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

Abstract

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.

1 Introduction and Overview

Let ${\Lambda }$ be a stationary random point process in ${\mathbb {R}}^d$, $d\ge 2$, and let $n_\Lambda = \sum _{{\lambda }\in {\Lambda }} \delta _\lambda$ be its counting measure. We take the probability space of ${\Lambda }$ to be$(\Omega ,{\mathcal {F}},{\mathbb {P}})$, where $\Omega$ is the space of locally finite configurations in ${\mathbb {R}}^d$ and ${\mathcal {F}}$ is the $\sigma$-algebra generated by the events

$$\begin{aligned} \big \{{\Lambda }\in \Omega :n_{\Lambda }(B)=k\big \}, \quad k\in {\mathbb {Z}}_{\ge 0},\quad B \text { Borel subset of } {\mathbb {R}}^d. \end{aligned}$$

Stationarity of ${\Lambda }$ amounts to invariance of ${\mathbb {P}}$ under translations, i.e. under the maps $T_x:\Omega \rightarrow \Omega$, where $T_x{\Lambda }={\Lambda }-x$. Denote by $c_\Lambda$ the (first) intensity of $\Lambda$, i.e., assume that ${\mathbb {E}}[n_\Lambda ]=c_\Lambda m$, where m is the Lebesgue measure, and consider the following question:

Question 1

For which stationary point processes ${\Lambda }$ does there exist a stationary random vector field $V_{\Lambda }$ with ${\text {div}} V_\Lambda = n_{\Lambda }- c_{\Lambda }m$ in the sense of distributions?

Probably, the first instance of such a field is due to Chandrasekhar, who noted in [7, Ch. IV] that, for the Poisson point process ${\Lambda }$ in ${\mathbb {R}}^3$, the stationary vector field $V_{\Lambda }$ can be defined by the regularized series

$$\begin{aligned} V_{\Lambda }(x) = \lim _{R\rightarrow \infty }\, \sum _{|{\lambda }|\le R} \frac{x-{\lambda }}{|x-{\lambda }|^3} - \kappa c_{\Lambda }x, \end{aligned}$$

where the summation is over ${\lambda }\in {\Lambda }$ (here and elsewhere, we skip ${\Lambda }$ under the summation sign), and $\kappa = 4\pi /3$ is the volume of the unit ball in ${\mathbb {R}}^3$. Chatterjee-Peled-Peres-Romik [8, Proposition 1] gave the rigorous proof of this for the Poisson point process in ${\mathbb {R}}^d$ with $d \ge 3$.

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 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 [4, 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 $V_\Lambda$ written in complex coordinates is nothing but $(F'/F)(z) - {{\bar{z}}}$ 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 [20, 21, 26, 27] and the references therein.

Since the higher dimensional version of the question does not bring any essentially new difficulties comparing with the planar case,¹ 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 $\Lambda$ does there exist a random meromorphic function $f_\Lambda$ with poles exactly at $\Lambda$, all simple with unit residue, such that the random function $f_{\Lambda }(z) - \pi c_{\Lambda }{{\bar{z}}}$ is stationary?

In this paper we will provide an answer to Question 2 for point processes with a finite second moment, i.e., ${\mathbb {E}}[n_{\Lambda }(B)^2]<\infty$ for any bounded Borel set $B\subset {\mathbb {C}}$. Such stationary point processes admit a spectral measure $\rho _{\Lambda }$; see Sect. 2 for the details and examples. In Sect. 4.2, we will construct a random analogue $\zeta _{\Lambda }$ of the Weierstrass zeta function from the theory of elliptic functions. The function $\zeta _{\Lambda }$ is meromorphic with poles exactly at ${\Lambda }$, all simple with unit residue, but in general it is not stationary. One of our findings is Theorem 5.1. In the simplifying case when the point process ${\Lambda }$ is ergodic, it states that the following three conditions are equivalent:

• (i) The spectral condition $\displaystyle {\int _{|\xi |\le 1}\frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^2}<\infty }$ holds.

• (ii) The sum $\displaystyle {\sum _{{\lambda }\in {\Lambda },\,1\le |{\lambda }|\le R}{\lambda }^{-1}}$ converges in $L^2(\Omega ,{\mathbb {P}})$ as $R\rightarrow \infty$.

• (iii) For some random constant $\Psi \in L^2(\Omega ,{\mathbb {P}})$ the field $\zeta _{\Lambda }(z)-\Psi -\pi c_{\Lambda }{\bar{z}}$ is stationary.

Moreover, any solution $f_{\Lambda }$ to the problem in Question 2 with some very mild regularity (e.g., ${\mathbb {E}}[|f_{\Lambda }(0)|]<\infty$) coincides with $\zeta _{\Lambda }-\Psi$ 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 $\Psi$ is given in Lemma 3.3.

The spectral condition (i) can be thought of as a sum rule for the two-point measure of $\Lambda$, cf. Remark 5.3.

Curiously, if we do not impose any regularity on $f_{\Lambda }$, 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 [6]. However, when the spectral condition (i) fails, $f_{\Lambda }$ is necessarily quite “exotic” with wild growth at infinity and very heavy tails, cf. [6].

As an application of the ideas developed here, we study in [28] the variance of line integrals of $f_{\Lambda }$ along dilated rectifiable curves $R\,\Gamma$ in the large-R limit. When $\Gamma$ encloses a Jordan domain $\Omega$, this coincides with the “charge fluctuation” in $R\,\Omega$, 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 ${\Lambda }\mapsto V_{\Lambda }$ of $\Omega$ into the space of (Borel) measurable functions on ${\mathbb {C}}$ taking values in the extended complex plane ${\widehat{{\mathbb {C}}}}$, such that for all $z\in {\mathbb {C}}$,

$$\begin{aligned} V_{T_z{\Lambda }}(\cdot )=V_{\Lambda }(\cdot + z). \end{aligned}$$

Equivalently, $V_{\Lambda }$ is a stationary random vector field if it takes the form $V_\Lambda (z)=F(T_z \Lambda )$ for some measurable function $F:\Omega \rightarrow {\widehat{{\mathbb {C}}}}$.

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 and Linnik [17, Chap. XVIII, §2, 3] for stationary processes on ${\mathbb {Z}}$ and on ${\mathbb {R}}$, and by Davidovich [11] for stationary processes on ${\mathbb {Z}}^d$ and ${\mathbb {R}}^d$, $d\ge 2$. Aizenmann et al. [1, Theorem 3.1] gave a criterion for the existence of a stationary primitive for a stationary point process on ${\mathbb {R}}$. The relevant ergodic theoretic result is Schmidt’s coboundary theorem [25, Theorem 11.8].

In physics papers, Lebowitz and Martin [18] and Alastuey and 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 et al. [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 1950s 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 translation-invariance 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 ${\Lambda }$.

Organization. The article is organized as follows. In Sect. 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 Sect. 3 we analyze the convergence properties of reciprocal sums over $\Lambda$, e.g. $\sum _{1\le |\lambda -z|\le R}\frac{1}{\lambda ^j}$ 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 Sect. 4. This overall scheme works for any point process, but in general the field obtained only has stationary increments. In Sect. 5 we discuss the existence, uniqueness and covariance structure of the invariant field $V_\Lambda$ under the above-mentioned spectral condition. In Sect. 6 we conclude with a discussion of the existence and covariance structure of random potentials, that is, solutions to the equation $\Delta \Pi _{\Lambda }= 2\pi (n_{\Lambda }- c_{\Lambda }m)$.

Notation. We use the following notation frequently.

• ${\mathbb {C}}$, ${\mathbb {R}}$; the complex plane and the real line

• ${\mathbb {D}}$; the unit disk $\{|z|\le 1\}$

• $\partial =\partial _z$ and ${\bar{\partial }} =\partial _{{{\bar{z}}}}$; the Wirtinger derivatives

  $$\begin{aligned} \partial =\frac{1}{2}\left( \frac{\partial }{\partial x}-\textrm{i}\frac{\partial }{\partial y}\right) ,\qquad {\bar{\partial }}=\frac{1}{2}\left( \frac{\partial }{\partial x}+\textrm{i}\frac{\partial }{\partial y}\right) \end{aligned}$$

• m; the Lebesgue measure on ${\mathbb {C}}$

• ${\widehat{f}}$; the Fourier transform, with the normalization

  $$\begin{aligned} {\widehat{f}}(\xi )=\int _{{\mathbb {C}}}\,e^{-2\pi \textrm{i}x\cdot \xi }f(x)\,\textrm{d}m(x) \end{aligned}$$

• ${\mathfrak {D}}$, ${\mathcal {S}}$; the class of compactly supported $C^\infty$-smooth functions and the class of Schwartz functions, respectively

• ${\mathbb {E}}$, $\textsf{Cov}$, $\textsf{Var}$; the expectation, covariance and variance (with respect to an underlying probability space $(\Omega ,{\mathcal {F}},{\mathbb {P}})$)

• ${\mathcal {F}}_\mathsf{{inv}}$; the sigma-algebra of translation-invariant events

• $T_a$; translation by $a\in {\mathbb {C}}$, acting on functions by $T_af(z)=f(z+a)$ and on sets by $T_aS=\{s-a:s\in S\}$

• $\delta _z$; unit point mass at $z\in {\mathbb {C}}$

• $\rho _{\Lambda }$; the spectral measure of the point process ${\Lambda }$

• $\kappa _{\Lambda }$, $\tau _{\Lambda }$; the truncated and reduced truncated covariance measures for ${\Lambda }$, respectively.

Oftentimes, we will treat sums and series where the summation variable ranges over a point process ${\Lambda }$. When this is clear from the context, we will abuse notation slightly and simply write

$$\begin{aligned} \sum _{|{\lambda }|\le R}h({\lambda })= \sum _{\lambda \in {\Lambda }\cap R\,{\mathbb {D}}}h({\lambda }). \end{aligned}$$

We use the standard Landau O-notation and the symbol $\lesssim$ interchangeably. For limiting procedures involving an auxiliary parameter a, we use the notation $f_a(x)=O_a(g(x))$ to indicate that the implicit constant may depend on a.

2 The Second-Order Structure of Stationary Point Processes

2.1 The Spectral Measure

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ with a finite second moment, that is, we assume that ${\mathbb {E}}[n_{\Lambda }(B)^2]<\infty$ for any bounded Borel set B. The spectral measure of ${\Lambda }$ is a non-negative locally finite measure $\rho _{\Lambda }$ on ${\mathbb {C}}$ such that the “Parseval formula” holds:

$$\begin{aligned} \textsf{Cov}\bigl [n_{\Lambda }(\varphi ), n_{\Lambda }(\psi ) \bigr ] = \int _{{\mathbb {C}}} {{\widehat{\varphi }}}(\xi ) \overline{{\widehat{\psi }}(\xi )}\, \textrm{d}\rho _{{\Lambda }}(\xi ) = \langle {\widehat{\varphi }}, {\widehat{\psi }} \rangle _{L^2(\rho _{\Lambda })}, \end{aligned}$$

(2.1)

where $\varphi , \psi \in {\mathfrak {D}}$, $n_{\Lambda }(\varphi )$ denotes the linear statistic

$$\begin{aligned} n_{\Lambda }(\varphi ) = \sum _{{\lambda }\in {\Lambda }} \varphi ({\lambda }), \end{aligned}$$

and ${\widehat{\varphi }}$, ${\widehat{\psi }}$ are the Fourier transforms, i.e.,

$$\begin{aligned} {\widehat{\varphi }}(\xi ) = \int _{{\mathbb {C}}} e^{-2\pi \textrm{i} \xi \cdot x}\varphi (x)\, \textrm{d}m(x)\,. \end{aligned}$$

Existence of the spectral measure follows from a version of the Bochner theorem, see Gelfand-Vilenkin [13, Ch III, §3]. In the physics literature the spectral measure is commonly assumed to have a density, known as the the structure function.

Similarly, one defines the spectral measure for stationary random measures, as well as for generalized stationary random processes (stationary random distributions).

It is also worth mentioning that the spectral measures of stationary point processes (as well as of stationary random measures) are always translation-bounded [9, Ch. 8], that is, for every $r>0$, $\sup _{z\in {\mathbb {C}}} \rho _{\Lambda }\bigl ( \{\xi :|\xi -z|\le r \} \bigr ) < \infty$. Hence, for every $a>2$,

$$\begin{aligned} \int _{{\mathbb {C}}} \frac{\textrm{d}\rho _{\Lambda }(\xi )}{1+|\xi |^a} < \infty . \end{aligned}$$

(2.2)

Remark 2.1

While the formula (2.1) initially holds for $\varphi ,\psi \in {\mathfrak {D}}$, it readily extends to more general test functions (and even some tempered distributions) by taking the closure in $L^2(\rho _{\Lambda })$. In fact, the Fourier image of ${\mathfrak {D}}$ is dense in $L^2(\rho _{\Lambda })$. This follows from the fact that ${\mathfrak {D}}$ is dense in the space ${\mathcal {S}}$ of Schwartz functions and that the Fourier transform is a topological isomorphism of ${\mathcal {S}}$. But in ${\mathcal {S}}$ any convergent sequence is bounded by $C(1+|\xi |^2)^{-2}$, so applying the bounded convergence theorem, we find that the relation (2.1) holds for any pair of test functions $\varphi ,\psi \in {\mathcal {S}}$. To see that the Fourier image is dense in the full $L^2$-space, it is sufficient to show that the closure of ${\mathcal {S}}$ in $L^2(\rho _{\Lambda })$ contains any bounded continuous function $f\in L^2(\rho _{\Lambda })$ with compact support. But this is again a direct consequence of the translation-boundedness of $\rho _{\Lambda }$ and the bounded convergence theorem applied to $f_j=f*h_j$, where $h_j(\xi )=j^{2} h(j\xi )$, $h\in {\mathcal {S}}$, $\int _{{\mathbb {C}}} h\textrm{d}m=1$, as $j\rightarrow \infty$.

2.2 The Reduced Covariance Measure

The spectral measure of a point process is the Fourier transform of “the reduced covariance measure” $\kappa _{\Lambda }$, which is a signed measure on ${\mathbb {C}}$ such that

$$\begin{aligned} \textsf{Cov}\bigl [ n_{\Lambda }(\varphi ), n_{\Lambda }(\psi ) \bigr ]&= \iint _{{\mathbb {C}}\times {\mathbb {C}}} \varphi (z) \overline{\psi (z')}\, \textrm{d}\kappa _{\Lambda }(z-z')\, \textrm{d}m(z) \\&=\int _{\mathbb {C}}\Bigl [ \int _{\mathbb {C}}\varphi (z) \overline{\psi (z-w)}\, \textrm{d}m(z) \Bigr ]\, \textrm{d}\kappa _{\Lambda }(w)\,, \end{aligned}$$

see Daley and Vere–Jones [9, Ch. 8] (their notation is slightly different from the one we use here). Recalling that

$$\begin{aligned} {\mathbb {E}}\bigl [n_{\Lambda }(\varphi ) n_{\Lambda }({{\overline{\psi }}}) \bigr ] = \iint _{{\mathbb {C}}\times {\mathbb {C}}} \varphi (z)\overline{\psi (z')}\, \textrm{d}\nu _{\Lambda }(z-z')\, \textrm{d}m(z) + c_{\Lambda }\, \int _{\mathbb {C}}\varphi (z)\overline{\psi (z)}\, \textrm{d}m(z), \end{aligned}$$

where $c_\Lambda$ is the first intensity of the point process $\Lambda$ (i.e., the mean number of points of $\Lambda$ per unit area) and $\nu _{\Lambda }$ is the reduced two-point measure of ${\Lambda }$, we get that $\kappa _{\Lambda }= \tau _{\Lambda }+ c_{\Lambda }\delta _0$, where $\tau _{\Lambda }= \nu _{\Lambda }- c_{\Lambda }^2\, m$ is the (reduced) truncated two-point measure of ${\Lambda }$. Note that in the physics literature it is tacitly assumed that the measures $\nu _{\Lambda }$ and $\tau _{\Lambda }$ 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 ${\mathbb {C}}$.

The total variation of any reduced covariance measure $\kappa _{\Lambda }$ (and therefore of the reduced truncated measure $\tau _{\Lambda }$) is also translation-bounded [9, Ch. 8].

It is worth mentioning that for many point processes the tails of the measure $\tau _{\Lambda }$ decay very fast, which means that on high frequencies the spectral measure is close to the Lebesgue measure $c_{\Lambda }m$. 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 $\kappa _{\Lambda }$ [21].

