Random zero sets for Fock type spaces

Abstract

Given a nondecreasing sequence $\Lambda =\{\lambda _n>0\}$ such that $\displaystyle \lim _{n\rightarrow \infty } \lambda _n=\infty ,$ we consider the sequence $\mathcal {N}_\Lambda :=\left\{ \lambda _ne^{i\theta _n},n\in \,\mathbb {N}\right\}$, where $\theta _n$ are independent random variables uniformly distributed on $[0,2\pi ].$ We discuss the conditions on the sequence $\Lambda$ under which $\mathcal N_\Lambda$ is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence $\Lambda$ with respect to the weight is found.

Appendix

Here we recall some notions and results which we have used in this note.

1.1 Bernstein-Hoeffding concentration inequality [14, Theorem 2.2.6]

Let $X_{1},X_{2},\dots ,X_{N}$ be independent random variables such that ${\displaystyle a_{i}\le X_{i}\le b_{i}}$ for every i. Then, for any $t>0$ we have

$$\begin{aligned} \displaystyle {\mathbb {P}} \left( \sum _{n=1}^N \left( X_n-{\mathbb {E}}(X_{n})\right) \ge t\right) \le \exp \left( -{\frac{2t^{2}}{\sum _{i=1}^{N}(b_{i}-a_i)^{2}}}\right) . \end{aligned}$$

1.2 Khinchine-Kolmogorov theorem on convergence of random series [1, Theorem 22.6]

Let $(Y_k)_{k\in {\mathbb {N}}}$ be a sequence of independent real-valued random variables with zero expectation. If

$$\begin{aligned} \displaystyle \sum {\textrm{Var}}\left( Y_k\right) <\infty , \end{aligned}$$

then the series $\displaystyle \sum Y_k$ converges a.s.

1.3 Kolmogorov maximal inequality [1, Theorem 22.4]

Let $Y_1, Y_2,\ldots , Y_N$ be a sequence of independent real-valued random variables with zero expectation and finite variances. For $\gamma >0$

$$\begin{aligned} {\mathbb {P}}\left( \max \limits _{1\le k\le N} \left| \sum _{m=1}^{k} Y_m\right| \ge \gamma \right) \le \frac{1}{\gamma ^2}\sum _{m=1}^N{\textrm{Var}}(Y_m). \end{aligned}$$

1.4 Weyl-type criterion of uniform distribution of random sequence [6, Theorem 2]

Let $X_{1},X_{2},\ldots$ be a sequence of independent real-valued random variables with characteristic functions $\phi _1,\phi _2,\ldots$. Then this sequence is uniformly distrubuted modulo 1 almost surely, that is

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{\#\{k: X_k-\lfloor X_k\rfloor <x\}}{N}=x,\;\;\;\forall x\in [0,1) \end{aligned}$$

with probability one, if and only if for every $k\in {\mathbb {N}}$

$$\begin{aligned} \lim _{N\rightarrow \infty }\frac{1}{N}{\sum _{n=1}^N\phi _n(2\pi k )}=0. \end{aligned}$$

1.5 Definition of a set of disks with zero linear density [11, Chapter II, §1]

A set D of disks $D_j$ in the complex plane is said to have zero linear density, if

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{1}{r}\sum _{|c_j|<r} \lambda _j=0, \end{aligned}$$

where $\lambda _j$ is the radius and $c_j$ is the center of $D_j$.

1.6 Levin-Pfluger theorem I [11, Chapter II, §1, Theorem 1]

Let a discrete set ${\mathcal {N}}\subset {\mathbb {C}}$ have an angular density of index $\rho (r)$, where $\rho (r)$ is a proximate order with $\rho =\lim _{r\rightarrow \infty }\rho (r)\notin {\mathbb {N}}$. Let $\Delta$ be a nondecreasing function such that for all but a countable set of angles

$$\begin{aligned} \Delta (\beta )-\Delta (\alpha )=\lim _{t\rightarrow \infty }\frac{n(t,\alpha ,\beta )}{t^{\rho (t)}}. \end{aligned}$$

Then for $z\in {\mathbb {C}}\setminus E$, where E is a set of disks with zero linear density, the canonical product

$$\begin{aligned} W(z)=\prod _k G\left( \frac{z}{z_k};\lfloor \rho \rfloor \right) \end{aligned}$$

satisfies the asymptotic relation

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log |W(re^{i\theta })|}{r^{\rho (r)}}=\frac{\pi }{\sin (\pi \rho )}\int _{\theta -2\pi }^{\theta }\cos \left( \rho (\theta -\psi -\pi )\right) {\textrm{d}}\Delta (\psi ). \end{aligned}$$

1.7 Levin-Pfluger theorem II [11, Chapter II, §1, Theorem 2]

Let a set ${\mathcal {N}}\subset {\mathbb {C}}$ have an angular density of index $\mathcal \rho (r)$, where $\rho (r)$ is a proximate order with $\rho =\lim _{r\rightarrow \infty }\rho (r)\in {\mathbb {N}}$. Let $\Delta$ be a nondecreasing function such that for all but a countable set of angles

$$\begin{aligned} \Delta (\beta )-\Delta (\alpha )=\lim _{t\rightarrow \infty }\frac{n(t,\alpha ,\beta )}{t^{\rho (t)}}. \end{aligned}$$

Let the following limits exist and be finite

$$\begin{aligned} S:= & {} \sum z_n^{-\rho },\\ \delta:= & {} \lim _{r\rightarrow \infty }r^{\rho -\rho (r)}\sum _{|z_n|> r} \frac{1}{ z_n^{\rho }}. \end{aligned}$$

Then for $z\in {\mathbb {C}}\setminus E$, where E is a set of disks with zero linear density, the entire function

$$\begin{aligned} W(z)=e^{S\cdot z^{\rho }}\prod _k G\left( \frac{z}{z_k};\rho \right) \end{aligned}$$

satisfies the following asymptotic relation

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{\log |W(re^{i\theta })|}{r^{\rho (r)}}= & {} -\int _{\theta -2\pi }^{\theta }(\psi -\theta )\sin \left( \rho (\psi -\theta )\right) {\textrm{d}}\Delta (\psi ) \\{} & {} +\frac{|\delta |}{\rho } \cos \left( \rho (\theta -\arg \delta )\right) . \end{aligned}$$