Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in distribution to the Gaussian White Noise. More generally, our results hold for the real zeros of a random real section of a line bundle of degree d over a real projective curve, in the complex Fubini--Study model.


Introduction
Kostlan polynomials. A real Kostlan polynomial of degree d is a univariate random polynomial of the form d k=0 a k d k X k , where the coefficients (a k ) 0 k d are independent N (0, 1) random variables. Here and in the following we use the standard notation N (m, σ 2 ) for the real Gaussian distribution of mean m and variance σ 2 . These random polynomials are also known as elliptic polynomials in the literature (see [29] for example). The roots of such a polynomial form a random subset of R that we denote by Z d . Kostlan proved (cf. [16]) that for all d ∈ N, the average number of roots of this random polynomial is E[Card(Z d )] = d 1 2 , where Card(Z d ) is the cardinality of Z d . It was later proved by Dalmao (see [11]) that Var(Card(Z d )) ∼ σ 2 d 1 2 as d → +∞, where σ is some explicit positive constant. Dalmao also proved that Card(Z d ) satisfies a Central Limit Theorem as d → +∞.
In this paper, we study the higher moments of Card(Z d ) in the large degree limit. Let p ∈ N, we denote by m p (Card(Z d )) the p-th central moment of Card(Z d ). A consequence of our main result (Theorem 1.12) is that, as d → +∞, we have: where σ is the constant appearing in Dalmao's variance estimate, and (µ p ) p∈N is the sequence of moments of the standard real Gaussian distribution N (0, 1). This results allows us to prove a strong Law of Large Numbers: d − 1 2 Card(Z d ) → 1 almost surely. We also prove that d − 1 2 Card(Z d ) concentrates around 1 in probability, faster than any negative power of d. Finally, we recover Dalmao's Central Limit Theorem by the method of moments. The original proof used the Wiener-Ito expansion of the number of roots. In fact, we improve this result by proving a Central Limit Theorem for the counting measure of Z d (see Theorem 1.9