2.3 The Conditional Intensity of ${\Lambda }$

We denote by ${\mathcal {F}}_\textsf{inv} \subset {\mathcal {F}}$ the sigma-algebra of translation-invariant events in ${\mathcal {F}}$. The random variable

$$\begin{aligned} {\mathfrak {c}}_{\Lambda }=\pi ^{-1}{\mathbb {E}}\bigl [ n_{\Lambda }({\mathbb {D}}) \big | {\mathcal {F}}_\textsf{inv} \bigr ], \end{aligned}$$

(2.3)

is known as the conditional intensity of ${\Lambda }$. Clearly, ${\mathbb {E}}[{\mathfrak {c}}_{\Lambda }] = c_{\Lambda }$, the first intensity of the point process. Furthermore, by the ergodic theorem, one can show that

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{n_{\Lambda }(R\,{\mathbb {D}})}{\pi R^2}={\mathfrak {c}}_{\Lambda }\end{aligned}$$

(2.4)

both almost surely and in $L^2(\Omega ,{\mathbb {P}})$ (see [10, Theorem 12.2.IV]). It is also not difficult to show (see [10, Exercise 12.2.9]) that $\textsf{Var}[{\mathfrak {c}}_{\Lambda }] = \rho _{{\Lambda }} (\{0\})$. Putting these pieces together, we arrive at

$$\begin{aligned} \textsf{Var}[{\mathfrak {c}}_{\Lambda }] = \rho _{{\Lambda }} (\{0\}) = \lim _{R\rightarrow \infty } \textsf{Var}\Bigl [ \frac{n_{\Lambda }(R\,{\mathbb {D}})}{\pi R^2} \Bigr ]. \end{aligned}$$

(2.5)

Hence, the random variable ${\mathfrak {c}}_{\Lambda }$ does not degenerate to the deterministic intensity $c_{\Lambda }$ if and only if $n_{\Lambda }(R\,{\mathbb {D}})$ asymptotically has the variance of the maximal possible order $R^4$, i.e., “hyper-fluctuates”, and this in turn is equivalent to $\rho _{\Lambda }(\{0\})>0$. Note that if the point process ${\Lambda }$ is ergodic, then the sigma-algebra ${\mathcal {F}}_\textsf{inv}$ contains only events of probability 0 or 1, and therefore, ${\mathfrak {c}}_{\Lambda }$ 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 ${\Lambda }$ to be any ergodic point process with spectral measure $\rho _{\Lambda }$, and denote by L a positive non-degenerate random variable with $\textsf{Var}[L]<\infty$. Then ${\Lambda }'=L^{-\frac{1}{2}}{\Lambda }$ is a point process with finite second moment. In view of (2.4), we get that

$$\begin{aligned} {\mathfrak {c}}_{{\Lambda }'} =\lim _{R\rightarrow \infty }\frac{n_{{\Lambda }'}(R\,{\mathbb {D}})}{\pi R^2}= \lim _{R\rightarrow \infty }\frac{n_{{\Lambda }}(L^{\frac{1}{2}}R\,{\mathbb {D}})}{\pi R^2} =c_{\Lambda }L, \end{aligned}$$

and hence (2.5) gives that $\rho _{{\Lambda }'}(\{0\})=c_{\Lambda }^2\,\textsf{Var}[L]$.

2.4 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 $c_{\Lambda }^2$ ($c_{\Lambda }$ is the intensity of the Poisson process), the truncated two-point function vanishes, $\rho _{\Lambda }= c_{\Lambda }^2 m$, 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 $N\times N$ 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

$$\begin{aligned} r(z_1, \ldots , z_k) = \pi ^{-k} e^{-\sum _{i=1}^k |z_i|^2} \det (e^{z_i{{\bar{z}}}_j} )_{1\le i, j \le k}, \end{aligned}$$

see, for instance, [4, Sect. 4.3.7]. This immediately yields the simple expression $-\pi ^{-2}e^{-\pi |z|^2}$ for the truncated two-point function and that $c_{\Lambda }=\pi ^{-1}$, which, in turn, yields that the spectral measure is absolutely continuous with the density $\pi ^{-1}(1-e^{-\pi |\xi |^2})$.

Zeroes of the Gaussian Entire Function. The Gaussian Entire Function (GEF, for short) is defined by the random Taylor series

$$\begin{aligned} F(z) = \sum _{n\ge 0} \zeta _n \frac{z^n}{\sqrt{n!}} \end{aligned}$$

with independent standard complex Gaussian coefficients $\zeta _n$. The most basic facts about GEFs and their zeroes can be found in [4, 22]. The two-point function and the spectral measure of the zero point process were explicitly computed by Forrester-Honner [12] and Nazarov-Sodin [23]. The intensity is given by $c_{\Lambda }=\pi ^{-1}$, the truncated two-point function equals h(|z|), where

$$\begin{aligned} h(r)=\frac{1}{2}\, \frac{\textrm{d}^2}{\textrm{d}r^2}\, r^2 (\coth r - 1), \end{aligned}$$

while the spectral measure is absolutely continuous with the density

$$\begin{aligned} \pi ^3 |\xi |^4\, \sum _{\ell \ge 1} \frac{1}{\ell ^3}\, e^{-\pi ^2|\xi |^2/\ell }. \end{aligned}$$

“Stationarized” random Gaussian perturbation of the lattice. This is a stationary point process defined as ${\Lambda }^a = \bigl \{\nu + \zeta ^a_\nu + U\bigr \}_{\nu \in {\mathbb {Z}}^2}$, where $\zeta ^a_\nu$ 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 $\zeta ^a_\nu$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 ${\mathbb {Z}}^2\setminus \{0\}$,

$$\begin{aligned} \rho _{{\Lambda }^a}=(1-e^{-2a\pi ^2|\xi |^2})m +\sum _{\nu \in {\mathbb {Z}}^2\setminus \{0\}} e^{-2a\pi ^2|\nu |^2} \delta _\nu . \end{aligned}$$

Moreover, the reduced covariance measure $\kappa _{{\Lambda }^a}$ is given by

$$\begin{aligned} \kappa _{{\Lambda }^a}=\delta _0-m+\sum _{\nu \ne 0}(2a\pi )^{-1}e^{-|z-\nu |^2/(2a)}m. \end{aligned}$$

This can be obtained by a direct computation on the spatial side, or by taking the inverse Fourier transform of $\rho _{{\Lambda }^a}$. In the limit as $a\rightarrow 0$, we obtain the randomly shifted lattice ${\Lambda }= \bigl \{\nu + U\bigr \}_{\nu \in {\mathbb {Z}}^2}$ with the purely atomic spectral measure $\rho _{\Lambda }= \sum _{\nu \in {\mathbb {Z}}^2\setminus \{0\}} \delta _\nu$.

3 Reciprocal Sums over Stationary Point Processes

3.1 Convergence of Three Series

Recall the notation $(\Omega , {\mathcal {F}}, {\mathbb {P}})$ for the probability space on which the point process ${\Lambda }$ is defined. To define the random Weierstrass zeta function, we need three lemmas.

Lemma 3.1

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ having a finite second moment. Then almost surely and in $L^2(\Omega , {\mathbb {P}})$,

$$\begin{aligned} \sum _{|{\lambda }|\ge 1} \frac{1}{|{\lambda }|^3} < \infty . \end{aligned}$$

For $R>1$, we set

$$\begin{aligned} \Psi _\ell (R) = \sum _{1\le |{\lambda }|\le R} \frac{1}{{\lambda }^\ell }, \qquad \ell = 1, 2. \end{aligned}$$

(3.1)

The behavior of these two sums as $R\rightarrow \infty$ will be important for us.

Lemma 3.2

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ having a finite second moment. Then there exists a random variable $\Psi _2(\infty )\in L^2(\Omega , {\mathbb {P}})$ such that

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\bigl [ |\Psi _2(R)- \Psi _2(\infty ) |^2 \bigr ] = 0. \end{aligned}$$

The convergence of $\Psi _1(R)$ in $L^2(\Omega , {\mathbb {P}})$ requires an additional property of the spectral measure $\rho _{\Lambda }$ of ${\Lambda }$.

Lemma 3.3

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ with spectral measure $\rho _{\Lambda }$, and assume that

$$\begin{aligned} \int _{0<|\xi |\le 1}\frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^2}<\infty . \end{aligned}$$

Then there exists a random variable $\Psi _1(\infty )\in L^2(\Omega , {\mathbb {P}})$ such that

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\bigl [ |\Psi _1(R)- \Psi _1(\infty ) |^2 \bigr ] = 0. \end{aligned}$$

It is worth mentioning that the conditional convergence of the series $\sum _{\Lambda }{\lambda }^{-\ell }$ with $\ell = 1, 2$, when ${\Lambda }$ is the limiting Ginibre process, or the zero process of GEF appear as auxiliary results in Ghosh–Peres [15, Sects. 8 and 10].

3.2 Proof of the Three Lemmas

The Bessel function of order $\nu \in {\mathbb {Z}}_+$ is given by

$$\begin{aligned} J_\nu (x) = \frac{1}{2\pi } \int _{-\pi }^{\pi } e^{\textrm{i} (x\sin \theta -\nu \theta )} \textrm{d} \theta . \end{aligned}$$

We will frequently use two basic properties of the Bessel function, namely the asymptotic formulas

$$\begin{aligned} J_\nu (x) = \frac{1}{\nu !} \left( \frac{x}{2}\right) ^\nu + o(x^\nu ),\qquad \text {as }\, x\rightarrow 0, \end{aligned}$$

(3.2)

and

$$\begin{aligned} \sup _{|x|\ge 1} |x|^{3/2}\left| J_\nu (x) - \sqrt{\frac{2}{\pi x}} \cos \left( x - \frac{\nu \pi }{2} - \frac{\pi }{4}\right) \right| < \infty . \end{aligned}$$

(3.3)

For the proof of both facts see, for instance, [32, Ch.7].

3.2.1 Proof of Lemma 3.1

Denote by ${\widetilde{\Psi }}_3(R) = \sum _{1\le |{\lambda }| \le R} |{\lambda }|^{-3}$. We will show that the limit ${\widetilde{\Psi }}_3(\infty )$ exists both almost surely and in $L^2(\Omega ,{\mathbb {P}})$. We have

$$\begin{aligned} {\mathbb {E}}[{\widetilde{\Psi }}_3(R)]&= \int _{\{1\le |x|\le R\}} \frac{{\mathbb {E}}[\textrm{d}n_{\Lambda }(x)]}{|x|^3} \\ {}&= {\mathbb {E}}[{\mathfrak {c}}_{\Lambda }] \int _{\{1\le |x|\le R\}} \frac{\textrm{d}m(x)}{|x|^3} \lesssim \int _{1}^{\infty } \frac{\textrm{d}s}{s^2} < \infty , \end{aligned}$$

which implies that ${\widetilde{\Psi }}_3(R)$ converges almost surely, as the sum consists of positive terms.

To show that ${\widetilde{\Psi }}_3(\infty ) \in L^2(\Omega ,{\mathbb {P}})$, we use the Parseval identity (2.1) to move to the spectral side, which gives

$$\begin{aligned} \textsf{Var}\left[ {\widetilde{\Psi }}_3(R)\right]&= \int _{{\mathbb {C}}} \left| \int _{\{1\le |x|\le R\}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{|x|^3}\right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ) \\&= \int _{{\mathbb {C}}} \left| \int _{1}^{R}\left( \int _{-\pi }^{\pi }e^{-2\pi \textrm{i} |\xi |t \cos \theta } \textrm{d}\theta \right) \frac{\textrm{d}t}{t^2}\right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ) \\&= \int _{{\mathbb {C}}}\left| 2\pi \int _{1}^{R} \frac{J_0(2\pi |\xi |t)}{t^2} \textrm{d}t\right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ). \end{aligned}$$

The integrand above is bounded uniformly in R. Thus, we need to check how fast it decays as $|\xi |$ becomes large. For this, we use the asymptotic formula (3.3) for the Bessel function and see that, for $|\xi |\ge 1$,

$$\begin{aligned} \left| \int _{1}^{R} \frac{J_0(2\pi |\xi |t)}{t^2} \textrm{d}t\right|&\lesssim \frac{1}{|\xi |^{3/2}} \int _{1}^{R} \frac{\textrm{d}t}{t^{7/2}} + \frac{1}{|\xi |^{1/2}}\left| \int _{1}^{R}\frac{\cos \left( 2\pi |\xi |t -\frac{\pi }{4}\right) }{t^{5/2}} \, \textrm{d}t\right| \\&\lesssim \frac{1}{|\xi |^{3/2}} + \frac{1}{|\xi |^{3/2}}\left[ 1+ \frac{1}{R^{5/2}} + \int _{1}^{R}\frac{\textrm{d}t}{t^{7/2}} \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} \sup _{R\ge 1} \left| \int _{1}^{R} \frac{J_0(2\pi |\xi |t)}{t^2} \textrm{d}t\right| ^2 \lesssim \min \{1, |\xi |^{-3}\} \end{aligned}$$

and the function on the RHS is $\textrm{d}\rho _{{\Lambda }}$-integrable. Hence, we can apply the dominated convergence theorem and deduce that ${\widetilde{\Psi }}_3(R)$ converge in $L^2(\Omega ,{\mathbb {P}})$ as well. $\square$

3.2.2 Proof of Lemma 3.2

Lemma 3.2 states that the random variables $\Psi _2(R) = \sum _{1\le |{\lambda }|\le R} {\lambda }^{-2}$ converge in $L^2(\Omega ,{\mathbb {P}})$ as $R\rightarrow \infty$ to a limiting random variable $\Psi _2(\infty ) \in L^2(\Omega ,{\mathbb {P}})$. This will follow once we show that $\{\Psi _2(R)\}_{R\ge 1}$ satisfies the Cauchy criterion in $L^2(\Omega ,{\mathbb {P}})$, which we do by a computation. For any $R\ge 1$,

$$\begin{aligned} {\mathbb {E}}\left[ \Psi _2(R)\right] = c_{\Lambda }\int _{\{1\le |{\lambda }|\le R\}} \frac{\textrm{d}m({\lambda })}{{\lambda }^2} = 0\,, \end{aligned}$$

and by the Parseval formula (2.1), applied with $\varphi (x)= \displaystyle {x^{-2}}{1\hspace{-2.5pt}\textrm{l}}_{\{R\le |x|\le R^\prime \}}$ for any $R^\prime > R$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \left| \Psi _2(R^\prime ) - \Psi _2(R)\right| ^2\right]&= \int _{{\mathbb {C}}} \left| \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x^2} \right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ) \\&= \int _{{\mathbb {C}}\setminus \{0\}} \left| \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x^2} \right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ). \end{aligned}$$

We may rewrite the inner integral on the RHS as

$$\begin{aligned} \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x^2}&= e^{-2\textrm{i}\chi }\int _{R}^{R^\prime } \left( \int _{-\pi }^{\pi }e^{-2\pi \textrm{i} |\xi |r \cos \theta } e^{-2\textrm{i}\theta }\textrm{d}\theta \right) \frac{\textrm{d} r}{r} \\&= -2\pi e^{-2\textrm{i}\chi } \int _{R}^{R^\prime } \frac{J_2(2\pi |\xi | r)}{r} \, \textrm{d}r \\&= e^{-2\textrm{i}\chi } \left( \frac{J_1(2\pi R^\prime |\xi |)}{R^\prime |\xi |} - \frac{J_1(2\pi R|\xi |)}{R|\xi |}\right) , \end{aligned}$$

where $\chi =\arg \xi$, and where in the last equality we used that $(J_1(x)/x)^\prime = -J_2(x)/x$. Thus,

$$\begin{aligned} {\mathbb {E}}\left[ \left| \Psi _2(R^\prime ) - \Psi _2(R)\right| ^2\right]&= \int _{{\mathbb {C}}\setminus \{0\}} \left| \frac{J_1(2\pi R^\prime |\xi |)}{R^\prime |\xi |} - \frac{J_1(2\pi R|\xi |)}{R|\xi |}\right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ). \end{aligned}$$

