# Random Carleson Sequences for the Hardy space on the Polydisc and the Unit
Ball

Nikolaos Chalmoukis  0000-0001-5210-8206 Dipartimento di Matematica e
Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 55,
20125, Milano, Italy<EMAIL_ADDRESS>,  Alberto Dayan
0000-0002-7346-4354 Fachrichtung Mathematik, Universität des Saarlandes, 66123
Saarbrücken, Germany<EMAIL_ADDRESS>and  Giuseppe Lamberti
0009-0009-0503-0421 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400
Talence, France<EMAIL_ADDRESS>
###### Abstract.

We study the Kolmogorov $0-1$ law for a random sequence with prescribed radii
so that it generates a Carleson measure almost surely, both for the Hardy
space on the polydisc and the Hardy space on the unit ball, thus providing
improved versions of previous results of the first two authors and of a
separate result of Massaneda. In the polydisc, the geometry of such sequences
is not well understood, so we proceed by studying the random Gramians
generated by random sequences, using tools from the theory of random matrices.
Another result we prove, and that is of its own relevance, is the $0-1$ law
for a random sequence to be partitioned into $M$ separated sequences with
respect to the pseudo-hyperbolic distance, which is used also to describe the
random sequences that are interpolating for the Bloch space on the unit disc
almost surely.

###### Key words and phrases:

Carleson sequences, Hardy spaces, Polydisc, Unit ball, random sequences

###### 2020 Mathematics Subject Classification:

Primary: 46E22; Secondary: 30E05; 47B32; 66G25

N. Chalmoukis is a member of Indam–Gnampa and partially supported by the
Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call
for H.F.R.I. Research Projects to support Faculty Members & Researchers”
(Project Number: 4662) and the Indam–Gnampa project “Function theory in
several complex and quaternionic variables” CUP_E53C22001930001.

A. Dayan is partially supported by the Emmy Noether Program of the German
Research Foundation (DFG Grant 466012782)

##  1\. Introduction

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of analytic functions
on a domain $X$ of $\mathbb{C}^{d}$, $d\in\mathbb{N}$. A positive regular
measure $\mu$ on $X$ is a Carleson measure for $\mathcal{H}$ if $\mathcal{H}$
embeds continuously inside $L^{2}(X,\mu)$, namely if there exists a constant
$C_{\mu}$ such that

$\|f\|_{L^{2}(X,\mu)}\leq C_{\mu}\leavevmode\nobreak\
\|f\|_{\mathcal{H}},\qquad\forall f\in\mathcal{H}.$ (CM)

Carleson measures have been studied in various settings, as they have
important applications in harmonic analysis. In this note, we will consider
measures generated by sequences, which are intimately connected to the study
of interpolating sequences (see for example [1, 2, 9, 4]). Let $k$ be the
reproducing kernel of $\mathcal{H}$, and let $Z=(z_{n})_{n}$ be a sequence in
$X$. Define the measure $\mu_{Z}$ as

$\mu_{Z}:=\sum_{n\in\mathbb{N}}\|k_{z_{n}}\|^{-2}\leavevmode\nobreak\
\delta_{z_{n}}.$

Thanks to the reproducing property of the kernels $(k_{z_{n}})_{n}$, condition
(CM) then becomes

$\sum_{n\in\mathbb{N}}|\langle f,\hat{k}_{n}\rangle|^{2}\leq
C_{\mu}\leavevmode\nobreak\ \|f\|^{2}_{\mathcal{H}},\qquad\forall
f\in\mathcal{H},$

where $\hat{k}_{n}:=k_{z_{n}}/\|k_{z_{n}}\|$ is the normalized kernel at the
point $z_{n}$. Therefore, (CM) is equivalent to the boundedness of the frame
operator $T\colon\mathcal{H}\to\mathcal{H}$, defined formally as

$T(f):=\sum_{n\in\mathbb{N}}\langle f,\hat{k}_{n}\rangle\leavevmode\nobreak\
\hat{k}_{n},$

which, in turn, is equivalent to the Gram matrix

$G:=(\langle\hat{k}_{n},\hat{k}_{j}\rangle)_{n,j\in\mathbb{N}}$ (1)

inducing a bounded operator $G\colon\ell^{2}\to\ell^{2}$ (see [1, Chapter 9]).
We will refer to sequences that generate a Carleson measure for $\mathcal{H}$
as Carleson sequences. In the literature these sequences are sometimes called
Carleson-Newman sequences.