below).
Equivalently, one can define Z d as the set of zeros, on the real projective line RP 1 , of the homogeneous Kostlan polynomial  1) is the hyperplane line bundle over CP 1 . Then, Z d is the real zero set of this random section. In this paper, we study more generally the real zeros of random real sections of positive Hermitian line bundles over real algebraic curves.
Framework and background. Let us introduce our general framework. More details are given in Section 2.1 below. Let X be a smooth complex projective manifold of dimension 1, that is a smooth compact Riemann surface. Let E and L be holomorphic line bundles over X . We assume that X , E and L are endowed with compatible real structures and that the real locus of X is not empty. We denote by M this real locus, which is then a smooth closed (i.e. compact without boundary) submanifold of X of real dimension 1.
Let h E and h L denote Hermitian metrics on E and L respectively, that are compatible with the real structures. We assume that (L, h L ) has positive curvature ω, so that L is ample and ω is a Kähler form on X . The form ω induces a Riemannian metric g on X , hence on M . Let us denote by |dV M | the arc-length measure on M defined by g.
For all d ∈ N, we denote by RH 0 (X , E ⊗ L d ) the space of global real holomorphic sections of E ⊗ L d → X . Let s ∈ RH 0 (X , E ⊗ L d ), we denote by Z s = s −1 (0) ∩ M the real zero set of s. Since s is holomorphic, if s = 0 its zeros are isolated and Z s is finite. In this case, we denote by ν s the counting measure of Z s , that is ν s = x∈Zs δ x , where δ x stands for the unit Dirac mass at x. For any φ ∈ C 0 (M ), we denote by ν s , φ = x∈Zs φ(x). Quantities of the form ν s , φ are called the linear statistics of ν s . Note that ν s , 1 = Card(Z s ), where 1 is the constant unit function.
For any d ∈ N, the space RH 0 (X , E ⊗ L d ) is finite-dimensional, and its dimension can be computed by the Riemann-Roch Theorem. Moreover, the measure |dV M | and the metrics h E and h L induce a Euclidean L 2 -inner product on this space (see Equation (2.1)). Let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ), see Section 2. 3. Then, ν s d is an almost surely well-defined random Radon measure on M . We denote by Z d = Z s d and by ν d = ν s d for simplicity. In this setting, the linear statistics of ν d were studied in [2,14,19,21], among others. In particular, the exact asymptotics of their expected values and their variances are known. Theorem 1.1 (Gayet-Welschinger). For every d ∈ N, let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Then the following holds as d → +∞: where the error term O(d − 1 2 ) does not depend on φ. That is: . Theorem 1.1 is [14, Theorem 1.2] with n = 1 and i = 0. See also [19,Theorem 1.3] when E is not trivial. In [19], the case the linear statistics is discussed in Section 5. 3. We use the following notation for the central moments of ν d .
Of course, m p (ν d ) is only interesting for p 2. For p = 2, the bilinear form m 2 (ν d ) encodes the covariance structure of the linear statistics of ν d . In particular, The large degree asymptotics of m 2 (ν d ) has been studied in [21,Theorem 1.6].
There exists σ > 0 such that, for all φ 1 and φ 2 ∈ C 0 (M ), the following holds as d → +∞: In [2], Ancona derived a two terms asymptotic expansion of the non-central moments of the linear statistics of ν d . As a consequence, he proved the following (cf. [2, Theorem 0.5]).
Remark 1. 6. In Ancona's paper the line bundle E is trivial. The results of [2] rely on precise estimates for the Bergman kernel of L d . These estimates still hold if we replace L d by E ⊗ L d , where E is any fixed real Hermitian line bundle (see [22,Theorem 4.2.1]). Thus, all the results in [2] are still valid for random real sections of E ⊗ L d → X .
Main results. In this paper, we prove a strong Law of Large Numbers (Theorem 1.7) and a Central Limit Theorem (Theorem 1.9) for the linear statistics of the random measures (ν d ) d∈N defined above. These results are deduced from our main result (Theorem 1.12), which gives the precise asymptotics of the central moments (m p (ν d )) p 3 (cf. Definition 1.2), as d → +∞.
Recall that all the results of the present paper apply when Z d is the set of real roots of a Kostlan polynomial of degree d. If one considers homogeneous Kostlan polynomials in two variables, then Since RP 1 is obtained from the Euclidean unit circle S 1 by identifying antipodal points, it is a circle of length π. In this case, |dV M | is the Lebesgue measure on this circle, normalized so that Vol(M ) = π. If one wants to consider the original Kostlan polynomials in one variable, then Z d ⊂ M = R, where R is seen as a standard affine chart in RP 1 . In this case, the measure |dV M | admits the density t → (1 + t 2 ) −1 with respect to the Lebesgue of R, and Vol(M ) = π once again. Theorem 1.7 (Strong Law of Large Numbers). Let X be a real projective curve whose real locus M is non-empty. Let E → X and L → X be real Hermitian line bundles such that L is positive. Let (s d ) d 1 be a sequence of standard Gaussian vectors in d 1 RH 0 (X , E ⊗ L d ). For any d 1, let Z d denote the real zero set of s d and let ν d denote the counting measure of Z d .
Then, almost surely, That is, almost surely: In particular, almost surely, Definition 1.8 (Standard Gaussian White Noise). The Standard Gaussian White Noise on M is a random Gaussian generalized function W ∈ D (M ), whose distribution is characterized by the following facts, where · , · (D ,C ∞ ) denotes the usual duality pairing between D (M ) and C ∞ (M ): In particular, we have W , Here and in the following, we avoid using the term "distribution" when talking about elements of D (M ) and rather use the term "generalized function". This is to avoid any possible confusion with the distribution of a random variable. Theorem 1.9 (Central Limit Theorem). Let X be a real projective curve whose real locus M is non-empty. Let E → X and L → X be real Hermitian line bundles such that L is positive. Let (s d ) d 1 be a sequence of standard Gaussian vectors in d 1 RH 0 (X , E ⊗ L d ). For any d 1, let Z d denote the real zero set of s d and let ν d denote the counting measure of Z d .
Then, the following holds in distribution, in the space D (M ) of generalized functions on M : Before stating our main result, we need to introduce some additional notations.
Definition 1.10 (Moments of the standard Gaussian). For all p ∈ N, we denote by µ p the p-th moment of the standard real Gaussian distribution. Recall that, for all p ∈ N, we have µ 2p = (2p)! 2 p p! and µ 2p+1 = 0.
Definition 1.11 (Partitions). Let A be a finite set, a partition of A is a family I of non-empty disjoint subsets of A such that I∈I I = A. We denote by P A (resp. P k ) the set of partitions of A (resp. {1, . . . , k}). A partition into pairs of A is a partition I ∈ P A such that Card(I) = 2 for all I ∈ I. We denote by PP A , (resp. PP k ) the set of partitions into pairs of A (resp. {1, . . . , k}). Note that P ∅ = {∅} = PP ∅ .
Theorem 1.12 (Central moments asymptotics). Let X be a real projective curve whose real locus M is non-empty. Let E → X and L → X be real Hermitian line bundles such that L is positive. For all d ∈ N, let s d ∈ RH 0 (X , E ⊗ L d ) be a standard Gaussian vector, let Z d denote the real zero set of s d and let ν d denote its counting measure. For all p 3, for all φ 1 , . . . , φ p ∈ C 0 (M ), the following holds as d → +∞: where · denotes the integer part, · ∞ denotes the sup-norm, σ is the same positive constant as in Theorem 1.3, the set PP p is defined by Definition 1.11, and the error term O d In particular, for all φ ∈ C 0 (M ), we have: Remark 1. 13. If p is odd, then the first term vanishes in the asymptotic expansions of Theorem 1. 12. Indeed, in this case PP p = ∅ and µ p = 0. Hence, if p is odd, for all φ 1 , . . . , φ p ∈ C 0 (M ), we have Other interesting corollaries of Theorem 1.12 include the following.
Corollary 1.14 (Concentration in probability). In the setting of Theorem 1.12, let (ε d ) d 1 denote a sequence of positive numbers and let φ ∈ C 0 (M ). Then, for all p ∈ N * , as d → +∞, we have: In particular, for all p ∈ N * , as d → +∞, we have: Corollary 1.14 and Theorem 1.
) for any p ∈ N * and C > 1 π Vol(M ). In the same spirit, Gayet and Welschinger proved in [13, Theorem 2] that there exists D > 0 such that In the other direction, the following corollary bounds the probability that Z d is empty. About the proofs. The proof of Theorem 1.12 relies on several ingredients. Some of them are classical, such as Kac-Rice formulas and estimates for the Bergman kernel of E ⊗ L d , other are new, such as the key combinatorial argument that we develop in Section 3. Kac-Rice formulas are a classical tool in the study of the number of real roots of random polynomials (see [1,6] for example). More generally, they allow to express the moments of local quantities associated with the level sets of a Gaussian process, such as their volume or their Euler characteristic, only in terms of the correlation function of the process. Even if these formulas are well-known, it is the first time, to the best of our knowledge, that they are used to compute the exact asymptotics of central moments of order greater than 3. The Kac-Rice formulas we use in this paper were proved in [2]. We recall them in Proposition 2. 23. They allow us to write the p-th central moment m p (ν d )(φ 1 , . . . , φ p ) as the integral over M p of φ : (x 1 , . . . , Here we are cheating a bit: the random set Z d being almost surely discrete, the Kac-Rice formulas yield the so-called factorial moments of ν d instead of the usual moments. This issue is usual (compare [1,6]), and it will not trouble us much since the central moments can be written as linear combinations of the factorial ones. For the purpose of this sketch of proof, let us pretend that we have indeed: This simplified situation is enough to understand the main ideas of the proof. The correct statement is given in Lemma 3.5 below.