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 $\rho _{\Lambda }$, we get

$$\begin{aligned}&\lim _{R\rightarrow \infty }\sup _{R^{\prime }\ge R} {\mathbb {E}}\left[ \left| \Psi _2(R^\prime ) - \Psi _2(R)\right| ^2\right] \\&\quad \lesssim \mathop {{\overline{\lim }}}_{R\rightarrow \infty } \left[ \rho _{{\Lambda }}(\{0<|\xi |\le R^{-1/2}\}) + \int _{\{|\xi |\ge R^{-1/2}\}} \frac{\textrm{d} \rho _{{\Lambda }}(\xi )}{(1+R|\xi |)^3}\right] = 0. \end{aligned}$$

That is, $\Psi _2(R)$ is a Cauchy sequence in $L^2(\Omega ,{\mathbb {P}})$. $\square$

3.2.3 Proof of Lemma 3.3

We start by computing ${\mathbb {E}}[|\Psi _1(R^\prime ) - \Psi _1(R)|^2]$ for $R^\prime >R$. For any $R\ge 1$ we have

$$\begin{aligned} {\mathbb {E}}\left[ \Psi _1(R)\right] =c_{\Lambda }\int _{\{1\le |{\lambda }|\le R\}} \frac{\textrm{d} m({\lambda })}{{\lambda }} = 0 \end{aligned}$$

and so, for any $R^\prime >R$, the Parseval formula (2.1) gives that

$$\begin{aligned} {\mathbb {E}}\left[ \left| \Psi _1(R^\prime ) - \Psi _1(R)\right| ^2\right]&= \int _{{\mathbb {C}}}\left| \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x}\right| ^2 \textrm{d} \rho _{\Lambda }(\xi ) \\&= \int _{{\mathbb {C}}\setminus \{0\}} \left| \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x}\right| ^2 \textrm{d} \rho _{\Lambda }(\xi ). \end{aligned}$$

Since

$$\begin{aligned} e^{-2\pi \textrm{i} \xi \cdot x} =e^{-\pi \textrm{i} (\xi {\bar{x}} + \bar{\xi } x)} = {\bar{\partial }}_{x} \left( \frac{e^{-2\pi \textrm{i} \xi \cdot x}}{-\pi \textrm{i} \xi }\right) , \end{aligned}$$

we can use the Cauchy-Green formula to obtain

$$\begin{aligned} \int _{\{R \le |x|\le R^\prime \}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x}&= \frac{1}{2\pi \xi } \left( \int _{\{|x|=R^\prime \}} e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d}x}{x} -\int _{\{|x|=R\}}e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d}x}{x}\right) \\ {}&= \frac{\bar{\xi }}{|\xi |} \frac{J_0(2\pi R^\prime |\xi |)-J_0(2\pi R|\xi |)}{\xi }, \end{aligned}$$

and plugging this into the above formula for ${\mathbb {E}}\left[ \left| \Psi _1(R^\prime ) - \Psi _1(R)\right| ^2\right]$ gives

$$\begin{aligned} {} {\mathbb {E}}\left[ \left| \Psi _1(R^\prime ) - \Psi _1(R)\right| ^2\right] = \int _{{\mathbb {C}}\setminus \{0\}} \left( J_0(2\pi R^\prime |\xi |) -J_0(2\pi R|\xi |)\right) ^2 \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^2}. \end{aligned}$$

(3.4)

Since $J_0$ is bounded, we can use (3.4) and the asymptotic formula (3.3) for $J_0$ to get

$$\begin{aligned}&\lim _{R\rightarrow \infty }\sup _{R^{\prime }\ge R} {\mathbb {E}}\left[ \left| \Psi _1(R^\prime ) - \Psi _1(R)\right| ^2\right] \\&\quad \lesssim \mathop {{\overline{\lim }}}_{R\rightarrow \infty } \left[ \int _{\{0<|\xi |\le R^{-1/4}\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^2} + \frac{1}{R} \int _{\{|\xi |\ge R^{-1/4}\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^3}\right] \\&\quad \lesssim \mathop {{\overline{\lim }}}_{R\rightarrow \infty }\left[ R^{-3/4}\int _{\{0<|\xi |\le 1\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^2} + \frac{1}{R}\int _{\{|\xi |\ge 1\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^3}\right] = 0. \end{aligned}$$

That is, $\{\Psi _1(R)\}_{R\ge 1}$ is Cauchy in $L^2(\Omega ,{\mathbb {P}})$. This completes the proof. $\square$

3.3 Translation Properties of Reciprocal Sums

The random variables $\Psi _\ell (R)$ 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 ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ with finite second moment, with conditional intensity ${\mathfrak {c}}_{\Lambda }= \pi ^{-1}{\mathbb {E}}[n_{\Lambda }({{\mathbb {D}}}) \mid {\mathcal {F}}_{\textsf{inv}}]$. Then, for any $u,v,z\in {\mathbb {C}}$, we have

$$\begin{aligned} \lim _{R\rightarrow \infty } \textsf{Var}\left[ \sum _{|{\lambda }-u|\le R} \frac{1}{z-{\lambda }} - \sum _{|{\lambda }-v|\le R} \frac{1}{z-{\lambda }} + \pi {\mathfrak {c}}_{\Lambda }(\overline{u-v})\right] = 0. \end{aligned}$$

Although we will need this in the paper, we remark that the convergence is locally uniform in u, v, z.

In the case when ${\Lambda }$ is the Poisson point process, Lemma 3.4 was proved by Chatterjee, Peled, Peres and Romik in [8, Lemma 8] where they obtain the analogous result for $d\ge 3$, but the same proof works also for $d=2$.

Proof

By stationarity of ${\Lambda }$, 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 $R{\mathbb {D}}$. By the Cauchy-Green formula it holds that $\int _{\{|x-u|\le R\}} \frac{\textrm{d}m(x)}{x} = \pi {{\bar{u}}}$, and thus

$$\begin{aligned} {\mathbb {E}}\Big [\sum _{|{\lambda }-u|\le R} \frac{1}{{\lambda }}\Big ] = {\mathbb {E}}[{\mathfrak {c}}_{\Lambda }] \left( \int _{\{|x-u|\le R\}} \frac{\textrm{d}m(x)}{x}\right) = \pi c_{\Lambda }\, {{\bar{u}}}. \end{aligned}$$

Introduce the notation

$$\begin{aligned} X&= \sum _{|{\lambda }-u|\le R} \frac{1}{{\lambda }} - \sum _{|{\lambda }-v|\le R} \frac{1}{{\lambda }} - \pi c_{\Lambda }(\overline{u-v}), \quad \text {and} \ \ Y= \pi \big ({\mathfrak {c}}_{{\Lambda }} - c_{\Lambda }\big ). \end{aligned}$$

Clearly, ${\mathbb {E}}[X] = 0$ and ${\mathbb {E}}[X\,|\, Y] = Y(\overline{u-v})$, since the law of the point process ${\Lambda }$, conditioned on Y, has the intensity ${\mathfrak {c}}_{\Lambda }$. Thus,

$$\begin{aligned} \textsf{Var}\left[ X - Y(\overline{u-v})\right]&= {\mathbb {E}}\left[ \left| X-Y(\overline{u-v})\right| ^2\right] \\&= {\mathbb {E}}\left[ \left| X-{\mathbb {E}}[X\mid Y]\right| ^2\right] \\&= {\mathbb {E}}\left| X\right| ^2 - {\mathbb {E}}\left| Y(\overline{u-v})\right| ^2 \qquad \text {(Pythagoras' theorem)} \\&= \textsf{Var}[X] - \pi ^2|u-v|^2\, \textsf{Var}({\mathfrak {c}}_{{\Lambda }}) \\&= \textsf{Var}[X] - \pi ^2|u-v|^2\rho _{{\Lambda }}(\{0\}), \qquad \text {(by (}2.5\text {))}. \end{aligned}$$

Plugging in the definition of X and Y yields that

$$\begin{aligned} \begin{aligned}&\textsf{Var} \left[ \sum _{|{\lambda }-u|\le R} \frac{1}{{\lambda }} - \sum _{|{\lambda }-v|\le R} \frac{1}{{\lambda }} - \pi {\mathfrak {c}}_{\Lambda }(\overline{u-v})\right] \\&\quad = \int _{{\mathbb {C}}}\left| \left( \int _{\{|x-u|\le R\}} - \int _{\{|x-v|\le R\}}\right) e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x} \right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ) - \pi ^2 |u-v|^2 \rho _{{\Lambda }}(\{0\}) \\&\quad = \int _{{\mathbb {C}}\setminus \{0\}}\left| \left( \int _{\{|x-u|\le R\}} - \int _{\{|x-v|\le R\}}\right) e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x} \right| ^2 \textrm{d} \rho _{{\Lambda }}(\xi ) = I_1 + I_2, \end{aligned} \end{aligned}$$

(3.5)

where,

$$\begin{aligned} I_1&{\mathop {=}\limits ^{\textrm{def}}} \int _{\{0<|\xi |\le R^{-1/4}\}} \left| \left( \int _{\{|x-u|\le R\}} - \int _{\{|x-v|\le R\}}\right) e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x} \right| ^2 \textrm{d}\rho _{{\Lambda }}(\xi ), \\ I_2&{\mathop {=}\limits ^{\textrm{def}}} \int _{\{|\xi |\ge R^{-1/4}\}} \left| \left( \int _{\{|x-u|\le R\}}- \int _{\{|x-v|\le R\}}\right) e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x} \right| ^2 \textrm{d}\rho _{{\Lambda }}(\xi ). \end{aligned}$$

Since the integrand is bounded by some constant $C=C(u,v)>0$ (independent of R), we can bound $I_1$ as

$$\begin{aligned} I_1 \lesssim _{u,v} \rho _{{\Lambda }}\left( \{0<|\xi |\le R^{-1/4}\}\right) \xrightarrow {R\rightarrow \infty } 0. \end{aligned}$$

To bound the second integral $I_2$, we use the Cauchy-Green formula to compute the inner integral:

$$\begin{aligned} \int _{\{|x-u|\le R\}} e^{-2\pi \textrm{i} \xi \cdot x} \frac{\textrm{d} m(x)}{x}&= \frac{1}{-\textrm{i}\pi \xi } \int _{\{|x-u|\le R\}} \frac{{\bar{\partial }}_{x}\left( e^{-2\pi \textrm{i} \xi \cdot x}\right) }{x} \textrm{d} m(x) \\ {}&= \frac{1}{-\textrm{i}\xi } \left( \frac{1}{2\pi \textrm{i}} \int _{\{|z-u|= R\}} e^{-2\pi \textrm{i} \xi \cdot z}\frac{\textrm{d} z}{z} - 1\right) \\ {}&= \frac{1}{2\pi \xi } \int _{\{|z-u|= R\}} e^{-2\pi \textrm{i} \xi \cdot z}\frac{\textrm{d} z}{z} - \frac{1}{\textrm{i}\xi }. \end{aligned}$$

Plugging back the above in the definition of $I_2$ gives us

$$\begin{aligned} I_2 = \int _{\{|\xi |\ge R^{-1/4}\}} \left| \frac{g_u^R(\xi ) - g_v^R(\xi )}{2\pi \xi }\right| ^2 \textrm{d}\rho _{{\Lambda }}(\xi ), \end{aligned}$$

where

$$\begin{aligned} g_u^{R}(\xi ) {\mathop {=}\limits ^{\textrm{def}}} \int _{\{|z-u| = R\}}e^{-2\pi \textrm{i} \xi \cdot z} \frac{\textrm{d} z}{z} = e^{-2\pi \textrm{i} \xi \cdot u} \int _{0}^{2\pi } e^{-2\pi \textrm{i} |\xi |R \cos \theta } \frac{Re^{\textrm{i}\theta }}{Re^{\textrm{i}\theta } + u} \textrm{d} \theta . \end{aligned}$$

By the standard stationary phase bound (see Proposition A.1 in the Appendix), for any $u\in {\mathbb {C}}$ we have $\left| g_u^R(\xi )\right| \lesssim _u (1+R|\xi |)^{-1/2}$. Hence, we see that

$$\begin{aligned}&I_2 \lesssim _{u,v} \int _{\{|\xi |\ge R^{-1/4}\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^2(1+R|\xi |)} \\&\quad \lesssim \frac{1}{\sqrt{R}} \, \rho _{{\Lambda }}({\mathbb {D}}) + \frac{1}{R}\int _{\{|\xi | \ge 1\}} \frac{\textrm{d}\rho _{{\Lambda }}(\xi )}{|\xi |^3} \xrightarrow {R\rightarrow \infty } 0. \end{aligned}$$

Plugging back the bounds on $I_1$ and $I_2$ into (3.5), we get that

$$\begin{aligned} \lim _{R\rightarrow \infty }\textsf{Var}\left[ \sum _{|{\lambda }-u|\le R} \frac{1}{{\lambda }} - \sum _{|{\lambda }-v|\le R} \frac{1}{{\lambda }} - \pi {\mathfrak {c}}_{\Lambda }(\overline{u-v})\right] \le \mathop {{\overline{\lim }}}_{R\rightarrow \infty } \left( I_1 + I_2\right) = 0 \end{aligned}$$

which gives the lemma. $\square$

4 Fields and Potentials with Stationary Increments

4.1 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 ${\Lambda }$ is a non-degenerate lattice in ${\mathbb {C}}$, then

$$\begin{aligned} \zeta _{\Lambda }(z) = \frac{1}{z} + \sum _{{\lambda }\in {\Lambda }\setminus \{0\}} \Bigl ( \frac{1}{z-{\lambda }} + \frac{1}{{\lambda }} + \frac{z}{{\lambda }^2} \Bigr ). \end{aligned}$$

In our context, it is more convenient to use a different normalization, which goes back to Eisenstein. Letting

$$\begin{aligned} \Psi _2&= \lim _{R\rightarrow \infty }\, \sum _{0<|{\lambda }|\le R} \frac{1}{{\lambda }^2}\,, \\ \zeta _{\Lambda }(z)&= \frac{1}{z} + \sum _{{\lambda }\in {\Lambda }\setminus \{0\}} \Bigl ( \frac{1}{z-{\lambda }} + \frac{1}{{\lambda }} + \frac{z}{{\lambda }^2} \Bigr ) - \Psi _2 z\,, \end{aligned}$$

and noting that, for each R,

$$\begin{aligned} \sum _{0<|{\lambda }|\le R} \frac{1}{{\lambda }} = 0, \end{aligned}$$

we find that

$$\begin{aligned} \zeta _{\Lambda }(z)= \lim _{R\rightarrow \infty } \sum _{|{\lambda }|\le R} \frac{1}{z-{\lambda }}. \end{aligned}$$

Let $c_{\Lambda }$ be the inverse area of the fundamental domain of ${\Lambda }$. Then, it is not difficult to show (see Taylor [30, Appendix]) that the function $\zeta _{\Lambda }(z) -\pi c_{\Lambda }{{\bar{z}}}$ is ${\Lambda }$-invariant. Note that in this case we have the limiting relation

$$\begin{aligned} c_{\Lambda }= \lim _{R\rightarrow \infty } \frac{n_{\Lambda }(R\,{\mathbb {D}})}{\pi R^2}. \end{aligned}$$

It is worth to mention that in [30] Taylor computed the Fourier expansions of the functions $\zeta _{\Lambda }(z+a)-\zeta _{\Lambda }(z)$ and $\zeta _{\Lambda }(z) -\pi c_{\Lambda }{{\bar{z}}}$.

4.2 The Random Weierstrass Zeta Function

We return to the probabilistic setting, and let ${\Lambda }$ be a stationary point process with finite second moment, and recall the quantities $\Psi _1(R)$ and $\Psi _2(R)$ defined in (3.1). Lemmas 3.1 and 3.2 in Sect. 3 allow us to define the random meromorphic function

$$\begin{aligned} \zeta _\Lambda (z) {\mathop {=}\limits ^\textrm{def}} \sum _{|{\lambda }|<1} \frac{1}{z-{\lambda }} + \sum _{|{\lambda }|\ge 1} \Bigl ( \frac{1}{z-{\lambda }} + \frac{1}{{\lambda }}+ \frac{z}{{\lambda }^2} \Bigr ) - \Psi _2(\infty ) z, \end{aligned}$$

(4.1)

where $\Psi _2(\infty )=\lim _{R\rightarrow \infty }\Psi _2(R)$. Note that, for any $R>1$,

$$\begin{aligned} \zeta _{\Lambda }(z) = \sum _{|{\lambda }|\le R} \frac{1}{z-{\lambda }} + \Psi _1(R) + \bigl ( \Psi _2(R) - \Psi _2(\infty ) \bigr ) z + \sum _{|{\lambda }|> R} \frac{z^2}{{\lambda }^2(z-{\lambda })}. \end{aligned}$$

(4.2)

Again by Lemmas 3.1 and 3.2, the last two terms on the right-hand side of (4.2) tend to zero as $R\rightarrow \infty$, where the convergence is in $L^2(\Omega , {\mathbb {P}})$ and is locally uniform in z. This hints that the function $\zeta _{\Lambda }$ is not too far from being a stationary one.

