , and we are done.
Note also that the above proof shows that the assumption (α) in Theorem 5.4 is stronger than necessary.What is really needed is that V Λ − V Λ has some uniformly bounded moment.
Remark 5.6.If we assume that the point process Λ is ergodic, i.e., that F inv is trivial, then any random function V Λ which satisfies conditions (α), (β) and (γ) differs from V Λ by a deterministic constant.Indeed, the function G Λ = V Λ − V Λ was shown to be a constant in the above proof, and measurable with respect to F inv .
If the field V Λ satisfies only conditions (α) and (β) of Theorem 5.4, then it is defined up to a random entire function with translation-invariant distribution.As was discovered by Weiss [33] such entire functions do exist.Developing his idea, one can show that, somewhat paradoxically, for any stationary process Λ, there exists a random field V Λ satisfying conditions (α) and (β).It is worth mentioning that these "exotic" random fields behave quite wildly (cf.Buhovsky-Glücksam-Logunov-Sodin [5]), as opposed to the "tame" ones from Theorem 5.1.

Fluctuations
The relations Theorem 5.7.Let Λ be a stationary point process in C having a finite second moment. Then, and as a consequence dρ Λ (ξ).Moreover, we have Proof.We begin by determining the spectral measure of ∆ a ζ Λ .We have ∂∆ a ζ Λ = π(n TaΛ − n Λ ), and the spectral measure for π(n Moving to the Fourier side, we get that for all ϕ ∈ D, and hence, It remains to analyze ρ ∆aζ Λ ({0}).This will involve a computation which we defer to Appendix B.1.In particular, these computations will reveal that the atom is only present in the rather exotic case when Λ "hyperfluctuates", i.e., when ρ Λ has an atom at the origin to begin with.With this, we conclude the proof of the first part.
Turning to the spectral measure of ℘ Λ , note that for any ϕ ∈ D, we have Moving to the Fourier side, we get that, for all ϕ ∈ D, To conclude the proof, it only remains to show that ρ ℘ Λ has no mass at the origin.We defer this computation to Appendix B.2.

5.3.2
The vector field V Λ Theorem 5.8.Suppose Λ is a stationary point process in C satisfying any of the equivalent conditions in Theorem 5.1.Then, It only remains to check that ρ V Λ ({0}) = 0, which we again postpone to Appendix B.3.
Theorem 5.8 yields a useful representation of the reduced covariance measure κ V Λ of the field V Λ .We denote by U µ the logarithmic potential of a signed measure µ, (provided that the integral on the RHS exists).
(τ 1 ) In this case, Thus, the density of κ V Λ equals −4π Assuming that the equivalent conditions of Theorem 5.1 hold, we will define a random potential Π Λ such that ∂ z Π Λ = 1 2 V Λ , and therefore, ∆Π Λ = 2π(n Λ − c Λ m) (both relations are understood in the sense of distributions).Since the field V Λ is stationary, this will yield that the potential Π Λ has stationary increments.The existence of the potential Π Λ with this property, in turn, shows that the vector field V Λ is stationary (and therefore, yields conditions (a) and (b) in Theorem 5.1).
Note that it is possible to prove an analogous result to Theorem 5.1, which states that the existence of a stationary potential is equivalent to the stronger spectral condition |ξ| 4 < ∞ (cf.Theorem 6.2 below).We will not pursue the details here.
We start with the entire function represented by the Hadamard product and note that, for each R > 1, where H(w) = − k≥3 w k /k.As R → ∞, the third and fourth factors on the RHS tend to 1 in L 2 (Ω, P) and locally uniformly in z, and therefore, We define Π Λ (z) Then, by a straightforward inspection, we get that Remark 6.1.Under the assumptions of Theorem 5.1, the quotient has a stationary distribution (as a function of z).An interesting characteristic of the point process Λ is the distribution of the phase To properly define this quantity, we fix a curve Γ connecting the points z and z + a, and consider the increment of the argument of This quantity was considered by Buckley-Sodin in [4] when F is replaced by the GEF.
Equivalently, one can consider the flux of the gradient field of the potential Π Λ through the curve Γ.
In [28], we study the asymptotic variance of this quantity under dilations of Γ.In the special case when Γ is a Jordan curve we recall that the change in argument coincides with the charge fluctuation around the mean in the domain enclosed by Γ.

The covariance structure of Π
Theorem 6.2.Suppose Λ is a stationary point process in C satisfying any of equivalent conditions in Theorem 5.1.Then, ∆ a Π Λ is stationary and Proof.We first claim that ∆ a Π Λ (z) can be written as where Q a is the L 2 (Ω, P)-limit which in particular says that ∆ a Π Λ is stationary.To verify (6.3)-( 6.4), we start with the representation (6.1), which in view of the spectral condition gives that where the limit is taken in L 2 (Ω, P).We readily rewrite this as where e R (Λ, z) is an "error term" given by where the new error term e R (Λ, z) is Turning to the spectral measure, note that by (6.2), we have which by the argument used in the proof of Theorem 5.7 shows that the desired equality for ρ ∆aΠ Λ holds outside the origin.Hence, it suffices to analyze the possible atom at the origin for the spectral measure.This is deferred to Appendix B.4.

