Complex determinantal processes and H1 noise

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity rho dnu, where nu is the corresponding invariant measure. We show that as rho converges to infinity, after centering, these processes converge to invariant H1 noise. More precisely, for all functions f in the interesection of H1(nu) and L1(nu) the distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with mean 0 and variance given by ||f||_H1^2 / (4 pi).


Introduction
Determinantal processes are point processes with a built-in pairwise repulsion. They were first considered by Macchi [10] as a model for fermions in quantum mechanics, and have since been understood to arise naturally in a number of contexts, from eigenvalues of random matrices to random spanning trees and non-intersecting paths, see [1,2,4,7,8,9,12,15,17].
A point process Z on C is determinantal if for disjoint sets D 1 , . . . , D k we have for each k ≥ 1. Here K(x, y) is a hermitian symmetric measurable function and µ is some reference measure. The integrand is often called the joint intensity or correlation function of the point process.
Conversely, if such a function K defines a self-adjoint integral operator K on L 2 (Λ, µ) which is locally trace class with all eigenvalues lying in [0, 1], then there exists a point process satisfying (1). In this case we speak of the determinantal process (K, µ). In a weak sense, the points repel one another because the determinant vanishes on the diagonal.
The processes we consider are defined by two properties. First, they correspond to a projection K to a subspace of analytic functions with respect to a radially symmetric reference measure. Second, their distribution is invariant under the symmetries of their underlying space Λ. The latter is either the complex plane C, 2-sphere S, or hyperbolic plane U. We will think of them as a subsets of C, or strictly speaking C ∪ {∞}, though for the sphere S usually there is no harm in ignoring the point at infinity.
These properties are uniquely satisfied by a family of processes indexed by a single density parameter ρ > 0 on each space, see Krishnapur [11,Theorem 3.0.5], who discovered several remarkable properties of these processes.
Here, as in the sequel, dz stands for Lebesgue area measure on C.
Note that the kernelǨ ρ is the projection onto the span of the orthonormal set { ρ k k! z k } ∞ k=0 in L 2 (C, µ ρ ), and so the above pair defines a determinantal process of infinitely many points in the complex plane, see, for example [7].
Our main theorem concerns the linear statistics for the point process.
Recall that for f : Λ → R, the ivariant measure ν = ν Λ and the intrinsic gradient ∇ ι we have We say f is in H 1 (ν) or L 1 (ν) if these corresponding norm is finite.
Then, as ρ → ∞, the distribution of converges to a mean zero normal with variance 1 4π f 2 H 1 (ν) .
Note that for both the limiting variance and the shift to make sense it is necessary to have f ∈ H 1 ∩ L 1 , so the theorem holds for the most general test functions possible.
The fact that the variance is of order 1 manifests the advertised repulsion. The H 1 -norm is conformally invariant, so one may replace the intrinsic gradient and intrinsic measure by the planar gradient and Lebesgue measure for the embedding.
This work is partially motivated by the recent results of Sodin and Tsilerson [16] on the three canonical Gaussian analytic functions (GAFs) with zero sets invariant under the symmetries of the plane, sphere, and hyperbolic plane. These processes are also indexed by a density parameter, and [16] establishes asymptotic normality for the corresponding linear statistics, with f ∈ C 2 0 . What is striking is that for GAFs the variance actually decays as the density tends to infinity: Thus, the zeros of typical GAFs are more orderly than their determinantal counterparts.
The determinantal processes studied here, while attractive solely on the basis of their invariance, also arise as matrix models. The planar case is really just the "infinite dimensional Ginibre ensemble". If A is an n×n matrix of iid standard complex Gaussians, then as n ↑ ∞ the point process of A-eigenvalues converges to the planar model, and ρ here corresponds to scaling. As for the spherical model, Krishnapur [11] has proved that it coincides with the eigenvalues of A −1 B, where A and B are independent ρ × ρ Ginibre matrices. Further, for integer ρ, Krishnapur provides strong evidence that the hyperbolic points have the same law as the singular points of In all three cases, Krishnapur provides natural random analytic functions for which Z is the set of zeros. Using an integration by parts argument, Theorem 1 can be interpreted to say that the log absolute value of these analytic functions converges to the Gaussian Free Field. See Section 3 in [13] for this relation in the Ginibre ensemble.
Theorem 1 also identifies the present as a companion paper to [13] which treats the limiting noise for the Ginibre eigenvalues. The eigenvalues also define a determinantal process in C, see [5]. In [13] it is shown that, along with an H 1 -noise in the interior of U similar to above, there is an H 1/2 (∂U) noise component in the corresponding n ↑ ∞ central limit theorem. The invariance and lack of boundary effects in the three models considered here makes for essentially different proofs that are shorter and rely less on combinatorial constructions.
The only overlap is the cumulant formula which is the starting point for both proofs.
The main Theorem 1 is proved in three steps. In Section 2 we establish some general conditions under which (smooth) linear statistics are asymptotically normal, without computing the asymptotic variance. For this, the fact that the kernels is an analytic projections and their specific decay properties are crucial. In Section 3 we check that these properties are satisfied by our models. In Section 4 we determine the asymptotic variance and extend the convergence to general test functions.