Theorem 4.1

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ having a finite second moment. Then the random meromorphic function $\zeta _{\Lambda }$, as defined in (4.1), has stationary increments. That is, for any $a\in {\mathbb {C}}$, the distribution of the random meromorphic function $\mathsf{\Delta }_a \zeta _{\Lambda }(z) = \zeta _{\Lambda }(z+a)-\zeta _{\Lambda }(z)$ is stationary.

Proof

We have

$$\begin{aligned} \zeta _{\Lambda }(z) = \sum _{|{\lambda }|\le R} \frac{1}{z-{\lambda }} + \Psi _1(R) + (\Psi _2(R)-\Psi _2(\infty ))z + \chi _R({\Lambda },z)z^2, \end{aligned}$$

where

$$\begin{aligned} \chi _R({\Lambda },z) = \sum _{|{\lambda }|> R} \frac{1}{{\lambda }^2(z-{\lambda })}. \end{aligned}$$

Hence, by Lemmas 3.1 and 3.2,

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\, \bigg |\zeta _{\Lambda }(z) - \sum _{|{\lambda }|\le R} \frac{1}{z-{\lambda }} - \Psi _1(R)\bigg |^2 = 0\,, \end{aligned}$$

for any $z\in {\mathbb {C}}$ fixed. The above, together with the “lunar lemma” (Lemma 3.4), gives us

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\, \bigg |\zeta _{\Lambda }(z) - \sum _{|{\lambda }-z|\le R} \frac{1}{z-{\lambda }} - \pi {\mathfrak {c}}_{\Lambda }{{\bar{z}}} - \Psi _1(R) \bigg |^2 = 0 \,, \end{aligned}$$

(4.3)

and therefore,

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\, \bigg |\zeta _{{\Lambda }}(z+a) - \zeta _{{\Lambda }}(z) - \sum _{|{\lambda }-z-a|\le R} \frac{1}{z+a-{\lambda }} + \sum _{|{\lambda }-z|\le R} \frac{1}{z-{\lambda }}-\pi {\mathfrak {c}}_{\Lambda }{{\bar{a}}}\bigg |^2=0 \end{aligned}$$

for all $z,a\in {\mathbb {C}}$.

Since ${\Lambda }$ is stationary and ${\mathfrak {c}}_{\Lambda }={\mathfrak {c}}_{T_a{\Lambda }}$, for each $R\ge 1$ the random functions

$$\begin{aligned} z \mapsto \sum _{|{\lambda }-z-a|\le R} \frac{1}{z+a-{\lambda }} -\sum _{|{\lambda }-z|\le R} \frac{1}{z-{\lambda }} + \pi {\mathfrak {c}}_{\Lambda }{{\bar{a}}} \end{aligned}$$

are stationary. But then the limiting random function $\zeta _{\Lambda }(z+a)-\zeta _{\Lambda }(z)$ is stationary as well (see Claim 4.2, which we record separately for later use). In fact, the proof of the claim shows that

$$\begin{aligned} \zeta _{\Lambda }(\cdot +a) - \zeta _{\Lambda }(\cdot )=H_a(T_z{\Lambda }) \end{aligned}$$

(4.4)

where $H_a$ is the $L^2(\Omega ,{\mathbb {P}})$-limit

$$\begin{aligned} H_a({\Lambda })=\lim _{R\rightarrow \infty }\sum _{|{\lambda }|\le R}\Big (\frac{1}{a-{\lambda }} +\frac{1}{{\lambda }}\Big ). \end{aligned}$$

(4.5)

This completes the proof, modulo the proof of Claim 4.2. $\square$

Claim 4.2

For $R> 0$, let $f_{R,{\Lambda }}$ be stationary random functions, and assume that there exist a random function $f_{\infty ,{\Lambda }}$ such that

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\left[ \left| f_{R,{\Lambda }}(z) - f_{\infty ,{\Lambda }}(z)\right| ^2\right] =0 \end{aligned}$$

for all $z\in {\mathbb {C}}$. Then $f_{\infty ,{\Lambda }}$ is stationary as well.

Proof

By the definition of stationarity, there exist random variables $h_R$ such that, for ${\mathbb {P}}$-a.e. ${\Lambda }\in \Omega$ and for any $z\in {\mathbb {C}}$, we have that

$$\begin{aligned} f_{R,{\Lambda }}(z)=h_R(T_z{\Lambda }). \end{aligned}$$

By assumption, there exists another random variable, $h_\infty$, such that

$$\begin{aligned} {\mathbb {E}}\left[ |h_{R}({\Lambda })-h_{\infty }({\Lambda })|^2\right] ={\mathbb {E}}\left[ |f_{R,{\Lambda }}(0)-h_\infty ({\Lambda })|^2\right] \rightarrow 0,\qquad R\rightarrow \infty . \end{aligned}$$

Moreover, by the invariance of ${\mathbb {P}}$ we have $(h_R-h_\infty )\circ T_z\rightarrow 0$ as well. We claim that for a.e. ${\Lambda }\in \Omega$ and for any $z\in {\mathbb {C}}$,

$$\begin{aligned} f_{\infty ,{\Lambda }}(z)=h_\infty (T_z{\Lambda }). \end{aligned}$$

(4.6)

Indeed, wherever $h_\infty$ is defined,

$$\begin{aligned} f_{\infty ,{\Lambda }}(z)-h_\infty (T_z{\Lambda })&=f_{\infty ,{\Lambda }}(z)-f_{R,{\Lambda }}(z) +f_{R,{\Lambda }}(z)-h_{R}(T_z{\Lambda }) + h_{R}(T_z{\Lambda })-h_\infty (T_{z}{\Lambda })\\&=f_{\infty ,{\Lambda }}(z)-f_{R,{\Lambda }}(z)+h_{R}(T_z{\Lambda })-h_\infty (T_{z}{\Lambda }). \end{aligned}$$

Since both $f_{\infty ,{\Lambda }}(z)-f_{R,{\Lambda }}(z)$ and $(h_R-h_\infty )\circ T_z$ tend to 0 in $L^2(\Omega ,{\mathbb {P}})$ and since $R>0$ was arbitrary, the right-hand side must vanish. Consequently, the representation (4.6) for $f_{\infty ,{\Lambda }}$ follows. $\square$

As a corollary to Theorem 4.1, we observe that the distribution of the random meromorphic function

$$\begin{aligned} \wp _{\Lambda }(z) {\mathop {=}\limits ^\textrm{def}} - \zeta '_{\Lambda }(z) = \lim _{R\rightarrow \infty }\, \sum _{|{\lambda }|\le R} \frac{1}{(z-{\lambda })^2} \end{aligned}$$

is stationary (as above, the convergence is in $L^2(\Omega , {\mathbb {P}})$ and is locally uniform in z). To obtain an equivariant representation of $\wp _{\Lambda }$ similar to (4.4) it suffices to note that

$$\begin{aligned} \mathop {{\overline{\lim }}}_{R\rightarrow \infty }{\mathbb {E}}\Big |\sum _{|{\lambda }|\le R}\frac{1}{{\lambda }^2} -\sum _{|{\lambda }-z|\le R}\frac{1}{{\lambda }^2}\Big |^2 \le \mathop {{\overline{\lim }}}_{R\rightarrow \infty }\frac{2}{(R-|z|)^4}{\mathbb {E}}\big |n_{\Lambda }({\mathbb {D}}(0,R)\setminus {\mathbb {D}}(z,R)\big )\big |^2 =0,\nonumber \\ \end{aligned}$$

(4.7)

where the last equality follows from the crude bound

$$\begin{aligned} {\mathbb {E}}\big [n_{\Lambda }\big ({\mathbb {D}}(0,R)\setminus {\mathbb {D}}(z,R)\big )\big ]^2 \lesssim _zR^2, \end{aligned}$$

(4.8)

which in turn is obtained by a simple covering argument and the triangle inequality. This yields the representation $\wp _{\Lambda }(z)=P(T_z{\Lambda })$, where

$$\begin{aligned} P({\Lambda })=\lim _{R\rightarrow \infty }\sum _{|{\lambda }|\le R}\frac{1}{{\lambda }^2}. \end{aligned}$$

(4.9)

Looking ahead a little, we note that the representation (4.2) of $\zeta _{\Lambda }$ suggests that existence of the $L^2(\Omega , {\mathbb {P}})$-limit

$$\begin{aligned} \lim _{R\rightarrow \infty } \Psi _1(R) = \lim _{R\rightarrow \infty }\, \sum _{1\le |{\lambda }|\le R} \frac{1}{{\lambda }} \end{aligned}$$

(4.10)

becomes equivalent to existence of a stationary vector field $V_{\Lambda }(z)$. In its turn, it appears that existence of the limit (4.10) is easy to check on the spectral side.

5 The Invariant Field

5.1 Existence

Theorem 5.1

Let $\Lambda$ be a stationary point process in ${\mathbb {C}}$ having a finite second moment, and denote by ${\mathfrak {c}}_{\Lambda }$ the conditional intensity of ${\Lambda }$ on translation-invariant events. Then the following are equivalent:

• (a) The spectral measure $\rho _{\Lambda }$ satisfies

  $$\begin{aligned} \int _{\{0<|\xi |\le 1\}} \frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^2} < \infty . \end{aligned}$$

• (b) The sum

  $$\begin{aligned} \Psi _1(R) = \sum _{1\le |{\lambda }| \le R} \frac{1}{{\lambda }} \end{aligned}$$

  converges in $L^2(\Omega , {\mathbb {P}})$ to the limit $\Psi _1(\infty )$.

• (c) For some constant $\Psi \in L^2(\Omega ,{\mathbb {P}})$, the random field $\zeta _{\Lambda }(z)-\Psi -\pi {\mathfrak {c}}_{\Lambda }{\bar{z}}$ is stationary.

If any of the three conditions (a)–(c) hold, we choose the particular normalization

$$\begin{aligned} V_\Lambda (z)=\zeta _{\Lambda }(z)-\Psi _1(\infty )-\pi {\mathfrak {c}}_{\Lambda }{\bar{z}} \end{aligned}$$

(5.1)

for the stationary random field.

We remark that in Condition (b), it is in fact sufficient to assume that $\Psi _1(R_j)$ is convergent in $L^2(\Omega ,{\mathbb {P}})$ along a sequence $R_j\rightarrow \infty$. Indeed, one can show that this condition directly implies (a). We will not pursue the details here.

Proof

The implication (a) $\Rightarrow$ (b) is exactly Lemma 3.3. It remains to prove the implications (b) $\Rightarrow$ (c) and (c) $\Rightarrow$ (a).

$\underline{\mathrm{(b)} \Rightarrow \mathrm{(c)}.}$ We let $\Psi =\Psi _1(\infty )$ and proceed to show that $V_\Lambda$, as given in (5.1), is stationary. By the relation (4.3) combined with Condition (b) in the theorem, we get

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\, \bigg |\, V_{\Lambda }(z) -\sum _{|{\lambda }- z|\le R} \frac{1}{z-{\lambda }} \, \bigg |^2 = 0\,. \end{aligned}$$

Since ${\Lambda }$ is stationary, the random functions

$$\begin{aligned} z\mapsto \sum _{|{\lambda }- z|\le R} \frac{1}{z-{\lambda }} \end{aligned}$$

are stationary for all $R\ge 1$. The stationarity of $V_{\Lambda }$ then follows from Claim 4.2.

$\underline{\mathrm{(c)} \Rightarrow \mathrm{(a)}.}$ First we prove that, for any $\varphi \in {\mathfrak {D}}$, the random variable

$$\begin{aligned} \zeta _{{\Lambda }}(\varphi ) {\mathop {=}\limits ^{\textrm{def}}} \int _{{\mathbb {C}}} \zeta _{{\Lambda }} \varphi \, \textrm{d}m \end{aligned}$$

is in $L^2(\Omega , {\mathbb {P}})$. Let R be large enough so that $\varphi$ is supported in $\frac{1}{2}R{\mathbb {D}}$. Then

$$\begin{aligned} \zeta _{{\Lambda }}(\varphi )= & {} \sum _{|{\lambda }|\le R} {\mathcal {C}}_{\varphi }({\lambda }) +\int _{{\mathbb {C}}} \chi _{R}({\Lambda },z) z^2 \,\varphi (z)\, \textrm{d}m(z) \nonumber \\{} & {} + \Psi _1(R)\int _{{\mathbb {C}}} \varphi (z)\, \textrm{d}m(z) + \bigl (\Psi _2(R)-\Psi _2(\infty )\bigr )\int _{{\mathbb {C}}}z\, \varphi (z)\,\textrm{d}m(z), \end{aligned}$$

(5.2)

where $C_\varphi$ is the Cauchy transform of $\varphi$, and

$$\begin{aligned} \chi _R({\Lambda },z) = \sum _{|{\lambda }|> R} \frac{1}{{\lambda }^2(z-{\lambda })}. \end{aligned}$$

Since the Cauchy transform ${\mathcal {C}}_\varphi$ is a bounded on ${\mathbb {C}}$, $\Big |\sum _{|{\lambda }|\le R} {\mathcal {C}}_{\varphi }({\lambda })\Big | \lesssim _{\varphi } n_{{\Lambda }}(R{\mathbb {D}})$, which implies that the first term on the right-hand side of (5.2) belongs to $L^2(\Omega ,{\mathbb {P}})$. By Lemma 3.1, we know that

$$\begin{aligned} \sup _{z\in \textsf{spt}(\varphi )} {\mathbb {E}}\left| \chi _R({\Lambda },z)\right| ^2 \le 2\sum _{|{\lambda }|\ge R} \frac{1}{|{\lambda }|^3}<\infty \end{aligned}$$

which implies that the second term in the sum is in $L^2(\Omega ,{\mathbb {P}})$. The random variable $\Psi _1(R)$ satisfies the bound

$$\begin{aligned} |\Psi _1(R)|\le \sum _{1\le |{\lambda }|\le R}\frac{1}{|{\lambda }|}\le n_{\Lambda }({\mathbb {D}}(0,R)), \end{aligned}$$

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 $L^2(\Omega ,{\mathbb {P}})$. Finally, Lemma 3.2 tells us that the last term on the right-hand side of (5.2) is in $L^2(\Omega ,{\mathbb {P}})$ as well, and all together we get that $\zeta _{{\Lambda }}(\varphi )\in L^2(\Omega ,{\mathbb {P}})$.

Let now ${\mathcal {V}}_{\Lambda }(z)=\zeta _{\Lambda }(z)-\Psi -\pi {\mathfrak {c}}_{\Lambda }{\bar{z}}$. Then ${\mathcal {V}}_{\Lambda }(\varphi )\in L^2({\mathbb {P}})$ for any test function $\varphi$, which implies that ${\mathcal {V}}_{\Lambda }$ has a spectral measure $\rho _{{\mathcal {V}}_{\Lambda }}$. In view of [34, Theorem 5], the identity ${\bar{\partial }} {\mathcal {V}}_{\Lambda }=\pi (n_{\Lambda }-{\mathfrak {c}}_{\Lambda }m)$ gives the relation

$$\begin{aligned} \rho _{{\Lambda }} = |\xi |^2 \, \rho _{{\mathcal {V}}_{\Lambda }}\quad \text {on } \, {\mathbb {C}}\setminus \{0\}. \end{aligned}$$

(5.3)

Since $\rho _{{\mathcal {V}}_{\Lambda }}$ is locally finite, the result then follows by solving for $\rho _{{\mathcal {V}}_{\Lambda }}$ in (5.3). For the reader’s convenience, we sketch a proof of (5.3).

