Rigidity for zero sets of Gaussian entire functions

In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call Admissible (as defined by Hayman). A notable example is the Gaussian Entire Function (GEF), whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set, a phenomenon discovered not long ago by Ghosh and Peres for the GEF. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is 'fully rigid'. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.


Introduction
Zero sets of random analytic functions, and especially Gaussian ones, have attracted the attention of researchers from various areas of mathematics in the last two decades. Given a sequence {a n } n≥0 of non-negative numbers, we consider the random Taylor series f (z) = n≥0 ξ n a n z n , where {ξ n } n≥0 is a sequence of independent and identically distributed standard complex Gaussians. A necessary and sufficient condition for f to be almost surely (a.s.) an entire function is lim n→∞ a 1/n n = 0, and in this case we call f a Gaussian entire function. We will only consider transcendental entire functions, that is, sequences a n which contain infinitely many non-zero terms. Denote by Z f = f −1 {0} the zero set of f ; its properties are determined by the covariance kernel a 2 n z n .
One model that was particularly well studied is the Gaussian Entire Function (GEF), given by the Taylor series with ξ n as before. It is well-known that its zero set is invariant under the isometries of the complex plane, and this fact has lead to quite an intensive study of its properties. Not long ago, Ghosh and Peres [GP17] established a rigidity phenomenon for the GEF. More precisely, if K ⊂ C is any compact set, then the number of zeros in K, and their first moment (i.e. sum), are uniquely determined if we are given the precise location of all zeros of F in C\K. For more details on the GEF see the book [Hou+09], and the ICM lecture notes [NS10].
In this note we explore the rigidity phenomenon for other Gaussian entire functions. In particular, we show that if G is admissible and has an infinite order of growth, then the zero set Z f is "fully rigid", that is, the restriction of Z f to any compact set K is determined by the restriction to the complement C \ K. Typical examples include the kernel functions G(z) = exp (exp (z)) , and G(z) = n≥0 z n log n (n + 2) .
When G is entire, admissible, and has a finite order of growth at least 1, there is always "partial rigidity", where finitely many moments of Z f restricted to K are determined by its restriction to C \ K. In the special case a 2 n = 1 (n!) α , α ∈ (0, 1) this was proved in [GK15]. In order to prove rigidity, we wish to study linear statistics of the zero set Z f , which, for a test function h with compact support, are given by the random variables for details see Section 2. Another approach to full rigidity can be found in [GL18].

Admissible kernel functions
Given an entire function G(z) = n≥0 a 2 n z n , we put Remark 1. Since the Taylor coefficients of G are non-negative, the function t → log G(e t ) is convex and therefore a (r) is non-decreasing.
Let us briefly explain the meaning of each condition. The first condition implies that the function G grows sufficiently fast as r → ∞. More precisely, Actually, we will require a condition which implies a much more rapid growth (see Remark 2). The second condition implies that near θ = 0, the function θ → log G re iθ can be wellapproximated by a Taylor polynomial of degree 2 (in θ), and away from θ = 0, the function is negligibly small compared to G (r). The class of admissible functions has many nice closure properties, for example if f is admissible, then so is e f , for details and examples, see [Hay56].

Rigidity of the zero set
Ghosh and Peres [GP17] introduced the following property for point processes, in the context of zero sets of Gaussian entire functions.
Definition 1. A random point processes Z taking values in C, is said to be rigid of level n ∈ N + ∪ {+∞}, if for any bounded open set D ⊂ C and any integer 0 ≤ k < n, there exists a map S k from the set of all locally finite point configurations in C \ D to the complex plane, such that In addition, if the point process is rigid of level +∞ we call it fully rigid.
For a fully rigid process, given the locations of all the points outside a given compact set K, we can recover the precise location of the points inside K, using our knowledge of all moments. More precisely, if the number of points in D is N = S 0 , then the points are the elementary symmetric polynomials of the roots (with e 0 ≡ 1). It is well known that the polynomials {e ℓ } can be computed in terms of the (power) sums {S k } (see [Sta99,Chap. 7]).
In Section 3 we will construct explicitly Gaussian entire functions whose zero sets are fully rigid point processes, using the following result.
Theorem 1. Let f be a Gaussian entire function with covariance kernel G(zw). If G is admissible and satisfies then g is fully rigid.
Remark 2. The condition (3) implies that G is of lower order B, that is Remark 3. For the specific choice a n = (n!) − 1 2B with B an integer, which in particular implies (3), Ghosh and Krishnapur [GK15] showed that the zero set process is rigid of level ⌈B⌉ + 1. However, assuming only (3), it seems likely that the best one can get is rigidity of level ⌈B⌉.
Remark 4. By inspecting the proof of Theorem 1 below, one can check that the expression b(r) in condition (2) can be replaced with b(r) 1/4 .