General conditions for asymptotic normality
Taking a broader perspective, this section shows that under certain conditions satisfied by the models we are considering, linear functionals are asymptotically normal.
Let B be a compact subset of C. Consider the following set-up. K ρ : B 2 → R is a set of kernels indexed by ρ, which ranges in an unbounded subset of the positive reals. The kernels here are the Hermitian and are with respect to Lebesgue measure; more precisely, if K ρ denotes the kernels outlined above, and κ = dµ ρ (z)/dz is the density of the reference measure, then here and below

Kernel Properties
The eventual asymptotic normality rests on the following asymptotic properties of K ρ as ρ → ∞; all limit statements and o(·) notations refer to this limit. Throughout, c denotes a numerical constants which may change from line to line.
• Uniform bound (UB). It holds • Interaction decay (ID). The above bounding function satisfies • Limited local analytic projection (LLAP) property. Assume that B ⊂ C. Fix Similarly, • Covariance (CO). For any function F with bounded third derivatives and compact support in the interior of B we have Cov ρ (∂ z,z F, zz) = o(1).
Note of course that Cov ρ (f, g) indicates the covariance of z∈Z f (z) and z∈Z g(z) in the K ρ -process. The main proposition of this section (proved as Proposition 8) is: That these conditions are satisfied by the planar, spherical and hyperbolic models is delayed to the next section. Here we provide a lemma that sheds more light as to how condition LLAP arises.
If K ρ satisfies UB and ID, then K ρ satisfies the LLAP.
Proof. Note that since for each z, the function y → yK ρ (y, z) is analytic, it follows from the analytic projection property that Thus, for (7) it suffices to show that where recall B 2 ⊂ B o . Setting s = y − z, there is a polynomial q of degree p so that |y p | ≤ q(|s|) for all choices of z ∈ B 2 , y ∈ B c . Also, q(|s|) ≤ c|s| 3 as soon as |s| bounded away from zero. So, for z a positive distance from S \ B, we have that and by UB, ID, the absolute value of the left hand side of (9) is bounded above by The proof of (8) is identical since K ρ is hermitian symmetric.