Claim 5.2

Assume that ${\mathcal {V}}$ is a stationary random function such that for some random constant $c\in L^2(\Omega ,{\mathbb {P}})$

$$\begin{aligned} {\bar{\partial }}{\mathcal {V}}=\pi (n_{\Lambda }-c\,m) \end{aligned}$$

in the sense of distributions, and suppose moreover that ${\mathcal {V}}(\varphi )\in L^2(\Omega ,{\mathbb {P}})$ for any $\varphi \in {\mathfrak {D}}$. Then

$$\begin{aligned} \rho _{{\Lambda }} = |\xi |^2 \, \rho _{{\mathcal {V}}_{\Lambda }}\qquad \text {on } \, {\mathbb {C}}\setminus \{0\}. \end{aligned}$$

(5.4)

Proof of Claim 5.2

Fix $\varphi \in {\mathfrak {D}}$. Since ${\bar{\partial }} {\mathcal {V}}= \pi (n_{\Lambda }- c m)$, we have

$$\begin{aligned} \pi (n_{\Lambda }- c m)(\varphi ) = - {\mathcal {V}}({\bar{\partial }} \varphi ), \end{aligned}$$

whence,

$$\begin{aligned} \textsf{Var} \big [(n_{\Lambda }- c m)(\varphi )\big ] = \frac{1}{\pi ^2} \textsf{Var} \big [{\mathcal {V}}({\bar{\partial }} \varphi )\big ]. \end{aligned}$$

Rewriting both sides in terms of the corresponding spectral measures (note that the spectral measure of $n_{\Lambda }- c m$ may differ from $\rho _{\Lambda }$ by at most an atom at the origin) and using that $\widehat{{\bar{\partial }} \varphi } (\xi ) = \pi \textrm{i} \xi \, {\widehat{\varphi }}(\xi )$, we get

$$\begin{aligned} \int _{{\mathbb {C}}\setminus \{0\}} |{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{{\Lambda }}+a|{\widehat{\varphi }}(0)|^2 = \int _{{\mathbb {C}}} |\xi |^2|{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{{\mathcal {V}}_{{\Lambda }}}, \end{aligned}$$

(5.5)

for some constant a. Next, we recall that the Fourier transforms of functions in ${\mathfrak {D}}$ are dense in the Schwartz space ${\mathcal {S}}$ (see Remark 2.1). For an arbitrary compact set $K\subset {\mathbb {C}}\setminus \{0\}$, we approximate its indicator function ${1\hspace{-2.5pt}\textrm{l}}_K$ by a uniformly bounded sequence $(\varphi _n) \subset {\mathcal {S}}$, converging to ${1\hspace{-2.5pt}\textrm{l}}_K$ pointwise. Passing to the limit in (5.5), we get

$$\begin{aligned} \rho _{{\Lambda }}(K) = \int _{K} |\xi |^2\, \textrm{d} \rho _{{\mathcal {V}}_{{\Lambda }}}(\xi ), \end{aligned}$$

which gives (5.3). $\square$

With the proof of Claim 5.2 complete, we are done with the proof of Theorem 5.1. $\square$

Let us note that the possible atom at the origin of the spectral measure $\rho _{\Lambda }$ 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 $V_{\Lambda }$ 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 $\kappa _{\Lambda }$ yields the spectral condition (a) in Theorem 5.1:

($\kappa _1$):

existence of the 1st moment: $\displaystyle \int _{\mathbb {C}}|s|\, \textrm{d}|\kappa _{\Lambda }|(s) < \infty$;

($\kappa _2$):

the zeroth sum-rule: $\kappa _{\Lambda }({\mathbb {C}})=0$.

Indeed, existence of the first moment of $\kappa _{\Lambda }$ yields that the spectral measure $\rho _{\Lambda }$ is absolutely continuous with a non-negative $C^1$-smooth density h. By condition ($\kappa _2$), h vanishes at the origin. Since h is continuously differentiable, we conclude that $h(\xi ) = O(|\xi |)$ as $\xi \rightarrow 0$, which yields the spectral condition (a).

The zeroth sum-rule ($\kappa _2$) is known to imply suppressed fluctuations of $n_{\Lambda }$ (see, for instance, [21, Sect. 1C]). Ghosh and Lebowitz [14] observed that the combination of $(\kappa _2)$ with a stronger than $(\kappa _1)$ decay of correlations yields an interesting geometric property of ${\Lambda }$ known as number-rigidity.

5.2 Uniqueness

Theorem 5.4

Let ${\mathcal {V}}_{\Lambda }$ be a generalized random function satisfying the following properties:

($\alpha$) It is stationary.

($\beta$) There exists a random constant c such that ${\mathcal {V}}_{\Lambda }(z) +\pi c{\bar{z}}$ is meromorphic with with poles exactly at $\Lambda$, all simple and with unit residue.

($\gamma$) For any test function $\varphi \in {\mathfrak {D}}$, the random variable

$$\begin{aligned} {\mathcal {V}}_{\Lambda }(\varphi ) = \int _{\mathbb {C}}{\mathcal {V}}_{\Lambda }\varphi \, \textrm{d}m \end{aligned}$$

lies in $L^2(\Omega ,{\mathbb {P}})$.

Then the spectral condition (a) of Theorem 5.1 holds, and the random fields ${\mathcal {V}}_{\Lambda }$ and $V_{\Lambda }$ differ by a constant in $L^2(\Omega ,{\mathbb {P}})$ which is measurable with respect to ${\mathcal {F}}_{\textsf {inv} }$, the sigma-algebra of translation invariant events.

Proof

By Claim 5.2, the spectral measure $\rho _{{\mathcal {V}}}$ of ${\mathcal {V}}_{\Lambda }$ agrees with $|\xi |^{-2}\rho _{\Lambda }$ outside the origin, and hence the spectral condition (a) in Theorem 5.1 holds. We may therefore speak about the random function $V_{\Lambda }$, and we have

$$\begin{aligned} {1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi )\textrm{d}\rho _{V_{\Lambda }}(\xi ) ={1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi )\textrm{d}\rho _{{\mathcal {V}}_{\Lambda }}(\xi ). \end{aligned}$$

We next observe that

$$\begin{aligned} {\mathbb {E}}\bigg |\int _{\{|z|=R\}} {\mathcal {V}}_{\Lambda }(z)\, \textrm{d}z\bigg |^2&=\int _{{\mathbb {C}}}|{{\widehat{\nu }}}_R|^2 \textrm{d}\rho _{{\mathcal {V}}_{\Lambda }}(\xi )\\&=\int _{{\mathbb {C}}}|{{\widehat{\nu }}}_R|^2 \textrm{d}\rho _{V_{\Lambda }}(\xi ) + \big (\rho _{{\mathcal {V}}_{\Lambda }}(\{0\})-\rho _{V_{\Lambda }}(\{0\})\big )|{\widehat{\nu }}_R(0)|^2\, , \end{aligned}$$

where $\nu _R$ is the current of integration $\nu _R(f):=\int _{\{|z|=R\}}f(z)\textrm{d}z$ with respect to the differential $\textrm{d}z$ along $|z|=R$. Hence

$$\begin{aligned} {\widehat{\nu }}_R(\xi )=\int _{\{|z|=R\}}e^{-2\pi i \xi \cdot z}\textrm{d}z, \end{aligned}$$

so it follows that ${\widehat{\nu }}_R(0)=0$. As a consequence, the atom at the origin does not matter, so by repeating the same calculation backwards we arrive at

$$\begin{aligned} {\mathbb {E}}\, \bigg |\int _{\{|z|=R\}} {\mathcal {V}}_{\Lambda }(z)\, \textrm{d}z\bigg |^2 ={\mathbb {E}}\, \bigg |\int _{\{|z|=R\}} V_{\Lambda }(z)\, \textrm{d}z\bigg |^2. \end{aligned}$$

Because $\int _{|z|=R} {{\bar{z}}} \, \textrm{d}z = 2\pi \textrm{i} R^2$, the residue theorem gives us that

$$\begin{aligned} \frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}}{\mathcal {V}}_{\Lambda }(z)\,\textrm{d}z=\frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}}({\mathcal {V}}_{\Lambda }(z)+\pi c {{\bar{z}}})\,\textrm{d}z-\pi c R^2=n_{{\Lambda }}(R{\mathbb {D}})-\pi cR^2\nonumber \\ \end{aligned}$$

(5.6)

and the same holds with ${\mathcal {V}}_{\Lambda }$ replaced by $V_{\Lambda }$ and c replaced by $\pi {\mathfrak {c}}_{\Lambda }$. But then

$$\begin{aligned} {\mathbb {E}}\Big [\Big |\frac{n_{\Lambda }(R{\mathbb {D}})}{R^2} - \pi c \, \Big |^2\Big ]= {\mathbb {E}}\Big [\Big |\frac{n_{\Lambda }(R{\mathbb {D}})}{R^2} - \pi {\mathfrak {c}}_{\Lambda }\Big |^2\Big ], \end{aligned}$$

and juxtaposing this identity with the fact that

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\Big [\Big |\frac{n_{\Lambda }(R{\mathbb {D}})}{R^2} -\pi {\mathfrak {c}}_{\Lambda }\Big |^2\Big ] = 0, \end{aligned}$$

(5.7)

we get that $c= {\mathfrak {c}}_{\Lambda }$, almost surely.

Let $G_{\Lambda }= V_{\Lambda }- {\mathcal {V}}_{\Lambda }$, so that $G_{\Lambda }$ is a stationary random entire function. If we choose the test-function $\varphi$ to be radial with total integral 1, then

$$\begin{aligned} \left( V_{\Lambda }-{\mathcal {V}}_{\Lambda }\right) (T_z\varphi )&= \int _{\mathbb {C}}\varphi (z+w)G_{\Lambda }(w) \, \textrm{d}m(w)\\&=\int _{0}^\infty \varphi (r)r\int _{0}^{2\pi }G_{\Lambda }(z+r e^{i\theta }) \, \textrm{d}\theta \, \textrm{d}r \\&=2\pi \, G_{\Lambda }(z)\int _{0}^\infty \varphi (r)r\,\textrm{d}r=2\pi \, G_{\Lambda }(z), \end{aligned}$$

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 $L^2(\Omega ,{\mathbb {P}})$, and the variance of each term is independent of z. Hence $G_{\Lambda }$ is a random entire function with

$$\begin{aligned} \sup _{z\in {\mathbb {C}}} {\mathbb {E}}\,|G_{\Lambda }(z)|^2 < \infty . \end{aligned}$$

(5.8)

Armed with this, we get the bound

$$\begin{aligned} {\mathbb {E}}\,\bigg |\int _{{\mathbb {C}}} \frac{|G_{\Lambda }(z)|}{1+|z|^{5/2}}\, \textrm{d}m(z)\bigg | \lesssim \int _{1}^{\infty } \frac{\textrm{d}t}{t^{3/2}} <\infty , \end{aligned}$$

which, together with positivity, implies that the random variable

$$\begin{aligned} \int _{{\mathbb {C}}} \frac{|G_{\Lambda }(z)|}{1+|z|^{5/2}}\, \textrm{d}m(z) \end{aligned}$$

is finite almost surely. The mean value property implies that

$$\begin{aligned} |G_{\Lambda }(\zeta )|&\le \frac{1}{\pi |\zeta |^2} \int _{\{|z-\zeta |\le |\zeta |\}} |G_{\Lambda }(z)| \, \textrm{d}m(z) \\&\lesssim \frac{|\zeta |^{5/2}}{|\zeta |^2} \int _{{\mathbb {C}}} \frac{|G_{\Lambda }(z)|}{1+|z|^{5/2}} \textrm{d} m(z) \lesssim |\zeta |^{1/2} \end{aligned}$$

for all $|\zeta |\ge 1$, so in view of Liouville’s theorem $G_{\Lambda }$ is almost surely constant.

Finally, to see that $G_{\Lambda }(0)$ is measurable with respect to ${\mathcal {F}}_\mathsf{{inv}}$, we note that

$$\begin{aligned} G_{\Lambda }(0) = \frac{1}{\pi R^2} \int _{\{|z|\le R\}} \Big (V_{\Lambda }(z) - {\mathcal {V}}_{\Lambda }(z)\Big ) \, \textrm{d}m(z) \end{aligned}$$

for all $R\ge 1$. By Wiener’s ergodic theorem [3, Theorem 3], we get that

$$\begin{aligned} \lim _{R\rightarrow \infty } {\mathbb {E}}\, \bigg | \frac{1}{\pi R^2} \int _{\{|z|\le R\}} \Big ({\mathcal {V}}_{\Lambda }(z) - V_{\Lambda }(z)\Big ) \, \textrm{d}m(z) - {\mathbb {E}}\big [{\mathcal {V}}_{\Lambda }(0) - V_{\Lambda }(0) \mid {\mathcal {F}}_\mathsf{{inv}}\big ]\bigg | = 0\,, \end{aligned}$$

which immediately implies that $G_{\Lambda }(0) = {\mathbb {E}}\big [G_{\Lambda }(0) \mid {\mathcal {F}}_\mathsf{{inv}}\big ]$ in $L^1(\Omega ,{\mathbb {P}})$. That is, $G_{\Lambda }(0)$ is measurable with respect to ${\mathcal {F}}_\mathsf{{inv}}$, and we are done. $\square$

Remark 5.5

We will note that, following [6], one can significantly relax the condition ($\gamma$) in Theorem 5.4 (cf. [6, Theorem 3A]). We will not pursue this here.

Note also that the above proof shows that the assumption $(\alpha )$ in Theorem 5.4 is stronger than necessary. What is really needed is that $V_{\Lambda }-{\mathcal {V}}_{\Lambda }$ has some uniformly bounded moment.

Remark 5.6

If we assume that the point process ${\Lambda }$ is ergodic, i.e., that ${\mathcal {F}}_{\textsf{inv}}$ is trivial, then any random function ${\mathcal {V}}_{\Lambda }$ which satisfies conditions ($\alpha$), ($\beta$) and ($\gamma$) differs from $V_{\Lambda }$ by a deterministic constant. Indeed, the function $G_{\Lambda }= V_{\Lambda }- {\mathcal {V}}_{\Lambda }$ was shown to be a constant in the above proof, and measurable with respect to ${\mathcal {F}}_{\textsf{inv}}$.

If the field $V_{\Lambda }$ satisfies only conditions ($\alpha$) and ($\beta$) 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. Develo** his idea, one can show that, somewhat paradoxically, for any stationary process ${\Lambda }$, there exists a random field $V_{\Lambda }$ satisfying conditions ($\alpha$) and ($\beta$). It is worth mentioning that these “exotic” random fields behave quite wildly (cf. Buhovsky et al. [6]), as opposed to the “tame” ones from Theorem 5.1.

5.3 Fluctuations

The relations $\partial _{{{\bar{z}}}} V_{\Lambda }= \pi (n_{\Lambda }- {\mathfrak {c}}_{\Lambda })$ and $\partial _{z} V_{\Lambda }= -\wp _{\Lambda }$ allow one to readily relate the spectral measures and the reduced covariance measures of these functions to the ones of the point process ${\Lambda }$.

5.3.1 The Functions $\mathsf \Delta _a \zeta _{\Lambda }$ and $\wp _{\Lambda }$

Here we only assume that ${\Lambda }$ is a stationary random planar point process having finite second moment. Then, by Theorem 4.1, the random meromorphic functions $\mathsf \Delta _a \zeta _{\Lambda }(z) = \zeta _{\Lambda }(z+a) - \zeta _{\Lambda }(z)$, $a\in {\mathbb {C}}$, and $\wp _{\Lambda }(z) = \lim _{a\rightarrow 0} \frac{1}{a}\, \mathsf \Delta _a \zeta _{\Lambda }(z)$ are stationary. Since their second moments are infinite pointwise, we treat them as generalized stationary random processes on the space ${\mathfrak {D}}$ of test-function by $\mathsf \Delta _a \zeta _{\Lambda }(\varphi ) = \zeta _{\Lambda }(T_{-a}\varphi - \varphi )$, where $T_w\varphi (z) = \varphi (z+w)$, and by $\wp _{\Lambda }(\varphi ) = - (\partial _z \zeta _{\Lambda })(\varphi ) = \zeta _{\Lambda }(\partial _z \varphi )$.