The density D p d is a polynomial in the Kac-Rice densities (R k d ) 1 k p appearing in Definition 2. 21. As such, it only depends on the correlation function of the Gaussian process (s d (x)) x∈M , which is the Bergman kernel of E ⊗ L d . This kernel admits a universal local scaling limit at scale d − 1 2 , which is exponentially decreasing (cf. [22,23]). In [2], the author used these Bergman kernel asymptotics and Olver multispaces (see [25]) to prove estimates for the (R k d ) 1 k p in the large d limit. These key estimates are recalled in Propositions 2.24 and 2.26 below. They allow us to show that D p , uniformly in x ∈ M p such that one of the components of x is far from the others. By this we mean that x = (x 1 , . . . , x p ) and there exists i ∈ {1, . . . , p} such that, for all j = i, the distance from x i to x j is larger than b p Thanks to our estimates on D p d , we show that if I contains a singleton then the integral over M p Hence it contributes only an error term in Theorem 1.12. Moreover, denoting by |I| the cardinality of I (i.e. the number of clusters), the volume of M p this is also an error term in Theorem 1. 12. Thus, the main contribution in m p (ν d )(φ 1 , . . . , φ p ) comes from the integral of φD p d over the pieces M p I indexed by partitions I ∈ P p without singletons and such that |I| Concerning the corollaries, Corollaries 1.14 and 1.15 follow from Theorem 1.12 and Markov's Inequality for the 2p-th moment. The strong Law of Large Numbers (Theorem 1.7) is deduced from Theorem 1.12 for p = 6 by a Borel-Cantelli type argument. The Central Limit Theorem (Theorem 1.9) for the linear statistics is obtained by the method of moments. The functional version of this Central Limit Theorem is then obtained by the Lévy-Fernique Theorem (cf. [12]), which is an extension of Lévy's Continuity Theorem adapted to generalized random processes.
Higher dimension. In this paper, we are concerned with the real roots of a random polynomial (or a random section) in an ambient space of dimension 1. There is a natural higher dimensional analogue of this problem. Namely, one can consider the common zero set Z d ⊂ RP n of r independent real Kostlan polynomials in n + 1 variables, where r ∈ {1, . . . , n}. More generally, we consider the real zero set Z d of a random real section of E ⊗ L d → X in the complex Fubini-Study model, where X is a real projective manifold of complex dimension n whose real locus M is non-empty, L is a positive line bundle as above, and E is a rank r real Hermitian bundle with 1 r n. Then, for d large enough, Z d is almost surely a smooth closed submanifold of codimension r in the smooth closed n-dimensional manifold M . In this setting, M is equipped with a natural Riemannian metric that induces a volume measure |dV M | on M and a volume measure ν d on Z d . As in the 1-dimensional case, ν d is an almost surely well-defined random Radon measure on M . In this higher dimensional setting, we have the following analogues of Theorem 1.1 and 1.3 (see [19,20,21]): where σ n,r > 0 is a universal constant depending only on n and r. In [21], Letendre and Puchol proved some analogues of Corollaries 1.14 and 1.15 for any n and r. They also showed that the strong Law of Large Numbers (Theorem 1.7) holds if n 3.
Most of the proof of Theorem 1.12 is valid in any dimension and codimension. In fact, the combinatorics are simpler when r < n. The only things we are missing, in order to prove the analogue of Theorem 1.12 for any n and r, are higher dimensional versions of Propositions 2.24 and 2. 26. The proofs of these propositions (see [2]) rely on the compactness of Olver multispaces, which holds when n = 1 but fails for n > 1. This seems to be only a technical obstacle and the authors are currently working toward the following.
In particular, for all φ Proving this conjecture for n = 2 and p = 4 is enough to prove that the strong Law of Large Numbers (Theorem 1.7) holds for n = 2, which is the only missing case. This conjecture also implies the Central Limit Theorem (Theorem 1.9) in dimension n and codimension r, with the same proof as the one given in Section 4.2. Note that a Central Limit Theorem for the volume of the common zero set of r Kostlan polynomials in RP n was proved in [4,5].
Other related works. The complex roots of complex Kostlan polynomials have been studied in relation with Physics in [10]. More generally, complex zeros of random holomorphic sections of positive line bundles over projective manifolds were studied in [27] and some subsequent papers by the same authors. In [27], they computed the asymptotics of the expected current of integration over the complex zeros of such a random section, and proved a Law of Large Numbers similar to Theorem 1. 7. In [28], they obtained a variance estimate for this random current, and proved that it satisfies a Central Limit Theorem. This last paper extends the results of [29] for the complex roots of a family of random polynomials, including elliptic ones. In [9], Bleher, Shiffman and Zelditch studied the p-points zero correlation function associated with random holomorphic sections. These functions are the Kac-Rice densities for the non-central p-th moment of the linear statistics in the complex case. The results of [2] are also valid in the 1-dimensional complex case, see [2, Section 5]. Thus, Theorem 1.12 can be extended to the complex case, with the same proof.
In Corollaries 1.14 and 1.15, we deduce from Theorem 1.12 some concentration in probability, faster than any negative power of d. However, our results are not precise enough to prove that this concentration is exponentially fast in d. In order to obtain such a large deviation estimate, one would need to investigate how the constants involved in the error terms in Theorem 1.12 grow with p. Some large deviations estimates are known for the complex roots of random polynomials. As far as real roots are concerned, the only result of this kind we are aware of is [7].
The Central Limit Theorem (Theorem 1.9) was already known for the roots of Kostlan polynomials, see [11]. In the wider context of random real geometry, Central Limit Theorems are known in several settings, see [4,5,26] and the references therein. The proofs of all these results rely on Wiener chaos techniques developed by Kratz-Leòn [17]. Our proof of Theorem 1.9 follows a different path, relying on the method of moments, for two reasons. First, to the best of our knowledge, Wiener chaos techniques are not available for random sections of line bundles. Second, these techniques are particularly convenient when dealing with random models with slowly decaying correlations, for example Random Waves models. In these cases, one of the first chaotic components, usually the second or the fourth, is asymptotically dominating. Hence, one can reduce the study of, say, the number of zeros to that of its dominating chaotic component, which is easier to handle. In the complex Fubini-Study setting we are considering in this paper, the correlations are exponentially decreasing, so that all chaotic components of the number of zeros have the same order of magnitude as the degree goes to infinity. In order to use Wiener chaoses to prove a Central Limit Theorem in our setting, one would have to study the joint asymptotic behavior of all the chaotic components, which seems more difficult than our method.
For real zeros in ambient dimension n = 1, Nazarov and Sodin [24] proved a strong Law of Large Numbers, as R → +∞, for the number of zeros of a Gaussian process lying in the interval [−R, R]. Finally, to the best of our knowledge, Theorem 1.12 gives the first precise estimate for the central moments of the number of real zeros of a family of random processes.
Organization of the paper. This paper is organized as follows. In Section 2 we introduce the object of our study and recall some useful previous results. More precisely, we introduce our geometric framework in Section 2. 1. In Section 2.2 we introduce various notations that will allow us to make sense of the combinatorics involved in our problem. The random measures we study are defined in Section 2. 3. Finally, we state the Kac-Rice formulas for higher moments in Section 2.4 and we recall several results from [2] concerning the density functions appearing in these formulas. In Section 3, we prove our main result, that is the moments estimates of Theorem 1. 12. Section 4 is concerned with the proofs of the corollaries of Theorem 1. 12. We prove the Law of Large Numbers (Theorem 1.7) in Section 4.1, the Central Limit Theorem (Theorem 1.9) in Section 4.2 and the remaining corollaries (Corollaries 1.14 and 1.15) in Section 4.3.