Cumulants
Recall that for any random variable X, the cumulants Cum k (X), k = 1, 2, . . . , are the coefficients in the expansion of the logarithmic generating function, and X is Gaussian if and only of Cum k (X) = 0 for all k ≥ 3. In any determinantal process (K ρ , µ ρ ), the cumulants of the linear statistic f (z k ) have the explicit form, (10) where x m+1 = x 1 is understood, the integral ranges over m copies of the full space (here B), and again we are absorbing the reference measure µ ρ into the K ρ kernel. The structure behind formula (10) has been employed in the past to establish asymptotic normality for determinantal processes with various assumptions on the regularity of f . See in particular the pioneering work of Costin-Lebowitz [3] and the later papers of Soshnikov, [18] and [19].
While going through cumulants, the method here is quite different.
We define the multiple integrals: for f a function of x 1 , . . . x k , (the indices are mod k), and, as another shorthand, if the f i are all functions of one variable, we setK Note that the cumulant (10) is just a weighted sum of terms of the formK ρ (g 1 , . . . , g m ), obtained by partitioning {1, 2, . . . , k} into m parts I 1 , . . . , I m of sizes k 1 , . . . k m and setting g i = j∈I i f j . Hence, we more generally seek conditions for the vanishing of Cum ρ (f 1 , . . . , f k ), defined in the obvious way, for k ≥ 3.
The first step is a collection of estimates on the integrals (12). The most fundamental of these are Lemmas 8 and 9 below. The former allows one to reduce the dimension in certain instances; the latter allows for the replacement of the test functions f i by their cubic approximations.

Lemma 4. Assumptions L 1 B and UB imply that
Proof. The integral is bounded above by Changing variables, y i = x i − x i−1 for i ≥ 2 allows the remaining integral to be bounded above by and the claim follows from assumptions (L 1 B, UB). Proof. Now use the bound where the integral on the right is over the product of the supports of the f 1 to f k . By Lemmas 4 and 5 were developed for the following purpose.
Similarly, for f j (z) =z p we havẽ Proof. We prove the first claim with f j = z p ; the proof of the second claim is identical. By the cyclic nature ofK ρ , we may assume i = 1. Fix a compact set B 2 such that and Lemma 5 we have the following (restrictions are placed only on functions with indices adjacent to j): Also, But, by the LLAP assumption, we have that which concludes the proof.
We close this subsection by showing that the assumed conditions enable one to Taylor expand inside theK ρ integrals.
Lemma 7. Assume that UB, L 1 B and ID hold. Let f j , 1 ≤ j ≤ k have bounded third derivatives. Then we havẽ where we use the standard tensor notation for the full first and second derivatives.
Proof. Starting at ℓ = 2, we will step-by-step replace f 1 (x 1 )f 2 (x 2 ) · · · f ℓ (x ℓ ), ℓ ≥ 2 by an approximation g ℓ of degree 2 at x 1 : Note that all the g ℓ are bounded on B ℓ . For the step-by-step replacement procedure we need to bound d ℓ = g ℓ−1 f ℓ − g ℓ . Towards this end, let Certainly, Let for the range of y i . As g ℓ is produced from g ℓ−1 f * ℓ by dropping all terms that are of degree 3 or 4 in the y i , there is a constant c such that on the range of the y i . (Any monomial in y i of degree 3 or 4 and coefficient 1 is bounded above on the compact range by the right hand side for c large enough).
Now we write the difference By UB and L 1 B (16) is bounded above by with (14) and (15) used in the second line. Again by L 1 B and ID, this in turn is upper bounded by