Theorem 5.7

Let ${\Lambda }$ be a stationary point process in ${\mathbb {C}}$ having a finite second moment. Then,

$$\begin{aligned} \textrm{d}\rho _{\mathsf \Delta _a\zeta _{\Lambda }}(\xi ) = {1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi )\, \frac{|1-e^{2\pi \textrm{i} a\cdot \xi }|^2}{|\xi |^2}\, \textrm{d}\rho _{\Lambda }(\xi ) + \pi ^2|a|^2\rho _{\Lambda }(\{0\})\delta _0(\xi ), \end{aligned}$$

and as a consequence $\textrm{d}\rho _{\mathsf \Delta _a\zeta _{\Lambda }-\pi {\mathfrak {c}}_{\Lambda }{\bar{a}}}(\xi ) ={1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}{\setminus }\{0\}}(\xi )\, \frac{|1-e^{2\pi \textrm{i} a\cdot \xi }|^2}{|\xi |^2}\, \textrm{d}\rho _{\Lambda }(\xi )$. Moreover, we have

$$\begin{aligned} \textrm{d}\rho _{\wp _{\Lambda }}(\xi ) = \pi ^2 {1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi ) \textrm{d}\rho _{\Lambda }(\xi ). \end{aligned}$$

Proof

We begin by determining the spectral measure of $\mathsf{\Delta }_a\zeta _{\Lambda }$. We have ${\bar{\partial }} \mathsf{\Delta }_a \zeta _{\Lambda }=\pi (n_{T_a {\Lambda }}-n_{{\Lambda }})$, and the spectral measure for $\pi (n_{T_a {\Lambda }}-n_{{\Lambda }})$ equals $\pi ^2|1-e^{2\pi \textrm{i}\xi \cdot a}|^2\textrm{d}\rho _{\Lambda }(\xi )$. For any $\varphi \in {\mathfrak {D}}$, we have

$$\begin{aligned} {\Delta }_a\zeta _{{\Lambda }}({\bar{\partial }}\varphi ) = -\pi (n_{T_a{\Lambda }}-n_{\Lambda })(\varphi ), \end{aligned}$$

which in turn implies that $\textsf{Var}\big ({\Delta }_a\zeta _{{\Lambda }}({\bar{\partial }}\varphi )\big ) = \pi ^2\, \textsf{Var}\big ((n_{T_a {\Lambda }}-n_{\Lambda }) (\varphi )\big )$. Moving to the Fourier side, we get that for all $\varphi \in {\mathfrak {D}}$,

$$\begin{aligned} \int _{{\mathbb {C}}} |\xi |^2 |{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }} = \int _{{\mathbb {C}}} \big |1-e^{2\pi \textrm{i} a\cdot \xi }\big |^2 |{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{{\Lambda }}, \end{aligned}$$

and hence,

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }} = \frac{\big |1-e^{2\pi \textrm{i} a\cdot \xi }\big |^2}{|\xi |^2} \, \rho _{{\Lambda }}\, \qquad \text {on} \ \, {\mathbb {C}}\setminus \{0\}. \end{aligned}$$

(5.9)

It remains to analyze $\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\})$. 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 ${\Lambda }$ “hyperfluctuates”, i.e., when $\rho _{\Lambda }$ 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 $\wp _{\Lambda }$, note that for any $\varphi \in {\mathfrak {D}}$, we have

$$\begin{aligned} \wp _{{\Lambda }}({\bar{\partial }}\varphi ) = -\partial \zeta _{{\Lambda }}({\bar{\partial }}\varphi ) = -{\bar{\partial }} \zeta _{{\Lambda }} ( \partial \varphi ) = \pi n_{\Lambda }(\partial \varphi )\,, \end{aligned}$$

which in turn implies that $\textsf{Var}[\wp _{{\Lambda }}({\bar{\partial }}\varphi )] = \pi ^2\, \textsf{Var}[n_{\Lambda }(\partial \varphi )]$. Moving to the Fourier side, we get that, for all $\varphi \in {\mathfrak {D}}$,

$$\begin{aligned} \int _{{\mathbb {C}}} |\xi |^2 |{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{\wp _{\Lambda }} = \pi ^2 \int _{{\mathbb {C}}} |\xi |^2 |{\widehat{\varphi }}|^2 \, \textrm{d}\rho _{{\Lambda }} \end{aligned}$$

which gives that

$$\begin{aligned} \rho _{\wp _{{\Lambda }}} = \pi ^2 \rho _{{\Lambda }}\, \qquad \text {on} \ \, {\mathbb {C}}\setminus \{0\}\,. \end{aligned}$$

(5.10)

To conclude the proof, it only remains to show that $\rho _{\wp _{\Lambda }}$ has no mass at the origin. We defer this computation to Appendix B.2. $\square$

5.3.2 The Vector Field $V_{\Lambda }$

Theorem 5.8

Suppose $\Lambda$ is a stationary point process in ${\mathbb {C}}$ satisfying any of the equivalent conditions in Theorem 5.1. Then,

$$\begin{aligned} \textrm{d}\rho _{V_{\Lambda }}(\xi ) = {1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi )\, \frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^2}. \end{aligned}$$

Proof

Since ${\bar{\partial }} V_{\Lambda }=\pi (n_{\Lambda }-{\mathfrak {c}}_{\Lambda }m)$, it follows from Claim 5.2 that

$$\begin{aligned} \textrm{d}\rho _{V_{\Lambda }}(\xi )=\frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^2},\qquad \text {on } \; {\mathbb {C}}\setminus \{0\}. \end{aligned}$$

It only remains to check that $\rho _{V_{\Lambda }}(\{0\})=0$, which we again postpone to Appendix B.3. $\square$

Theorem 5.8 yields a useful representation of the reduced covariance measure $\kappa _{V_{\Lambda }}$ of the field $V_{\Lambda }$. We denote by $U^\mu$ the logarithmic potential of a signed measure $\mu$,

$$\begin{aligned} U^\mu (z) = \int _{\mathbb {C}}\log \frac{1}{|z-s|}\, \textrm{d}\mu (s) \end{aligned}$$

(provided that the integral on the RHS exists).

Proposition 5.9

Suppose that reduced covariance measure $\kappa _{\Lambda }$ satisfies assumptions $(\kappa _1)$ and $(\kappa _2)$ of Remark 5.3. Then the reduced covariance measure of $V_{\Lambda }$ equals

$$\begin{aligned} \textrm{d}\kappa _{V_{\Lambda }} = 2\pi U^{\kappa _{\Lambda }}\, \textrm{d}m. \end{aligned}$$

Proof

To prove the proposition, we show that $2\pi \widehat{U^{\kappa _{\Lambda }}} = {1\hspace{-2.5pt}\textrm{l}}_{\{\xi \ne 0\}} |\xi |^{-2} \widehat{\kappa _{\Lambda }}$. We treat both sides as Schwartz distributions and understand the Fourier transforms in the sense of distributions.

First, we note that $\Delta U^{\kappa _{\Lambda }} = -(2\pi )^{-1} \kappa _{\Lambda }$, and therefore, $\widehat{\kappa _{\Lambda }} = 2\pi |\xi |^2 \widehat{U^{\kappa _{\Lambda }}}$. Hence, on the test functions $\varphi$ with $0\notin \textrm{spt}(\varphi )$, the distributions $2\pi \widehat{U^{\kappa _{\Lambda }}}$ and $|\xi |^{-2} \widehat{\kappa _{\Lambda }}$ coincide. I.e., the distribution $\nu =2\pi \widehat{U^{\kappa _{\Lambda }}} - |\xi |^{-2} \widehat{\kappa _{\Lambda }}$ is supported by the origin, and therefore, is a (finite) linear combination of the delta-function and its partial derivatives. We need to show that $\nu =0$.

Fix a non-negative $C^\infty$-smooth function $\gamma$ with a compact support, normalized by $\displaystyle \int _{{\mathbb {C}}} \gamma \, \textrm{d}m =1$, and let $\gamma _a(z)=a^{-2}\gamma (z/a)$, $a>0$. This is a convolutor on the Schwartz space ${\mathcal {S}}'$ of tempered distributions, and, for any $f\in {\mathcal {S}}'$, $f *\gamma _a \rightarrow f$ in ${\mathcal {S}}'$, as $a\rightarrow 0$. The Fourier transform $\widehat{\gamma _a}={\widehat{\gamma }}(az)$ is a $C^\infty$-smooth, fast decaying multiplier on the Fourier side of ${\mathcal {S}}'$, boundedly tending to 1 pointwise, as $a\rightarrow 0$. Consider the product

$$\begin{aligned} 2\pi \widehat{U^{\kappa _{\Lambda }}} \cdot \widehat{\gamma _a} =|\xi |^{-2} h_{\Lambda }\cdot \widehat{\gamma _a} + \nu \cdot \widehat{\gamma _a}, \end{aligned}$$

where $h_{\Lambda }$ is the density of $\widehat{\kappa _{\Lambda }}$. The inverse Fourier transform of the LHS equals $2\pi U^{\kappa _{\Lambda }}*\gamma _a = U^{\kappa _{\Lambda }*\gamma _a}$. The measure $\kappa _{\Lambda }*\gamma _a$ enjoys the same properties $(\kappa _1)$ and $(\kappa _2)$ as $\kappa _{\Lambda }$. Hence, the logarithmic potential $U^{\kappa _{\Lambda }*\gamma _a}$ tends to 0 as $z\rightarrow \infty$. By the Riemann-Lebesgue lemma, the inverse Fourier transform of $|\xi |^{-2} h_{\Lambda }\cdot \widehat{\gamma _a}$ also tends to zero as $z\rightarrow \infty$. Hence, the same holds for the inverse Fourier transform of the distribution $\nu \cdot \widehat{\gamma _a}$. But the inverse Fourier transform of the distribution $\nu$ is a polynomial (of $\textrm{Re}(z)$ and $\textrm{Im}(z)$), and therefore, the inverse Fourier transform of the distribution $\nu \cdot \widehat{\gamma _a}$ is also a polynomial. We conclude that $\nu \cdot \widehat{\gamma _a} = 0$ for all $a>0$, and therefore, $\nu =0$. $\square$

Remark 5.10

Recalling that $\kappa _{\Lambda }= \tau _{\Lambda }+ c_{\Lambda }\delta _0$, we see that

$$\begin{aligned} U^{\kappa _{\Lambda }}(z) = c_{\Lambda }\log \frac{1}{|z|} + U^{\tau _{\Lambda }}(z), \end{aligned}$$

that is, under the assumptions of Remark 5.3, the covariance kernel of $V_{\Lambda }$ always blows up logarithmically at the origin.

Remark 5.11

In the case when the distribution of the point process ${\Lambda }$ is also rotationally invariant the density of the reduced covariance measure $\kappa _{V_{\Lambda }}$ has a simpler expression. We assume that $\textrm{d}\tau _{\Lambda }(s) = k(t)t\,\textrm{d}t\, \textrm{d}\theta$, $s=te^{\textrm{i}\theta }$. Then conditions $(\kappa _1)$ and $(\kappa _2)$ of Remark 5.3 can be re-written as

($\tau _1$):

$\displaystyle \int _0^\infty t^2|k(t)|\, \textrm{d}t < \infty$;

($\tau _2$):

$\displaystyle \int _0^\infty tk(t)\, \textrm{d}t = - (2\pi )^{-1}c_{\Lambda }$.

In this case,

$$\begin{aligned} U^{\kappa _{{\Lambda }}}(z)&= c_{\Lambda }\log \frac{1}{|z|} + \int _0^\infty \Bigl ( \log \frac{1}{|z-te^{\textrm{i}\theta }|} \, \textrm{d}\theta \Bigr ) k(t) t\, \textrm{d}t \\&= c_{\Lambda }\log \frac{1}{|z|} + \int _0^\infty 2\pi \bigl (\, \log \frac{1}{|z|} - \log _+\frac{t}{|z|}\, \bigr ) k(t) t\, \textrm{d}t \\&{\mathop {=}\limits ^{(\tau _2)}} - 2\pi \int _{|z|}^\infty \log \frac{t}{|z|} k(t)t\, \textrm{d}t\,. \end{aligned}$$

Thus, the density of $\kappa _{V_{\Lambda }}$ equals

$$\begin{aligned} - 4\pi ^2 \int _{|z|}^\infty \log \frac{t}{|z|} k(t)t\, \textrm{d}t. \end{aligned}$$

(5.11)

6 The Random Potential

6.1 The Potential $\Pi _{\Lambda }$

Assuming that the equivalent conditions of Theorem 5.1 hold, we will define a random potential $\Pi _{\Lambda }$ such that $\partial _z \Pi _{\Lambda }= \tfrac{1}{2}\, V_{\Lambda }$, and therefore, $\Delta \Pi _{\Lambda }= 2\pi (n_{\Lambda }- {\mathfrak {c}}_{\Lambda }\,m)$ (both relations are understood in the sense of distributions). Since the field $V_{\Lambda }$ is stationary, this will yield that the potential $\Pi _{\Lambda }$ has stationary increments. The existence of the potential $\Pi _\Lambda$ with this property, in turn, shows that the vector field $V_{\Lambda }$ 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

$$\begin{aligned} \int _{|\xi |>0}\frac{\textrm{d}\rho _{\Lambda }(\xi )}{|\xi |^4}<\infty \end{aligned}$$

(cf. Theorem 6.2). We will not pursue the details here.

We start with the entire function represented by the Hadamard product

$$\begin{aligned} F_{\Lambda }(z) = \exp \bigl [ -\Psi _1(\infty )z - \frac{1}{2}\, \Psi _2(\infty )z^2 \bigr ]\, \cdot \prod _{|{\lambda }|<1} ({\lambda }-z) \prod _{|{\lambda }|\ge 1} \left( \frac{{\lambda }-z}{{\lambda }}\, \exp \Bigl [\,\frac{z}{{\lambda }}\, + \frac{z^2}{2{\lambda }^2}\, \Bigr ]\right) , \end{aligned}$$

and note that, for each $R>1$,

$$\begin{aligned} F_{\Lambda }(z)= & {} \prod _{|{\lambda }|<1} ({\lambda }-z) \prod _{1\le |{\lambda }|\le R} \frac{{\lambda }-z}{{\lambda }}\\{} & {} \times \exp \Bigl [ (\Psi _1(R)-\Psi _1(\infty )) z + \frac{1}{2}\, (\Psi _2(R)-\Psi _2(\infty )) z^2 \Bigr ] \prod _{|{\lambda }|>R} e^{H(z/{\lambda })}, \end{aligned}$$

where $H(w) = - \sum _{k\ge 3} w^k/k$. As $R\rightarrow \infty$, the third and fourth factors on the RHS tend to 1 in $L^2(\Omega , {\mathbb {P}})$ and locally uniformly in z, and therefore,

$$\begin{aligned} F_{\Lambda }(z) = \prod _{|{\lambda }|<1} ({\lambda }-z) \lim _{R\rightarrow \infty }\, \prod _{1\le |{\lambda }|\le R} \frac{{\lambda }-z}{{\lambda }}. \end{aligned}$$

(6.1)

We define $\Pi _{\Lambda }(z) {\mathop {=}\limits ^\textrm{def}} \log |F_{\Lambda }(z)| - \tfrac{1}{2}\, \pi {\mathfrak {c}}_{\Lambda }|z|^2$. Then, by a straightforward inspection, we get that

$$\begin{aligned} \partial _z \Pi _{\Lambda }= \frac{1}{2}\, \bigl ( \zeta _{\Lambda }- \Psi _1(\infty ) - \pi {\mathfrak {c}}_{\Lambda }{{\bar{z}}} \bigr ) = \tfrac{1}{2}\, V_{\Lambda }. \end{aligned}$$

(6.2)

Remark 6.1

Under the assumptions of Theorem 5.1, the quotient

$$\begin{aligned} \Bigl |\, \frac{F_{\Lambda }(z+a)}{F_{\Lambda }(z)}\, \Bigr | \, \exp \bigl [ -\frac{1}{2}\, \pi {\mathfrak {c}}_{\Lambda }(z{{\bar{a}}} + a{{\bar{z}}} + |a|^2) \bigr ], \quad a\in {\mathbb {C}}, \end{aligned}$$

has a stationary distribution (as a function of z). An interesting characteristic of the point process $\Lambda$ is the distribution of the phase

$$\begin{aligned} \arg F_{\Lambda }(z+a)/F_{\Lambda }(z) = \arg F_{\Lambda }(z+a) - \arg F_{\Lambda }(z). \end{aligned}$$

To properly define this quantity, we fix a curve $\Gamma$ connecting the points z and $z+a$, and consider the increment of the argument of $F_{\Lambda }$ along $\Gamma$

$$\begin{aligned} \frac{1}{2\pi } \Delta _\Gamma \arg F_{\Lambda }= \textrm{Im}\, \frac{1}{2\pi }\, \int _\Gamma (\zeta _{\Lambda }(z) -\Psi _1(\infty ))\, \textrm{d}z. \end{aligned}$$

This quantity was considered by Buckley–Sodin in [5] when F is replaced by the GEF. Equivalently, one can consider the flux of the gradient field of the potential $\Pi _{\Lambda }$ through the curve $\Gamma$.