A The stationary phase bound
The purpose of this section is to prove the following simple stationary phase bound.It is standard, but we did not find a textbook reference for the version we need.
Proposition A.1.There exists a universal constant C, such that for any for ω ≥ 1 and 0 ≤ a < b ≤ 2π.
Proof.If the critical points of the phase (i.e.0, π) are bounded away from the end-points, we may isolate the end-points with the help of a cut-off function.The conclusion then follows by estimating the end-point contributions with the van der Corput Lemma (see Proposition 2, Ch.VIII in [29]), and the contributions from the interior of [a, b] with the standard stationary phase bound for an interior critical point (see Theorem 7.7.5 in [15]).The Gaussian integral satisfies for ω, b ′ > 0, so the first term on the RHS of (A.1) is bounded above by Cω − 1 2 f .The second term can be estimated as follows Hence, the proof is complete.

B Atoms at the origin for the spectral measures
In this section, we collect all computations concerning the atoms at the origin for the various spectral measures from Sections 5 and 6.This will finalize the proofs of Theorems 5.7, 5.8 and 6.2.

B.1 The atom of ρ ∆aζ Λ
We will first prove the remaining part of Theorem 5.7 concerning the spectral measure of To this end, note that by the relation (5.9), translation-boundedness of ρ Λ , and the asymptotic formula (3.3) for the Bessel function, the function ξ → sup R≥1 J 0 (2πR|ξ|) is squareintegrable with respect to ρ ∆aζ Λ .As a consequence where the last step follows from the dominated convergence theorem.We next claim that the right-hand side of (B.1) may be interpreted as a variance To see this, first note that where σ R denotes the normalized arc-length measure on the circle {|z| = R}.By considering a suitable mollifier h j (ξ) = j 2 h(jξ) with h ∈ S, C hdm = 1, and putting ϕ R,j = σ R * h j , we may write where the limit is taken in L 2 (Ω, P).The formula (B.2) then follows by a reverse application of the Parseval identity for the atom at the origin.Now, recall the formula which, since a.s.there are no points of Λ which lie on the circle {|z| = R}, implies that and by invoking the representation (4.3) of ζ Λ we see that the first two terms on the right-hand side tend to zero in L 2 (Ω, P) as R → ∞.Hence, we get lim Putting this together with the above representation for ρ ∆aζ Λ ({0}), we see that which completes the proof.
The starting point is the analogous formula to (B.1), namely We next observe that
Arguing as in Appendix B.1, we find that For any given R ≥ 1, a.s., there are no points of Λ which lie on the circle {|z| = R}.By a residue computation, we get

B.4 The atom of ρ ∆aΠ Λ
To conclude the proof of Theorem 6.2, it only remains to prove that We again use the method introduced in Appendix B.1, in particular the formula We start with the representation Since the corresponding bound holds also for f R,B , we get and hence the result follows by invoking the dominated convergence theorem.
Since Λ is stationary and c Λ = c TaΛ , for each R ≥ 1 the random functions z → |λ−z−a|≤R 1 z + a − λ − |λ−z|≤R 1 z − λ + πc Λ ā are stationary.But then the limiting random function ζ Λ (z + a) − ζ Λ (z) is stationary as well (see Claim 4.2 below, which we record separately for later use).In fact, the proof of the claim shows that

Proof.
The implication (a) ⇒ (b) is exactly Lemma 3.3.It remains to prove the implications (b) ⇒ (c) and (c) ⇒ (a).(b) ⇒ (c).We let Ψ = Ψ 1 (∞) and proceed to show that V Λ , as given in (5.1), is stationary.By the relation (4.3) combined with Condition (b) in the theorem, we get lim R→∞ E V Λ (z) − are stationary for all R ≥ 1.The stationarity of V Λ then follows from Claim 4.2.

1 a
one to readily relate the spectral measures and the reduced covariance measures of these functions to the ones of the point process Λ. 5.3.1 The functions ∆ a ζ Λ and ℘ Λ Here we only assume that Λ is a stationary random planar point process having finite second moment.Then, by Theorem 4.1, the random meromorphic functions ∆ a ζ Λ (z) = ζ Λ (z + a) − ζ Λ (z), a ∈ C, and ℘ Λ (z) = lim a→0 ∆ a ζ Λ (z) are stationary.Since their second moments are infinite pointwise, we treat them as generalized stationary random processes on the space D of test-function by ∆ a ζ Λ (ϕ) = ζ Λ (T −a ϕ − ϕ), where T w ϕ(z) = ϕ(z + w), and by ℘ Λ
(ρ Λ ).In fact, the Fourier image of D is dense in L 2 (ρ Λ ).This follows from the fact that D is dense in the space S of Schwartz functions and that the Fourier transform is a topological isomorphism of S. But in S any convergent sequence is bounded byC(1 + |ξ| 2 ) −2 ,so applying the bounded convergence theorem, we find that the relation (2.1) holds for any pair of test functions ϕ, ψ ∈ S. To see that the Fourier image is dense in the full L 2 -space, it is sufficient to show that the closure of S in L 2 (ρ Λ ) contains any bounded continuous function f ∈ L 2 (ρ Λ ) with compact support.But this is again a direct consequence of the translation-boundedness of ρ Λ and the bounded convergence theorem