Geometric setting
In this section, we introduce our geometric framework, which is the same as that of [2,14,19,20,21]. See also [9,27,28], where the authors work in a related complex setting. A classical reference for some of the material of this section is [15].
• Let (X , c X ) be a smooth real projective curve, that is a smooth complex projective manifold X of complex dimension 1, equipped with an anti-holomorphic involution c X . We denote by M the real locus of X , that is the set of fixed points of c X . Throughout the paper, we assume that M is not empty. In this case, M is a smooth compact submanifold of X of real dimension 1 without boundary, that is the disjoint union of a finite number of circles.
• Let (E, c E ) and (L, c L ) be two real holomorphic line bundles over (X , c X ). Denoting by π E (resp. π L ) the bundle projection, this means that E → X (resp. L → X ) is an holomorphic which defines a real Hermitian metric on E ⊗ L d → X . We assume that (L, h L ) is positive, in the sense that its curvature form ω is a Kähler form. Recall that ω is locally defined as 1 2i ∂∂ ln(h L (ζ, ζ)), where ζ is any local holomorphic frame of L. The Kähler structure defines a Riemannian metric g = ω(·, i·) on X , hence on M . The Riemannian volume form on X associated with g is simply ω. We denote by |dV M | the arc-length measure on M associated with g. For all k ∈ N * , we denote by |dV M | k the product measure on M k .
This is a complex vector space of complex dimension N d . By the Riemann-Roch Theorem, N d is finite and diverges to infinity as d → +∞. We denote by: • The volume form ω and the Hermitian metric h d induce an Hermitian L 2 -inner product · , · on H 0 (X , E ⊗ L d ). It is defined by: The restriction of this inner product to RH 0 (X , E ⊗ L d ) is a Euclidean inner product.
• For any section s ∈ RH 0 (X , E ⊗ L d ), we denote by Z s = s −1 (0) ∩ M its real zero set. Since s is holomorphic, if s = 0 its zeros are isolated. In this case, Z s is finite by compactness of M , and we denote by ν s = x∈Zs δ x , where δ x stands for the unit Dirac mass at x ∈ M . The measure ν s is called the counting measure of Z s . It is a Radon measure, that is a continuous linear form on the space (C 0 (M ), · ∞ ) of continuous functions equipped with the sup-norm. It acts on continuous functions by: for all φ ∈ C 0 (M ), ν s , φ = x∈Zs φ(x).
Example 2.1 (Kostlan scalar product). We conclude this section by giving an example of our geometric setting. We consider X = CP 1 , equipped with the conjugation induced by the one in C 2 . Its real locus is M = RP 1 . We take E to be trivial and L to be the dual of the tautological . Both E and L are canonically real Hermitian line bundle over CP 1 , and the curvature of L is the Fubini-Study form, normalized so that Vol CP 1 = π. The corresponding Riemannian metric on RP 1 is the quotient of the metric on the Euclidean unit circle, so that the length of RP 1 is π.
In this setting, H 0 (X , E ⊗ L d ) (resp. RH 0 (X , E ⊗ L d )) is the space of homogeneous polynomials of degree d in two variables with complex (resp. real) coefficients. If s ∈ RH 0 (X , E ⊗ L d ) is such a polynomial, then Z s is the set of its roots in RP 1 . Finally, up to a factor (d + 1)π which is irrelevant to us, the inner product of Equation (2.1) is defined by: for any homogeneous polynomials P and Q of degree d in two variables. In particular, the family d k X k Y d−k 0 k d is an orthonormal basis of RH 0 (X , E ⊗ L d ) for this inner product.