In [28], we study the asymptotic variance of this quantity under dilations of $\Gamma$. In the special case when $\Gamma$ is a Jordan curve we recall that the change in argument coincides with the charge fluctuation around the mean in the domain enclosed by $\Gamma$.

6.2 The Covariance Structure of $\Pi _{\Lambda }$

Let $\mathsf \Delta _a \Pi _{\Lambda }(z) {\mathop {=}\limits ^\textrm{def}} \Pi _{\Lambda }(z+a) - \Pi _{\Lambda }(z)$, for $a\in {\mathbb {C}}$.

Theorem 6.2

Suppose $\Lambda$ is a stationary point process in ${\mathbb {C}}$ satisfying any of equivalent conditions in Theorem 5.1. Then, $\mathsf{\Delta }_a\Pi _{\Lambda }$ is stationary and

$$\begin{aligned} \rho _{\mathsf \Delta _a \Pi _{\Lambda }}(\xi ) = \frac{1}{4}\, {1\hspace{-2.5pt}\textrm{l}}_{{\mathbb {C}}\setminus \{0\}}(\xi )\, \frac{|1-e^{2\pi \textrm{i}a\cdot \xi }|^2}{|\xi |^4}\, \rho _{\Lambda }(\xi ) +\frac{\pi ^2|a|^4}{4}\rho _{\Lambda }(\{0\})\,\delta _0(z). \end{aligned}$$

Proof

We first claim that $\mathsf{\Delta }_a\Pi _{\Lambda }(z)$ can be written as

$$\begin{aligned} \mathsf{\Delta }_a\Pi _{\Lambda }(z)=Q_a(T_z{\Lambda })-\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }|a|^2, \end{aligned}$$

(6.3)

where $Q_a$ is the $L^2(\Omega ,{\mathbb {P}})$-limit

$$\begin{aligned} Q_a({\Lambda })=\lim _{R\rightarrow \infty }\sum _{|{\lambda }|\le R}\Big (\log |z-a|-\log |z|\Big ), \end{aligned}$$

(6.4)

which in particular says that $\mathsf{\Delta }_a\Pi _{\Lambda }$ is stationary. To verify (6.3)–(6.4), we start with the representation (6.1), which in view of the spectral condition gives that

$$\begin{aligned} \mathsf {\Delta }_a\Pi _{\Lambda }(z)=\lim _{R\rightarrow \infty } \sum _{|{\lambda }|\le R}\big (\log |z-({\lambda }-a)|-\log |z-{\lambda }|\big ) -\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }\big (2\text {Re}\,({\bar{a}}z)+|a|^2\big ),\end{aligned}$$

where the limit is taken in $L^2(\Omega ,{\mathbb {P}})$. We readily rewrite this as

$$\begin{aligned} \mathsf{\Delta }_a\Pi _{\Lambda }(z)=\lim _{R\rightarrow \infty } \Big [\sum _{|{\lambda }-z|\le R}\big (\log |z-{\lambda }+a)|-\log |z-{\lambda }|\big ) +e_R({\Lambda },z)\Big ]-\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }|a|^2, \end{aligned}$$

where $e_R({\Lambda },z)$ is an “error term” given by

$$\begin{aligned} e_R({\Lambda },z)= & {} \sum _{|{\lambda }|\le R}\big (\log |z-{\lambda }+a|-\log |z-{\lambda }|\big ) \\ {}{} & {} - \sum _{|{\lambda }-z|\le R}\big (\log |z-{\lambda }+a|-\log |z-{\lambda }|\big ) - \pi {\mathfrak {c}}_{\Lambda }\textrm{Re}\,({\bar{a}}z). \end{aligned}$$

By Claim 4.2, it only remains to prove that ${\mathbb {E}}\big [|e_R({\Lambda },z)|^2\big ]\rightarrow 0$ as $R\rightarrow \infty$. To see this, we will relate $e_R({\Lambda },z)$ to the sums appearing in the Lunar lemma (Lemma 3.4). Note first that

$$\begin{aligned} \log |z-{\lambda }+a|-\log |z-{\lambda }|=\textrm{Re}\,\frac{a}{z-{\lambda }} +O\Big (\frac{1}{|z-{\lambda }|^{2}}\Big ) \end{aligned}$$

(6.5)

as $|z-{\lambda }|\rightarrow \infty$. Using this expansion, we rewrite $e_R({\Lambda },z)$ as

$$\begin{aligned} e_R({\Lambda },z)&=\textrm{Re}\,\Bigg [a\Bigg (\sum _{|{\lambda }|\le R}\frac{1}{z-{\lambda }} -\sum _{|{\lambda }-z|\le R}\frac{1}{z-{\lambda }}\Bigg )\Bigg ]-\pi {\mathfrak {c}}_{\Lambda }\textrm{Re}\,(a{\bar{z}}) +{\widetilde{e}}_R({\Lambda },z)\nonumber \\&=\textrm{Re}\,\Bigg [a\Bigg (\sum _{|{\lambda }|\le R}\frac{1}{z-{\lambda }} -\sum _{|{\lambda }-z|\le R}\frac{1}{z-{\lambda }}-\pi {\mathfrak {c}}_{\Lambda }{\bar{z}}\Bigg )\Bigg ]+{\widetilde{e}}_R({\Lambda },z) \end{aligned}$$

(6.6)

where the new error term ${\widetilde{e}}_R({\Lambda },z)$ is

$$\begin{aligned} {\widetilde{e}}_R({\Lambda },z)&=\sum _{|{\lambda }|\le R}l_a(z-{\lambda }) -\sum _{|{\lambda }-z|\le R}l_a(z-{\lambda }) \end{aligned}$$

(6.7)

$$\begin{aligned}&=\sum _{{\lambda }\in {\mathbb {D}}(0,R)\setminus {\mathbb {D}}(z,R)}l_a(z-{\lambda }) -\sum _{{\lambda }\in {\mathbb {D}}(z,R)\setminus {\mathbb {D}}(0,R)}l_a(z-{\lambda }), \end{aligned}$$

(6.8)

and where

$$\begin{aligned} l_a(w)=\log |w+a|-\log |w|-\textrm{Re}\,\Big (\frac{a}{w}\Big )=O(|w|^{-2}). \end{aligned}$$

The first term on the RHS of (6.6) tends to zero in $L^2(\Omega ,{\mathbb {P}})$ by Lemma 3.4. Moreover, for any point ${\lambda }$ in the symmetric difference

$$\begin{aligned} S_R\overset{\textsf{def}}{=}\big ({\mathbb {D}}(0,R)\setminus {\mathbb {D}}(z,R)\big )\cup \big ({\mathbb {D}}(z,R)\setminus {\mathbb {D}}(0,R)\big ), \end{aligned}$$

we have $|l_a(z-{\lambda })|=O(R^{-2})$, so by the upper bound (4.8) of ${\mathbb {E}}[n_{\Lambda }(S_R)]^2$, ${\widetilde{e}}_R({\Lambda },z)\rightarrow 0$ in $L^2(\Omega ,{\mathbb {P}})$ as $R\rightarrow \infty$. This completes the proof.

Turning to the spectral measure, note that by (6.2), we have

$$\begin{aligned} \mathsf{\Delta }_a \Pi _{\Lambda }(\partial \varphi )=-\frac{1}{2} V_{\Lambda }(T_a\varphi -\varphi ), \end{aligned}$$

which by the argument used in the proof of Theorem 5.7 shows that the desired equality for $\rho _{\mathsf \Delta _a \Pi _{\Lambda }}$ 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. $\square$

Appendices

Appendix 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 $f\in C^1({\mathbb {T}})$,

$$\begin{aligned} \left| \int _a^b e^{-\textrm{i}\omega \cos (\theta )}f(\theta )\textrm{d}\theta \right| \le C\,\omega ^{-\frac{1}{2}}\left( \Vert f\Vert _{L^\infty ({\mathbb {T}})} +\Vert f'\Vert _{L^1({\mathbb {T}})}\right) , \end{aligned}$$

for $\omega \ge 1$ and $0\le a<b\le 2\pi$.

Proof

If the critical points of the phase (i.e. 0, $\pi$) 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, Chap. 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 [16]).

Thus, it suffices to consider integrals $\int _a^b e^{-\textrm{i}\omega \cos (\theta )}f(\theta )\textrm{d}\theta$ for $a=0$ and small $b>0$. The change of variables $\theta =\theta (x)$ defined by $\cos (\theta )=1-x^2/2$ is smooth and bijective as a map $\theta :[0,b']\rightarrow [0,b]$, where $b'>0$. Moreover, $\theta (x)$, $\theta '(x)$ are both uniformly bounded, and $\theta (x)=x+O(x^2)$ uniformly on $[0,b']$ as $b\rightarrow 0$. We rewrite the integral using this change of variables

$$\begin{aligned} \int _0^b e^{-\textrm{i}\omega \cos (\theta )}f(\theta )\textrm{d}\theta =\int _0^{b'}e^{-\textrm{i}\omega x^2}f(\theta (x))\theta '(x)\textrm{d}x=: \int _0^{b'}e^{-\textrm{i}\omega x^2}g(x)\textrm{d}x \end{aligned}$$

where we have introduced the function $g(x)=f(\theta (x))\theta '(x)$, and where $b'=b+O(b^2)$ as $b\rightarrow 0$. It is clear from the properties of $\theta (x)$ that $|g|\le C|f|$ and that $|g'|\le C|f'|$, for some universal constant C.

An integration by parts shows that

$$\begin{aligned} \int _0^{b'}e^{-\textrm{i}\omega x^2}g(x)\textrm{d}x= g(b')\int _0^{b'}e^{-\textrm{i}\omega x^2}\textrm{d}x - \int _0^{b'}\left( \int _0^{x}e^{-\textrm{i}\omega y^2}\textrm{d}y\right) g'(x)\textrm{d}x. \end{aligned}$$

(A.1)

The Gaussian integral satisfies

$$\begin{aligned} \left| \int _0^{b'}e^{-\textrm{i}\omega x^2}\textrm{d}x\right| = \frac{\sqrt{\pi }}{\sqrt{2\omega }}\big |\textrm{erf}\big (b'\sqrt{\textrm{i}\omega }\big )\big | \le C\omega ^{-\frac{1}{2}} \end{aligned}$$

for $\omega ,b'>0$, so the first term on the RHS of (A.1) is bounded above by $C\omega ^{-\frac{1}{2}}\Vert f\Vert$. The second term can be estimated as follows

$$\begin{aligned} \left| \int _0^{b'}\left( \int _0^{x}e^{-\textrm{i}\omega y^2}\textrm{d}y\right) g'(x)\textrm{d}x\right| \le \Vert g'\Vert _{L^1(0,b')}\left| \int _0^{x}e^{-\textrm{i}\omega y^2}\textrm{d}y\right| \le C\Vert f'\Vert _{L^1({\mathbb {T}})}\omega ^{-\frac{1}{2}}. \end{aligned}$$

Hence, the proof is complete. $\square$

Appendix 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 Sects. 5 and 6. This will finalize the proofs of Theorems 5.7, 5.8 and 6.2.

1.1 B.1  The Atom of $\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}$

We will first prove the remaining part of Theorem 5.7 concerning the spectral measure of $\mathsf{\Delta }_a\zeta _{\Lambda }$, namely that

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\})=\pi ^2|a|^2\rho _{\Lambda }(\{0\}). \end{aligned}$$

To this end, note that by the relation (5.9), translation-boundedness of $\rho _{{\Lambda }}$, and the asymptotic formula (3.3) for the Bessel function, the function $\xi \mapsto \sup _{R\ge 1}J_0(2\pi R |\xi |)$ is square-integrable with respect to $\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}$. As a consequence

$$\begin{aligned} \nonumber \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\})&= \int _{{\mathbb {C}}}\lim _{R\rightarrow \infty } (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi ) \\ {}&=\lim _{R\rightarrow \infty } \int _{{\mathbb {C}}}(J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi ), \end{aligned}$$

(B.1)

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

$$\begin{aligned} \int _{{\mathbb {C}}} (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi )= \textsf{Var}\left[ \frac{1}{2{\pi \textrm{i}}} \int _{\{|z|=R\}} \frac{\mathsf{\Delta }_a\zeta _{\Lambda }(z)}{z} \, \textrm{d} z\right] . \end{aligned}$$

(B.2)

To see this, first note that

$$\begin{aligned} J_0(2\pi R|\xi |)=\frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}}e^{-2\pi \textrm{i} \xi \cdot z} \frac{\textrm{d} z}{z}={\widehat{\sigma }}_R(\xi ), \end{aligned}$$

where $\sigma _R$ denotes the normalized arc-length measure on the circle $\{|z|=R\}$. By considering a suitable mollifier $h_j(\xi )=j^2 h(j\xi )$ with $h\in {\mathcal {S}}$, $\int _{{\mathbb {C}}} h\textrm{d}m=1$, and putting $\varphi _{R,j}=\sigma _R*h_j$, we may write

$$\begin{aligned} \frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}} \frac{\mathsf{\Delta }_a\zeta _{\Lambda }(z)}{z} \, \textrm{d} z =\lim _{j\rightarrow \infty }\int _{{\mathbb {C}}}\mathsf{\Delta }_a\zeta _{\Lambda }(z) \varphi _{R,j}(z)\,\textrm{d}m(z), \end{aligned}$$

where the limit is taken in $L^2(\Omega ,{\mathbb {P}})$. The formula (B.2) then follows by a reverse application of the Parseval identity

$$\begin{aligned} \textsf{Var}\Big [\int _{{\mathbb {C}}}\mathsf{\Delta }_a\zeta _{\Lambda }(z) \varphi _{R,j}(z)\,\textrm{d}m(z)\Big ]&=\int _{{\mathbb {C}}}|{\widehat{\varphi }}_{R,j}(\xi )|^2\textrm{d} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi ) \\ {}&= \int _{{\mathbb {C}}}|{\widehat{\sigma }}_R(\xi )|^2\, |{\widehat{h}}_j(\xi )|^2\textrm{d} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi ) \end{aligned}$$

(cf. (2.1) and Remark 2.1). Combining (B.2) with (B.1), we obtain the representation

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\})&= \lim _{R\rightarrow \infty } \int _{{\mathbb {C}}} (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\xi ) \\&= \lim _{R\rightarrow \infty } \textsf{Var}\left[ \frac{1}{2{\pi \textrm{i}}} \int _{|z|=R} \frac{\mathsf{\Delta }_a\zeta _{\Lambda }(z)}{z} \, \textrm{d} z\right] \end{aligned}$$

for the atom at the origin.