Proof of the proposition
The above bounds onK ρ made use of UB, L 1 B , ID, and LLAP. If we add CO to the mix, the result is the following. To prove this, remember that Cum ρ (f 1 , . . . , f k ) is a weighted sum of terms in the form K ρ (g 1 , . . . , g m ) each g j being a product of the underlying f 's (here m ≤ k). Lemma 7 gives in which s i = z i − z 1 , and we use complex coordinates s i ,s i . For example, Further, the k-fold integrals on the right hand side of (17) are all of the formK •k ρ (h(z 1 )σ i σ j ) where h(z 1 ) is a C 3 compactly supported (in B) function of z 1 and σ η = s η ,s η or 1. Since functions of at most three of the z i -s are present in this integrand, Lemma 6 with p = 0 allows us to reduce it to an at most threefold integral: where d = 0, 1 or 2 is the number of distinct variables in σ i σ j .
Next, two applications of Lemma 6 gives Similarly, all except the s isj terms vanish. Even among those, only half of the terms with i = j survive, depending on the order of conjugation. Again by successive applications of Lemma 6K Apart from the error, the latter equals −Cov ρ (h, zz).
Therefore, aside from a constant term, the only O(1) contributions to the cumulant sum are of the form (18). The possible choices of h are: with F = f 1 f 2 · · · f k , and our fullK ρ formula reduces toK ρ (·, zz) −K ρ (· × zz) applied to Reverting back to the original test functions f 1 , f 2 , . . . , f k , this is a weighted sum of the Since Cum ρ (f 1 , . . . , f k ) is symmetric under permutations of indices of the f i 's, it suffices to show that the total weight over the cumulant sum for each one of the two types of terms F u and F uv vanishes. Also, as each g i is a product of k i of the f i , then G i is a sum of k i terms of type F u and k i (k i − 1) terms of type F uv . Similarly, when i = j, G ij is a sum of k i k j terms of type F uv . Thus, the total number of terms of each type is given by while each type appears in the cumulant sum with a different coefficient.
Finally we invoke property CO: after subtracting ∂ z,z F/k from each term of type F u it can be replaced by k − 1 terms of type F uv with the opposite sign. Thus, our final count is That is to say, each m ≤ k term in the cumulant sum is the same constant multiple of (19).
That this vanishes for k ≥ 3 when summed over the full cumulant expansion is the content of the next lemma. ϕ(k, m, k 1 , . . . , k m ) for all k ≥ 3.
Proof. First realize, if we denote ϕ ≡ 1 by 1, then which explains why the k ≥ 3 cumulant of any constant is zero. Now set so that the coefficient of the y term in the y-power series expansion of log f is and similarly, In order to obtain the pure quadratic sums we set The coefficient of the y term in this power series expansion reads These series produce the types of terms we are after up to the fact that our cumulant expressions do not have the first coeffiicient k 1 . To omit this, the above may be modified as in and finally Now we have an easy proof of the claim. Simply note that the right hand side of (20) equals the latter being a straightforward computation.
That is, ϕ ρ (z) = ρ φ e −ρ|z| 2 /2 , and it is immediate that conditions UB, L 1 B, and ID are satisfied for all z, w ∈ C.
Rounding out the basic properties we have: Proof. Fix p > 0, and assume that ρ ≥ 1 + p. We consider a truncated kernel, which is a projection to the space of polynomials of degree at most ρ − 1 − p with respect to the same measure as K ρ , that is µ ρ . That is, we introduce with again t = (1 − |z| 2 )(1 − |w| 2 ). This truncated kernel is shown to have LLAP, and then the truncation is shown to make a negligible difference.

Asymptotic variance and general test functions
Our goal is to prove asymptotic normality with explicit variances for any f ∈ L 1 ∩ H 1 . We do this by proving normality and determining the variance asymptotics for an · H 1 -dense set of functions and then giving a uniform variance bound for all functions.
First note the general formula valid for all bounded f, g of compact support: which, after symmetrization, reads Lemma 11. The subset of smooth functions with compact support not containing ∞ is Note that this subset is not dense in H 1 (U), only among H 1 ∩ L 1 functions: harmonic functions h in U are H 1 -orthogonal to any compactly supported f . This can be seen via an integration by parts, moving the gradient from f to produce a △h = 0.
Proof. Replacing f by (f ∧ b) ∨ (−b) and letting b → ∞ shows that bounded functions are dense. Then convolving a bounded f with a smooth probability density approaching δ 0 shows that bounded C 3 functions are dense.
First consider the planar or hyperbolic case, and use the invariant gradient ∇ ι , measure ν and distance dist ι . We may apply a sequence of smooth cutoff functions g r to f which are equal to 1 on the ball of radius r but are compactly supported and have |∇ ι g r | ≤ 1. Let these converge to 0 for bounded f ∈ L 1 ∩ H 1 as r → ∞.
For the sphere, we again consider smooth f , and note that adding a constant to f does not change its H 1 -norm (formally, the space H 1 consists of equivalence classes of functions which differ by a constant). Adding a constant does not change the fact that f ∈ L 1 either, as the invariant measure ν S is finite. So we may assume that f (∞) = 0, and by smoothness and compactness f (z) ≤ c f dist ι (∞, z). We now take g ε (z) = ((dist ι (z, ∞)/ε − 1) ∨ 0) ∧ 1, which is supported on points at least ε away from ∞. Also, |∇ ι g(z)| ≤ 1/ε and vanishes for z more than 2ε away from ∞. As before, we have Both terms converge to 0 when ε → 0, as required.
Lemma 12 (Asymptotic variance for a dense set). Let f and g be C 1 and of compact support in Λ, where Λ = C for the plane or the sphere or U for the hyperbolic plane. Then Proof. It suffices to compute lim ρ→∞ Var ρ (f ) for f ∈ C 1 0 as the covariance may be identified by substituting f + g for f . First, by Taylor's theorem with remainder there is a bounded non-negative function ε(r) tending to 0 as r ↓ 0 for which