Partitions, products and diagonal inclusions
In this section, we introduce some notations that will be useful throughout the paper, in particular to sort out the combinatorics involved in the proof of Theorem 1.12 (see Section 3). In all this section, M denotes a smooth manifold.

Notations 2.2. Let
A be a finite set.
• We denote by Card(A) or by |A| the cardinality of A.
• We denote by M A the Cartesian product of |A| copies of M , indexed by the elements of A.
If A is clear from the context or of the form {1, . . . , k} with k ∈ N * , we use the simpler notations x for x A and φ for φ A .
Recall the we defined the set P A (resp. P k ) of partitions of a finite set A (resp. of {1, . . . , k}) in the introduction, see Definition 1.11.

Definition 2.3 (Diagonals). Let
A be a finite set, we denote by ∆ A the large diagonal of M A , that is: Moreover, for all I ∈ P A , we denote by ∆ A,I = (x a ) a∈A ∈ M A ∀a, b ∈ A, (x a = x b ⇐⇒ ∃I ∈ I such that a ∈ I and b ∈ I) .
If A = {1, . . . , k}, we use the simpler notations ∆ k for ∆ A , and ∆ k,I for ∆ A,I .
In the following, we avoid using the notation ∆ A,I0(A) and use M A \ ∆ A instead.
Let us now go back to the setting of Section 2.1, in which M is the real locus of the projective manifold X . Let d ∈ N and let s ∈ RH 0 (X , E ⊗ L d ) \ {0}. In Section 2.1, we defined the counting measure ν s of the real zero set Z s of s. More generally, for any finite set A, we can define the counting measure of Z A s = (x a ) a∈A ∈ M A ∀a ∈ A, x a ∈ Z s and that of Z A s \ ∆ A . The latter is especially interesting for us, since this is the one that appears in the Kac-Rice formulas, see Proposition 2.23 below. Definition 2.6 (Counting measures). Let d ∈ N and let A be a finite set. For any non-zero s ∈ RH 0 (X , E ⊗ L d ), we denote by: where δ x is the unit Dirac mass at x = (x a ) a∈A ∈ M A and ∆ A is defined by Definition 2. 3. These measures are the counting measures of Z A s and Z A s \ ∆ A respectively. Both ν A s and ν A s are Radon measure on M A . They act on C 0 (M A ) as follows: for any φ ∈ C 0 (M A ), As usual, if A = {1, . . . , k}, we denote ν k s for ν A s and ν k s for ν A s . We have seen in Remark 2.5 that M A splits as the disjoint union of the diagonals ∆ A,I , with I ∈ P A . Taking the intersection with Z A s yields a splitting of this set. Using the diagonal inclusions of Definition 2.4, this can be expressed in terms of counting measures as follows.
Lemma 2.7. Let d ∈ N and let A be a finite set. For any s ∈ RH 0 (X , E ⊗ L d ) \ {0}, we have: Proof. Recall that M A = I∈P A ∆ A,I . Then, we have: Let I ∈ P A , recall that ι I defines a smooth diffeomorphism from M I \ ∆ I onto ∆ A,I . Moreover,