Acknowledgment
We thank Misha Sodin for encouraging us to work on this topic, very helpful discussions, and suggestions regarding the presentation of the result.

Proof of Theorem 1
The proof is based on the following theorem, which is a special case of [GP17, Theorem 6.1].
Theorem (Rigidity condition). Let n ∈ N + . If for any bounded open set D ⊂ C, any ε > 0, and every 0 ≤ k < n there exists a smooth test function Φ ε D,k such that Φ ε D,k = z k on D and V ar C Φ ε D,k dZ f < ε, then Z f is rigid of level n.
Let 0 ≤ k < B be an integer, fix η > 0 (depending on ⌈B⌉), and ϕ η : [0, ∞) → [0, 1] be a twice differentiable function which is equal to 1 on [0, 1], equal to 0 on e 1 η , ∞ , and such that Where here, and in the rest of the proof, c and C will denote positive numerical constants, independent of all the other parameters, which may differ between appearances. In addition, constants depending only on B are denoted by , and for L ≥ 1 write ϕ η,L (z) = ϕ η (z/L) and Φ η,L (z) = z k ϕ η,L (z). According to [Hou+09, equation 3.5.2], where m is the Lebesgue measure in the complex plane. Furthermore, by [Hou+09, Lemma 3.5.2], is the normalized covariance kernel. Since the Taylor coefficients of G are non-negative, |G(z)| ≤ G(|z|), and in particular |J (z, w)| ≤ 1. Thus Making the change of variables z = zL and w = wL, we find that the integral above is bounded by where χ(r) = ηr k−2 1 [1,e 1/η ] (r). Here we used (4) and the fact Φ η,L is harmonic in {|z| < L}.
Using polar coordinates z = re iθ 1 and w = se iθ 2 , we get rs dr ds.

Preliminary claims
Before bounding the integral I L , we need two simple claims.
Claim 1. We have for R sufficiently large.
By Remark 1 the function t → log G(e t ) is convex for t ∈ R, its derivative is a (e t ) and its second derivative is b (e t ). Thus, which immediately yield the claim.

Bounding the integral I L
By Claim 1 we have that The above integral is symmetric with respect to the variables r and s, and therefore, if we put M L := min where we used Claim 2. By the definition of χ, we get Fix k < B 0 < B. By the assumption in the statement of the theorem, there exist a constant c > 0 such that b (x) ≥ cx B 0 for any x ≥ 1. Therefore, Making the change of variables s = xr, we obtain To bound I 1 L , we use the elementary inequality log x > 1 3 (x − 1), for 1 ≤ x ≤ 2e, and for L ≥ 1 we get Next we turn to the integral I 2 L . Since x > 2e, for L ≥ 1 we get Thus we obtain We have established that, for L ≥ 1. Choosing η sufficiently small, so that C 6 B η 2 < ε and appealing to the rigidity condition of Ghosh and Peres, we find that Z f is rigid of level ⌈B⌉. This completes the proof of Theorem 1.

Examples
Here are some explicit examples for Theorems 1.

The Mittag-Leffler function
We consider the function f whose associated kernel function G is the Mittag-Leffler function where α ∈ (0, ∞) is a fixed parameter. Notice that G 1 (z) = e z and G 1 2 (z) = cosh √ z.
Thus, by Theorem 1, Z f is rigid of level ⌈α⌉. However, for α an integer, using a similar proof to [GK15] and the above asymptotic formula one can check that Z f is rigid of level α + 1.

The double exponent and the Lindelöf functions
Here we consider the function f associated with kernel function G(z) = e e z . Since e z is admissible, so is G. In addition, G has an infinite order of growth, with a (r) = re r , b (r) = r (r + 1) e r .
By Theorem 1, Z f is fully rigid. For α > 0, we consider the function f whose associated kernel function G is given by z n log αn (n + e) .
Again by Theorem 1, Z f is fully rigid.