Now, recall the formula

$$\begin{aligned} \frac{1}{2\pi \textrm{i}}\int _{\{|z|=R\}}\frac{1}{z-w}\frac{\textrm{d}z}{z}= {\left\{ \begin{array}{ll} 0,&{} |w|<R\\ -\frac{1}{w},&{} |w|>R \end{array}\right. } \end{aligned}$$

which, since a.s. there are no points of ${\Lambda }$ which lie on the circle $\{|z|=R\}$, implies that

$$\begin{aligned} \frac{1}{2\pi \textrm{i}}\int _{\{|z|=R\}}\frac{\mathsf{\Delta }_a\zeta _{\Lambda }(z)}{z}\textrm{d}z =\Big [\zeta _{\Lambda }(a) - \sum _{|{\lambda }-a|<R}\frac{1}{a-{\lambda }}\Big ] -\Big [\zeta _{\Lambda }(0)-\sum _{|{\lambda }|< R}\frac{1}{-{\lambda }}\Big ]. \end{aligned}$$

But the right-hand side equals

$$\begin{aligned} \Big [\zeta _{\Lambda }(a) - \sum _{|{\lambda }-a|<R}\frac{1}{a-{\lambda }}-\Psi _1(R)-\pi {\mathfrak {c}}_{\Lambda }{\bar{a}}\Big ] -\Big [\zeta _{\Lambda }(0)-\sum _{|{\lambda }|< R}\frac{1}{-{\lambda }}-\Psi _1(R)\Big ]+\pi {\mathfrak {c}}_{\Lambda }{\bar{a}} \end{aligned}$$

and by invoking the representation (4.3) of $\zeta _{\Lambda }$ we see that the first two terms on the right-hand side tend to zero in $L^2(\Omega ,{\mathbb {P}})$ as $R\rightarrow \infty$. Hence, we get

$$\begin{aligned} \lim _{R\rightarrow \infty }\textsf{Var}\Big [\frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}}\frac{\mathsf{\Delta }_a\zeta _{\Lambda }(z)}{z}\textrm{d}z\Big ]= \pi ^2 |a|^2\textsf{Var}[{\mathfrak {c}}_{\Lambda }]=\pi ^2 |a|^2\rho _{\Lambda }(\{0\}). \end{aligned}$$

Putting this together with the above representation for $\rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\})$, we see that

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\zeta _{\Lambda }}(\{0\}) = \pi ^2|a|^2\rho _{\Lambda }(\{0\}), \end{aligned}$$

which completes the proof. $\square$

1.2 B.2  The Atom of $\rho _{\wp _{\Lambda }}$

We next finalize the proof of Theorem 5.7, which amounts to verifying that $\rho _{\wp _{\Lambda }}(\{0\})=0$. The starting point is the analogous formula to (B.1), namely

$$\begin{aligned} \rho _{\wp _{{\Lambda }}}(\{0\}) = \int _{{\mathbb {C}}}\lim _{R\rightarrow \infty } \left| \frac{J_1(2\pi R|\xi |)}{\pi R|\xi |}\right| ^2\textrm{d} \rho _{\wp _{\Lambda }}(\xi ). \end{aligned}$$

We next observe that

$$\begin{aligned} \frac{J_1(2\pi R|\xi |)}{\pi R |\xi |}=\frac{1}{\pi R^2}{\widehat{{1\hspace{-2.5pt}\textrm{l}}}}_{R{\mathbb {D}}}(\xi ), \end{aligned}$$

and since the LHS is bounded above by an $L^2({\mathbb {C}},\rho _{\wp _{\Lambda }})$-integrable function, the dominated convergence theorem gives that

$$\begin{aligned} \rho _{\wp _{{\Lambda }}}(\{0\}) =\lim _{R\rightarrow \infty }\textsf{Var} \left[ \wp _{{\Lambda }}\Big (\frac{1}{\pi R^2}{1\hspace{-2.5pt}\textrm{l}}_{R{\mathbb {D}}}\Big )\right] \end{aligned}$$

Using the familiar formula,

$$\begin{aligned} \text {p.v.} \int _{R{\mathbb {D}}} \frac{1}{(z-{\lambda })^2} \, \textrm{d}m(z) = {\left\{ \begin{array}{ll} 0 &{} \text {for}\ |{\lambda }| < R, \\ \frac{\pi R^2}{{\lambda }^2} &{} \text {for} \ |{\lambda }|> R, \end{array}\right. } \end{aligned}$$

we get the representation

$$\begin{aligned} \wp _{\Lambda }({1\hspace{-2.5pt}\textrm{l}}_{R{\mathbb {D}}}) = \pi R^2(\Psi _2(\infty ) - \Psi _2(R)) \,. \end{aligned}$$

By Lemma 3.2, $\wp _{{\Lambda }}({1\hspace{-2.5pt}\textrm{l}}_{R{\mathbb {D}}}) \in L^2(\Omega ,{\mathbb {P}})$ and

$$\begin{aligned} \lim _{R\rightarrow \infty } \textsf{Var} \left[ \wp _{{\Lambda }}\Big (\frac{1}{\pi R^2}{1\hspace{-2.5pt}\textrm{l}}_{R{\mathbb {D}}}\Big )\right] = \lim _{R\rightarrow \infty } \textsf{Var} \left[ \Psi _2(\infty ) - \Psi _2(R)\right] = 0, \end{aligned}$$

which completes the proof. $\square$

1.3 B.3  The Atom of $\rho _{V_{\Lambda }}$

We turn to proving the last remaining statement of Theorem 5.8, i.e., that $\rho _{V_{\Lambda }}(\{0\})=0$. Arguing as in Appendix B.1, we find that

$$\begin{aligned} \rho _{V_{\Lambda }}(\{0\})&= \int _{{\mathbb {C}}}\lim _{R\rightarrow \infty } (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{V_{\Lambda }}(\xi ) \\&= \lim _{R\rightarrow \infty } \int _{{\mathbb {C}}} (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{V_{\Lambda }}(\xi ) \\&= \lim _{R\rightarrow \infty } \textsf{Var}\left[ \frac{1}{2{\pi \textrm{i}}} \int _{\{|z|=R\}} \frac{V_{\Lambda }(z)}{z} \, \textrm{d} z\right] . \end{aligned}$$

For any given $R\ge 1$, a.s., there are no points of ${\Lambda }$ which lie on the circle $\{|z|=R\}$. By a residue computation, we get

$$\begin{aligned} \frac{1}{2{\pi \textrm{i}}} \int _{\{|z|=R\}} \frac{V_{\Lambda }(z)}{z} \, \textrm{d} z&= \frac{1}{2{\pi \textrm{i}}} \int _{\{|z|=R\}} \frac{\zeta _{\Lambda }(z)-\Psi _1(\infty )}{z} \, \textrm{d} z \\&= \zeta _{\Lambda }(0)+\sum _{|{\lambda }|\le R}\frac{1}{{\lambda }}-\Psi _1(\infty ) \\ {}&= -\sum _{|{\lambda }|< 1} \frac{1}{{\lambda }} +\sum _{|{\lambda }|<R} \frac{1}{{\lambda }}-\Psi _1(\infty ) = \Psi _1(R)-\Psi _1(\infty ). \end{aligned}$$

But Lemma 3.3 implies that

$$\begin{aligned} \textsf{Var}\big [\Psi _1(R)-\Psi _1(\infty )\big ]=0 \end{aligned}$$

which completes the proof. $\square$

1.4 B.4  The Atom of $\rho _{\mathsf{\Delta }_a\Pi _{\Lambda }}$

To conclude the proof of Theorem 6.2, it only remains to prove that

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\Pi _{\Lambda }}(\{0\})=\frac{\pi ^2|a|^4}{4}\rho _{\Lambda }(\{0\}). \end{aligned}$$

We again use the method introduced in Appendix B.1, in particular the formula

$$\begin{aligned} \rho _{\mathsf{\Delta }_a\Pi _{\Lambda }}\left( \{0\}\right) = \lim _{R\rightarrow \infty } \int _{{\mathbb {C}}} (J_0(2\pi R|\xi |))^2 \, \textrm{d}\rho _{\mathsf{\Delta }_a\Pi _{\Lambda }}(\xi ) = \lim _{R\rightarrow \infty } \textsf{Var}\Big [\frac{1}{2\pi \textrm{i}} \int _{\{|z|=R\}}\mathsf{\Delta }_a\Pi _{\Lambda }(z)\frac{\textrm{d}z}{z}\Big ]. \end{aligned}$$

We start with the representation

$$\begin{aligned} \mathsf{\Delta }_a\Pi _{\Lambda }(z)=\lim _{S\rightarrow \infty } \sum _{|{\lambda }|\le S}\big (\log |z-({\lambda }-a)|-\log |z-{\lambda }|\big ) -\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }\big (2\textrm{Re}\,{\bar{a}}z)-|a|^2\big ) \end{aligned}$$

and the formula

$$\begin{aligned} \frac{1}{2\pi R}\int _{\{|z|=R\}}\log |z-w|\,|\textrm{d}z| ={\left\{ \begin{array}{ll}\log R&{}|w|\le R\\ \log |w|&{}|w|>R\end{array}\right. }. \end{aligned}$$

Combining these two, we get

$$\begin{aligned} \frac{1}{2\pi \textrm{i}}\int _{\{|z|=R\}}\mathsf{\Delta }_a\Pi _{\Lambda }(z)\frac{\textrm{d}z}{z}= & {} \lim _{S\rightarrow \infty }\left[ \sum _{|{\lambda }|\le S,\;|{\lambda }-a|\ge R}\log |{\lambda }-a| -\sum _{R\le |{\lambda }|\le S}\log |{\lambda }|\right] \\{} & {} +\sum _{|{\lambda }-a|\le R}\log R-\sum _{|{\lambda }|\le R}\log R -\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }|a|^2, \end{aligned}$$

where we have used the fact that $\textrm{Re}\,({\bar{a}}z)$ has zero mean over circles centered at the origin. By subtracting and adding back $\sum _{|{\lambda }-a|\ge R,\,|{\lambda }|\le S}\log |{\lambda }|$, we may rewrite the right-hand side as

$$\begin{aligned} \sum _{|{\lambda }-a|\ge R}\big (\log |{\lambda }-a|-\log |{\lambda }|\big ) +\left[ \sum _{|{\lambda }|\le R}\log \frac{|{\lambda }|}{R} -\sum _{|{\lambda }-a|\le R}\log \frac{|{\lambda }|}{R}\right] -\frac{1}{2}\pi {\mathfrak {c}}_{\Lambda }|a|^2. \end{aligned}$$

In view of the spectral condition, Lemma 3.3 on the convergence of the sequence $(\Psi _1(R))_{R\ge 1}$ gives that the first term tends to zero in $L^2(\Omega ,{\mathbb {P}})$. To obtain the result, if suffices to show that the variance of

$$\begin{aligned} \textsf{Var}\left[ \sum _{|{\lambda }|\le R}\log \frac{|{\lambda }|}{R} -\sum _{|{\lambda }-a|\le R}\log \frac{|{\lambda }|}{R}\right] = \textsf{Var}\left[ \sum _{{\lambda }\in A}\log \frac{|{\lambda }|}{R} -\sum _{{\lambda }\in B}\log \frac{|{\lambda }|}{R}\right] \end{aligned}$$

tends to zero as $R\rightarrow \infty$, where we use the notation A and B for the lunar domains

$$\begin{aligned} A=\{|{\lambda }|\le R,\;|{\lambda }-a|\ge R\}\quad \text {and}\quad B =\{|{\lambda }|\ge R,\;|{\lambda }-a|\le R\}. \end{aligned}$$

The corresponding linear statistic is

$$\begin{aligned} f_R({\lambda })=\big ({1\hspace{-2.5pt}\textrm{l}}_{A}({\lambda })-{1\hspace{-2.5pt}\textrm{l}}_{B}({\lambda })\big ) \log \frac{|{\lambda }|}{R}=:f_{R,A}(\xi )-f_{R,B}(\xi ), \end{aligned}$$

so the variance equals

$$\begin{aligned} \textsf{Var}\left[ \sum _{{\lambda }\in A}\log \frac{|{\lambda }|}{R} -\sum _{{\lambda }\in B}\log \frac{|{\lambda }|}{R}\right] =\int _{{\mathbb {C}}\setminus \{0\}} |{\widehat{f}}_R(\xi )|^2\textrm{d}\rho _{\Lambda }(\xi ), \end{aligned}$$

where we have used the fact that ${\widehat{f}}_{R}(0)=0$ to infer that the origin does not contribute to the variance. We need to estimate $|{\widehat{f}}_R(\xi )|^2$ from above, and by symmetry it suffices to consider only $f_{R,A}$.

Since $\log \frac{|{\lambda }|}{R}=O(R^{-1})$ on A, the naive bound $|{\widehat{f}}_{R,A}(\xi )|\le \Vert f_{R,A}\Vert _{L^1({\mathbb {C}})}$ gives that $|{\widehat{f}}_{R,A}(\xi )|^2=O(1)$ as $R\rightarrow \infty$ for $\xi \in {\mathbb {C}}$. In order to complete the proof, we need better estimates valid for large $|\xi |$. An application of Green’s formula shows that

$$\begin{aligned} {\widehat{f}}_{R,A}(\xi )&=\int _A\log \frac{|{\lambda }|}{R}e^{-2\pi \textrm{i} \xi \cdot {\lambda }}\textrm{d}m({\lambda })\\&=\frac{1}{\pi ^2|\xi |^2}\int _A\log \frac{|{\lambda }|}{R} \Delta \left( e^{-2\pi \textrm{i}\xi \cdot {\lambda }}\right) \textrm{d}m({\lambda })\\&=\frac{1}{\pi ^2|\xi |^2}\int _{\partial A}e^{-2\pi \textrm{i}\xi \cdot {\lambda }} \left[ -2\pi \textrm{i}\log \frac{|{\lambda }|}{R}\big \langle \xi , N({\lambda })\big \rangle +\left\langle \tfrac{1}{{\bar{{\lambda }}}},N({\lambda }) \right\rangle \right] |\textrm{d}{\lambda }|, \end{aligned}$$

where N denotes the outward unit normal to A. Parameterizing the two circular arcs that make up the boundary $\partial A$, we find that

$$\begin{aligned} {\widehat{f}}_{R,A}(\xi )=\frac{1}{\pi ^2|\xi |^2}\big (J_1(\xi ,R,a)+J_2(\xi ,R,a)\big ), \end{aligned}$$

where $J_1$ and $J_2$ are the integrals

$$\begin{aligned} J_1(R,\xi ,a)=\frac{1}{R}\int _{s_1}^{t_1}e^{-2\pi \textrm{i}R|\xi |\cos (\theta )}\textrm{d}\theta \end{aligned}$$

(B.3)

and

$$\begin{aligned} \begin{aligned} J_2(R,\xi ,a)&=-2\pi \textrm{i}|\xi |\int _{s_2}^{t_2}e^{-2\pi \textrm{i}R|\xi |\cos (\theta )} \log \big |1+\tfrac{ae^{-\textrm{i}\theta }}{R}\big |\cos (\theta )\textrm{d}\theta \\&\quad +\frac{1}{R}\int _{s_2}^{t_2}e^{-2\pi \textrm{i}R|\xi |\cos (\theta )} \textrm{Re}\,\Big [\tfrac{1}{1+R^{-1}{\bar{a}}e^{-\textrm{i}\theta }}\Big ]\textrm{d}\theta \end{aligned} \end{aligned}$$

(B.4)

respectively. Here, $s_i$ and $t_i$ are positive parameters with $0\le s_i\le t_i\le 2\pi$, depending on R and a and $\xi$. For large $|\xi |$, we may thus use the standard stationary phase bound

$$\begin{aligned} \left| \int _{\alpha }^\beta e^{-2\pi \textrm{i}\omega \cos (\theta )}f(\theta )\textrm{d}\theta \right| \lesssim \omega ^{-\frac{1}{2}}\big (\Vert f\Vert _{L^\infty ([\alpha ,\beta ])}+ \Vert f'\Vert _{L^1([\alpha ,\beta ])}\big ) \end{aligned}$$

from Proposition A.1 in the Appendix to obtain the estimates

$$\begin{aligned} \big |J_1(R,\xi ,a)\big |\lesssim |\xi |^{-\frac{1}{2}}R^{-\frac{3}{2}} \end{aligned}$$

and

$$\begin{aligned} \big |J_1(R,\xi ,a)\big |\lesssim _a |\xi |^\frac{1}{2}R^{-\frac{3}{2}}+\xi ^{-\frac{1}{2}}R^{-\frac{3}{2}} \end{aligned}$$

for the two contributions $J_1(R,\xi ,a)$ and $J_2(R,\xi ,a)$ for ${\widehat{f}}_{R,A}(\xi )$. Putting everything together, we arrive at

$$\begin{aligned} |{\widehat{f}}_{R,A}(\xi )|^2=\frac{1}{\pi ^4|\xi |^4} \big |J_1(\xi ,R,a)+J_2(\xi ,R,a)\big |^2\lesssim _a \min \big \{1,R^{-3}|\xi |^{-3}\big \}. \end{aligned}$$

Since the corresponding bound holds also for ${\widehat{f}}_{R,B}$, we get

$$\begin{aligned} \textsf{Var}\left[ n_{\Lambda }(f_R)\right] =\int _{{\mathbb {C}}\setminus \{0\}} \big |{\widehat{f}}_R(\xi )\big |^2\textrm{d}\rho _{\Lambda }(\xi ) \lesssim _a \int _{{\mathbb {C}}\setminus \{0\}} \min \big \{1,R^{-3}|\xi |^{-3}\big \}\textrm{d}\rho _{\Lambda }(\xi ), \end{aligned}$$

and hence the result follows by invoking the dominated convergence theorem. $\square$