Zeros of random real sections
Let us now introduce the main object of our study: the random Radon measure ν d encoding the real zeros of a random real section s d ∈ RH 0 (X , E ⊗ L d ). The model of random real sections we study is often referred to as the complex Fubini-Study model. It was introduced in this generality by Gayet and Welschinger in [13]. This model is the real counterpart of the model of random holomorphic sections studied by Shiffman and Zelditch in [27] and subsequent articles.
Definition 2.8 (Gaussian vectors). Let (V, · , · ) be a Euclidean space of dimension N , and let Λ denote a positive self-adjoint operator on V . Recall that a random vector X in V is said to be a centered Gaussian with variance operator Λ if its distribution admits the following density with respect to the normalized Lebesgue measure: We denote by X ∼ N (0, Λ) the fact that X follows this distribution. If X ∼ N (0, Id), where Id is the identity of V , we say that X is a standard Gaussian vector in V .
Remark 2. 9. Recall that if (e 1 , . . . , e N ) is an orthonormal basis of V , a random vector X ∈ V is a standard Gaussian vector if and only if X = N i=1 a i e i where the (a i ) 1 i N are independent identically distributed N (0, 1) real random variables.
In the setting of Section 2.1, for any d ∈ N, the space RH 0 (X , E ⊗ L d ) is endowed with the Euclidean inner product · , · defined by Equation (2.1). We denote by s d a standard Gaussian vector in RH 0 (X , E ⊗L d ). Almost surely s d = 0, hence its real zero set and the associated counting measure are well-defined. For is the homogeneous Kostlan polynomial of degree d defined in Section 1.
Note that the rotations of the circle M = RP 1 are induced by orthogonal transformations of R 2 . The group O 2 (R) acts on the space of homogeneous polynomials in two variables of degree d by composition on the right. Example 2.1 shows that the Kostlan inner product is invariant under this action, hence so is the distribution of s d . In other terms, the random process (s d (x)) x∈M is stationary.
Example 2.11. Let us give another example showing that the complex Fubini-Study model goes beyond Kostlan polynomials. Let X be a Riemann surface embedded in CP n . We assume that X is real, in the sense that it is stable under complex conjugation in CP n . Note that, in our setting, we can always realize X in such a way for some n large enough, by Kodaira's Embedding Theorem. Then M = X ∩ RP n is the disjoint union of at most g+1 smooth circles in RP n , where g is the genus of X . Figure 1 below shows two examples of this geometric situation.
(a) M is a circle even though X has genus 2.
(b) X is connected but M has three connected components M 1 , M 2 and M 3 . Figure 1: Two examples of a real Riemann surface X embedded in CP n and its real locus M .
We choose E to be trivial and L to be the restriction to X of the hyperplane line bundle O(1) → CP n . Then, the elements of RH 0 (X , E ⊗ L d ) are restrictions to X of homogeneous polynomials of degree d in n + 1 variables with real coefficients. This space is equipped with the inner product (2.1). In Equation (2.1), the Kähler form ω is the restriction X of the Fubini-Study form on CP n . However, the domain of integration is X , so that (2.1) is not the analogue in n + 1 variables of the Kostlan inner product of Example 2.1.
• On Figure 1a, the curve M is connected hence diffeomorphic to RP 1 . However, E ⊗ L d → X is not an avatar of O(d) → CP 1 for some d 1, since X CP 1 . In particular, the previous construction gives a random homogeneous polynomial s d on M RP 1 , which is Gaussian and centered but is not a Kostlan polynomial of some degree.
Unlike what happens in Example 2.10, rotations in M are not obtained as restriction of isometries of X (generically, the only isometries of X are the identity and the complex conjugation). Thus, there is no reason why the process (s d (x)) x∈M should be stationary.
• On Figure 1b, the real locus M had several connected components while X is connected.
Since the inner product (2.1) is defined by integrating on the whole complex locus X , the values of s d in different connected components of M are a priori correlated.

Lemma 2.12 (Boundedness of linear statistics). Let d 1 and s
For all φ ∈ C 0 (M ), the random variable ν d , φ is bounded. In particular, it admits finite moments of any order.
Hence it is enough to prove that Card(Z d ) is a bounded random variable.
The cardinality of Z d is the number of zeros of s d in M , which is smaller than the number of zeros of s d in X . Now, almost surely, s d = 0, and the complex zero set of s d defines a divisor which is Poincaré-dual to the first Chern class of E ⊗ L d (see [15, pp. 136 and 141]). Hence, almost surely: Remark 2. 13. In the case of polynomials, the proof is clearer: the number of real roots of a non-zero polynomial of degree d is bounded by the number of its complex roots, which is at most d.

Kac-Rice formulas and density functions
In this section, we recall some important facts about Kac-Rice formulas. These formulas are classical tools in the study of moments of local quantities such as the cardinality, or more generally the volume, of the zero set of a smooth Gaussian process. Classical references for this material are [1,6]. With a more geometric point of view, the following formulas were proved and used in [2,14,21], see also [19]. In the same spirit, Lerario and Stecconi derived a Kac-Rice formula for sections of fiber bundles, see [18,Theorem 23.6].
Remark 2.14. In some of the papers we refer to in this section, the line bundle E is taken to be trivial. That is the authors considers random real sections of L d instead of E ⊗ L d . As we already explained (see Remark 1.6), the proofs of the results we cite below rely on asymptotics for the Bergman kernel of E ⊗ L d , as d → +∞. These asymptotics do not depend on E, see [22,   Let us now consider the geometric setting of Section 2.1.
Here, ∇ is any connection on E ⊗ L d and See also [21,Theorem 5.5] in the case k = 2. Recall that ν k d was defined by Definition 2.6 and is the counting measure of the random set s −1 where R k d is the density function defined by Definition 2.21. Let k 2 and d ∈ N. If x ∈ ∆ k , then the evaluation map ev d x can not be surjective. Hence, the continuous map x → det ⊥ ev d x from M k to R vanishes on ∆ k , and one would expect R k d to be singular along the diagonal. Yet, Ancona showed that one can extend continuously R k d to the whole of M k and that the extension vanishes on ∆ k , see [2,Theorem 4.7]. Moreover, he showed that d − k 2 R k d is uniformly bounded on M k \ ∆ k as d → +∞. We will use this last fact repeatedly in the proof of Theorem 1.12, see Section 3 below.
Let d 1 and let s d ∈ RH 0 (X , E ⊗ L d ) be a standard Gaussian. A fundamental idea in our problem is that the values (and more generally the k-jets) of s d at two points x and y ∈ M are "quasi-independent" if x and y are far from one another, at scale d − 1 2 . More precisely, (s d (x)) x∈M defines a Gaussian process with values in R(E ⊗ L d ) whose correlation kernel is the Bergman kernel E d of E ⊗ L d . Recall that E d is the integral kernel of the orthogonal projection from the space of square integrable sections of E ⊗ L d onto H 0 (X , E ⊗ L d ), for the inner product defined by Equation (2.1). In particular, for any x, y ∈ X , we have y . For our purpose, it is more convenient to consider the normalized Bergman kernel For any x ∈ X , the map E d (x, x) is an endomorphism of the 1-dimensional space (E ⊗ L d ) x , hence can be seen as a scalar. Note that e d is the correlation kernel of the normalized process x∈M , which has unit variance and the same zero set as (s d (x)) x∈M .
The normalized Bergman kernel e d admits a universal local scaling limit at scale d − 1 2 around any point of X (cf. [ These facts were extensively used in [2,20,21] and we refer to these papers for a more detailed discussion of how estimates for the Bergman kernel are used in the context of random real algebraic geometry. An important consequence of these estimates that we use in the present paper is Proposition 2.26 below.
Definition 2.25. Let p ∈ N, we denote by b p = 1 C 1 + p 4 , where C > 0 is the same as above. That is C is the constant appearing in the exponential in [ Here we used the notations defined in Section 2.2 (see Notation 2.2), and by convention R 0 d = 1 for all d.
Proof. Proposition 2.26 is the same as [2, Proposition 3.2] but for two small points. We refer to [2] for the core of the proof of this proposition. Here, let us just explain what the differences are between Proposition 2.26 and [2, Proposition 3.2], and how these differences affect the proof.  Remarks 2. 27. We conclude this section with some comments for the reader who might be interested in the proof of [2, Proposition 3.2].
• In [2], estimates for the Bergman kernel (i.e. the correlation function of the random process under study) are obtained using peak sections. This alternative method yields the necessary estimates without having to use the results of [22,23].
• In [2], the proof of Proposition 3.2 is written for k = 2 and |A| = 1 = |B|, for clarity of exposition. The extension to k > 2 is non-trivial, and the proof for k > 2 requires in fact the full power of the techniques developed in [2, Section 4]. More recently, the authors developed similar techniques for smooth stationary Gaussian processes in R, see [3, Theorem 1.14].