Now examine the remaining integrand
where θ(z, w) is the angle between f (z) and w − z, under the change of variables w = z + ρ −1/2 w ′ . Pointwise, in each of the three models, we have where ψ(z) = 1 (plane), = 1 + |z| 2 (sphere), = 1 − |z| 2 (hyperbolic plane). This would result in the limiting formula for the variance: with θ ′ denoting the limiting angle between z and w ′ , On the other hand, z and w ranging in a bounded set and ||∇f || L ∞ < ∞, the right hand side of is integrable on B × C (this again uses (21), (22), and (23)). Therefore, dominated convergence validates (30) and completes the proof.
Lemma (12) with implies For the last equality, note that ∇f and ∇∂ zz (gzz) have disjoint support.

Now
Cov where the second term vanishes, and the first one converges to 0 by Lemma 5 and the fact that the arguments have disjoint support.
There is a universal c > 0 so that for all ρ > 1 we have Proof. By considering the negative and positive parts of f separately, we may assume f ≥ 0.
In each of the three models by the invariant version of (28) we have, for f bounded and compact support Var where K ι (z, w) = K(z, w)(η(z)η(w)) −1/2 and η = dν(z)/dz is the density of the invariant measure (2).
Repeating the identities in (28) for the invariant K ι , and using that for ρ fixed K ι is bounded, we get that (32) extends to all f ∈ L 2 (ν), in particular for bounded f ∈ L 1 (ν).
Now replace the nonnegative f ∈ L 1 (ν) by f n = f ∧ n. Let V(f ) denote the right hand side of (32). Since |f n (z) − f n (w)| is monotone increasing in n, the monotone convergence theorem gives V(f n ) → V(f ). We also have Var ρ (f n ) → Var ρ (f ): the mean converges to a finite limit as f ∈ L 1 (ν) and the second moment converges by the monotone convergence theorem. Thus (32) holds for all f ≥ 0 in L 1 (ν), although a priori both sides may be infinite.
A simple way to check (33) is to write |K ι (z, w)| directly as a function of the single variable |T z (w)|, where T z is the isometry taking z to 0; such an expression is clearly invariant. It is also possible to get (33) from the invariance of the process and the covariance formula (28).
Proof of Theorem 1. Corollary (13) shows that condition CO holds, and the other conditions have been checked in Section 3. For f ∈ C 3 of compact support Proposition 2 gives asymptotic normality and Lemma 12 gives the limiting variance, so we have Lemma 14 allows us to extend the preliminary conclusion (35) to the advertised result.
For any f ∈ H 1 (Λ) (and of appropriate support) there is a sequence of f ε ∈ C 3 0 with ||f − f ε || H 1 → 0 as ε → 0. Moreover, Lemma 14 implies that the family {Z ρ (f )} is tight and also that The right hand side can be made small at will. Now, choosing a subsequence ρ ′ over which Z ρ (f ) has a limit in distribution, we find the Fourier transform of that limit is as close as we like to that of a mean zero Gaussian with variance 1 4π Λ |∇ ι f | 2 . (The full limit for Z ρ (f ε ) exists for any ε > 0). Since this appraisal is the same for any subsequence ρ ′ , we have pinned down the unique distributional limit of Z ρ (f ).