Asymptotics of the central moments
The goal of this section is to prove Theorem 1. 12. In Section 3.1 we derive an integral expression of the central moments we want to estimate, see Lemma 3.5 below. Then, in Section 3.2, we define a decomposition of the manifolds M A , where M is as in Section 2.1 and A is a finite set. In Sections 3.3, 3.4 and 3.5, we compute the contributions of the various pieces of the decomposition defined in Section 3.2 to the asymptotics of the integrals appearing in Lemma 3. 5. Finally, we conclude the proof of Theorem 1.12 in Section 3.6.

An integral expression of the central moments
The purpose of this section is to derive a tractable integral expression for the central moments of the form m p (ν d )(φ 1 , . . . , φ p ), defined by Definition 1.2 and appearing in Theorem 1. 12. This is done in Lemma 3.5 below. Before stating and proving this lemma, we introduce several concepts and notations that will be useful to deal with the combinatorics of this section and the following ones.
Recall that we already defined the set P A of partitions of a finite set A (see Definition 1.11). The next concept we need to introduce is that of induced partition. In this paper, we will only consider the case where I ∈ P B and A is the reunion of some of the elements of I, that is: for any I ∈ I, either I ⊂ A or I ∩ A = ∅. This condition is equivalent to I A ⊂ I, and in this case we have I A = {I ∈ I | I ⊂ A}.
Another useful notion is the following order relation on partitions.

Definition 3.2 (Order on partitions). Let
A be a finite set and let I, J ∈ P A . We denote by J I and we say that J is finer than I if J is obtained by subdividing the elements of I. That is, for any J ∈ J there exists I ∈ I such that J ⊂ I. If J I and J = I, we say that J is strictly finer than I, which we denote by J < I.
One can check the following facts. Given a finite set A, the relation defines a partial order on P A such that I → |I| is decreasing. The partially ordered set (P A , ) admits a unique maximum equal to {A}. It also admits a unique minimum equal to {{a} | a ∈ A}, that we denote by I 0 (A) as in Remark 2. 5

. If
A subset A ∈ S I is said to be adapted to I.
The set S I will appear as an index set in the integral expression derived in Lemma 3.5 below. The only thing we need to know about it is the following natural result. We can now derive the integral expression of the central moments we are looking for. We will make use of the various notations introduced in Section 2.2.
Lemma 3.5. Let p 2 and let φ 1 , . . . , φ p ∈ C 0 (M ). Let d p ∈ N be given by Lemma 2.18, for all d d p we have: where, for any finite set B, any I ∈ P B and any x I = (x I ) I∈I ∈ M I , Here, S I is the one we defined in Definition 3. 3.
where the last equality comes from Lemma 2. 7. By the Kac-Rice formulas of Proposition 2.23, this equals: By Lemma 3.4, we can exchange the two sums and obtain the following: which concludes the proof.
Before going further, let us try to give some insight into what is going to happen. Our strategy is to compute the large d asymptotics of the terms of the form appearing in Lemma 3. 5. In the case where p is even, which is a bit simpler to describe, many of these terms will contribute a leading term of order d p 4 , and the others will only contribute a smaller order term. Terms of the form (3.1) can all be dealt with in the same way, hence we will not make any formal difference between them in the course of the proof. Note however, that the term indexed by I 0 (p) is simpler to understand, as shown in Example 3.6 below. This term is the one we considered in our sketch of proof in Section 1. At each step in the following, we advise the reader to first understand what happens for this term, before looking at the general case.  4. Hence, the term indexed by I 0 (p) in Lemma 3.5 is: Then, S I0(p) is just the set of all subsets of {1, . . . , p}, see Definition 3. 3. Furthermore, for any A ⊂ {1, . . . , p} the induced partition I 0 (p) A equals {{i} | i ∈ A}, hence is canonically in bijection with A. Finally, we obtain that:

Cutting M A into pieces
In this section, we define a way to cut the Cartesian product M A into disjoint pieces, for any finite subset A. The upshot is to use this decomposition of M A with A = I ∈ P p , in order to obtain the large d asymptotics of integrals of the form (3.1). For this to work, we need to define a splitting of M A that depends on the parameters p ∈ N and d 1.  x 1 x 2 x 3 x 4 x 5 x 6 (a) A configuration x = (x 1 , . . . , x 6 ) of six distinct points on a circle. The length of the segment is the cut-off distance bp ln d    9. Let A be a finite set, let I ∈ P A , let p ∈ N and let d 1, we define: Remarks 3. 10.
• For any p ∈ N and d 1, we have: M A = I∈P A M A,p I,d .
• Let I ∈ P A and let (x a ) a∈A ∈ M A,p I,d . For all I ∈ I, for any a and b ∈ I, the geodesic distance from x a to x b satisfies: •  ). We know that the components of x A indexed by a given I ∈ I are close together in M , hence belong to the same cluster. However, given I and J ∈ I, the points indexed by I and those indexed by J might be close enough in M that they belong to the same cluster. Thus, all we can say is that the clusters of x A are indexed by unions of elements of I, that is I p d (x A ) I. This shows that x A ∈ J I M A,p J ,d and concludes the proof.
We conclude this section by bounding from above the volume of M A,p I,d . What we are interested in here is an asymptotic upper bound as d → +∞ with A, I and p fixed.

Contribution of the partitions into pairs
Let p 2 and let φ 1 , . . . , φ p be test-functions. Recall that m p (ν d )(φ 1 , . . . , φ p ) is the sum of the integrals (3.2) for I ∈ P p and J ∈ P I , and that our final goal is to prove Theorem 1. 12. This theorem gives the asymptotics of m p (ν d )(φ 1 , . . . , φ p ) as d goes to infinity, up to an error term of We proved in Lemma 3.14 that the integral (3.2) only contributes an error term if |J | < p 2 . Besides, we proved in Corollary 3.17 that the integral (3.2) also contributes an error term if there exists j such that {{j}} ∈ J , in particular if |J | is too large. This last point will be made more precise in Section 3.6 below.
In this section, we study the integrals of the form (3.2) that will contribute to the leading term in the asymptotics of m p (ν d )(φ 1 , . . . , φ p ). These integrals are indexed by couples of partitions (I, J ) satisfying the following technical condition: the double partitions into pairs. Recall that we denoted by PP A the set of partition into pairs of the finite set A, see Definition 1.11, and that we denoted by I S the partition of S ⊂ A induced by a partition I ∈ P A , see Definition 3.1. We denote by C A the set of such double partitions into pairs of A. If A = {1, . . . , p} we simply denote C p = C A .
Let us take some time to comment upon this definition. Note that we will mostly use it with A = {1, . . . , p}. However, the general case will be useful in Lemmas 3. 21 19. Let A be a finite set and let (I, J ) ∈ C A . Let S be as in Definition 3. 18. We have |S| = |I S |, and since I S admits a partition J I S into pairs, this cardinality is even. Similarly, since I \ I S is a partition into pairs of A \ S, we have |A| = |S| + 2 |I \ I S |. Hence, |A| is even. In particular, C A is empty if |A| is odd. • Taking S = A, we must have I = I S = {{a} | a ∈ A}. Identifying canonically I with A, we can choose any J ∈ PP I PP A .
In the remainder of this section, we study the large d asymptotics of integrals of the form (3.2) where (I, J ) ∈ C p . For this, we need to show that the function D I d defined in Lemma 3.5 factorizes nicely on M I,p J ,d , up to an error term. The first step is Lemma 3.21 where we factor the contribution from the pairs of singletons in J , i.e the elements of J I S .

Proof.
Recall that, as discussed in Remark 3.19, the set C A is empty if |A| is odd. Hence Lemma 3.21 is true in this case. If |A| is even, the proof is by induction on (one half of) |A|.
Since A is adapted to I, we have I = I A {{i} | i / ∈ A} by Definition 3. 3. This implies that |I| = |I A | + p − |A|. Hence the error term in the previous equation is O(d |I| 2 − p 4 −1 ). Summing these terms over the subsets A ∈ S I such that A ⊂ {1, . . . , p} \ {s, t}, we proved that Equation (3.4) is satisfied uniformly for all x I ∈ M I,p J ,d .
With the same notations as in Lemma 3.21, the second step in our factorization of D I d is to factor the contributions from the singletons in J , that is from the {{I} | I ∈ I \ I S }. This is the purpose of the following lemma.
Lemma 3. 22. Let A be a finite set, let (I, J ) ∈ C A and let S ⊂ A be as in Definition 3. 18. Then, the following holds uniformly for all x I ∈ M I,p J ,d : Factorization of the integrand. Recall that, by Condition 1 in Definition 3.18, the partition I only contains singletons and pairs, and S is the union of the singletons of I. Let x I ∈ M I , we have: Let F I d : M I → R be the function defined by: for all x I ∈ M I . Here we use Notation 2.2, hence for any J = {{i}, {j}} ∈ J we have: Note that since I ∈ P p , we have |I| p so that |I| 2 − p 4 − 1 . Applying Lemma 3.22, we obtain: We want to prove that the integral of F I d over M I,p J ,d is equal to its integral over J∈J M J,p {J},d , up to an error term. Let K ∈ P I be such that K > J . We can bound the integral of F I d over Ω K = M I,p K,d ∩ J∈J M J,p {J},d , using the same method as in the proof of Lemma 3.14. We have: Using Proposition 2.24 in Equation (3.9), we prove that the function F I d is bounded uniformly over M I by ( . Then, by Lemma 3.13, we have: Recall that K > J , so that |K| < |J |, as discussed after Definition 3.2. Since (I, J ) ∈ C p , by Definition 3. 18