Remarkably, apart from such an operator theoretical characterization, some
well known spaces of analytic functions enjoy also a geometric
characterization for such measures. For instance, let

$X=\mathbb{D}^{d}:=\\{z=(z^{1},\dots,z^{d})\in\mathbb{C}^{d}\,|\,|z^{i}|<1\\}$

denote the $d$-dimensional polydisc, and let $\mathcal{H}=H^{2}_{d}$ be the
Hardy space on $\mathbb{D}^{d}$, that is, the reproducing kernel Hilbert space
on $\mathbb{D}^{d}$ with kernel

$s(z,w):=\prod_{i=1}^{d}\frac{1}{1-\overline{w^{i}}z^{i}},\qquad
z,w\in\mathbb{D}^{d}.$

When $d=1$, we use the standard notation $H^{2}$, rather than $H^{2}_{1}$.
Carleson showed in [7] that if $d=1$ a measure $\mu$ satisfies the embedding
condition (CM) for the Hardy space if and only if it satisfies the one-box
condition, namely if there exists $C_{\mu}>0$ such that

$\mu(S_{I})\leq C_{\mu}|I|$ (OB)

for all arcs $I\subseteq\mathbb{T}$, where $|I|$ is the arc-length measure and
$S_{I}$ is the Carleson square in $\mathbb{D}$ with basis $I$:

$S_{I}:=\\{z\in\mathbb{D}\setminus\\{0\\}\,|\,z/|z|\in I,1-|z|\leq|I|\\}.$

On the other hand, the geometry of Carleson sequences for $H^{2}_{d},d\geq 2$
seems to be much more complicated. In general no necessary and sufficient
condition is known for a measure to be Carleson in the polydisc. Regarding
this very interesting problem we refer the reader to the work of Chang [11]
and Carleson’s counterexample [8]. Consequently, in order to understand better
such sequences, we consider _random sequences with prescribed radii_ in the
polydisc, and we study the probability of such sequences to generate a
Carleson measure for $H^{2}_{d}$.

A random sequence with prescribed radii in $\mathbb{D}^{d}$ can be defined as
follows. Given a sequence of deterministic radii $(r_{n})_{n\in\mathbb{N}}$ in
$[0,1)^{d}$ and a sequence of i.i.d. random variables
$(\theta_{n})_{n\in\mathbb{N}}$ defined on the same probability space
$(\Omega,\mathcal{A},\mathbb{P})$ and uniformly distributed in the
$d$-dimensional torus $\mathbb{T}^{d}$, let
$\Lambda=(\lambda_{n})_{n\in\mathbb{N}}$ be the random sequence defined by

$\lambda_{n}(\omega):=(r_{n}^{1}e^{2\pi
i\theta^{1}_{n}(\omega)},\dots,r_{n}^{d}e^{2\pi
i\theta^{d}_{n}(\omega)}),\qquad n\in\mathbb{N},\omega\in\Omega.$

In order to state our results, it is convenient to introduce a certain
counting function. To do so we partition $\mathbb{D}^{d}$ in the dyadic
rectangular regions

$A_{m}:=\\{z\in\mathbb{D}^{d}\,|\,2^{-(m_{i}+1)}\leq
1-|z^{i}|<2^{-m_{i}},\,\,i=1,\dots,d\\},\qquad
m=(m_{1},\dots,m_{d})\in\mathbb{N}^{d},$

and denote by

$N_{m}:=\\#\Lambda\cap A_{m},\qquad m\in\mathbb{N}^{d}$

the (deterministic) number of points of $\Lambda$ in each $A_{m}$. Define
$|m|:=m_{1}+\dots+m_{d}$ for all multi-indices in $\mathbb{N}^{d}$. Observe
that being a Carleson sequence is a tail event since it is independent of any
finite number of random variables. Therefore, by Kolmogorov’s 0-1 Theorem [5,
Theorem 4.5], it has probability either $0$ or $1$. We would like to find
neccesary and sufficient conditions on the sequence $(r_{n})_{n}$ such that
the corresponding random sequence is a Carleson sequence almost surely. In
[19] Rudowicz proved that for $d=1$ a sequence is almost surely Carleson for
the Hardy space if

$\sum_{m\in\mathbb{N}}2^{-m}N_{m}^{2}<+\infty.$ (2)

Recently, in [10, Theorem 1.1], with the use of the one box characterization
of Carleson measres (OB), the authors improved Rudowicz’s sufficient
condition, showing that in order for a random sequence to be Carleson almost
surely it is sufficient that for some $\varepsilon>0$

$N_{m}=\mathcal{O}(2^{(1-\varepsilon)m}).$

Moreover, in [13, Theorem 1.3], a Rudowicz type sufficient condition has been
obtained for the Hardy space in the polydisc. Our first main result is the
sharp version of the $0-1$ law for random Carleson sequences in $H^{2}_{d}$,
for all $d\geq 1$.

######  Theorem 1.1.

Let $d$ be a positive integer and $\Lambda$ be a random sequence in
$\mathbb{D}^{d}$. Then,

$\mathbb{P}\left(\Lambda\,\,\text{is a Carleson sequence for
}H^{2}_{d}\right)=\begin{dcases}1&\text{if}\,\,N_{m}=\mathcal{O}(2^{(1-\varepsilon)|m|})\,\,\text{for
some }\varepsilon>0,\\\ 0&\text{otherwise}.\end{dcases}$ (3)

Notice that for $d\geq 2$ the one-box condition (OB) is unavailable, hence the
techniques used in [10] do not provide insights to the proof of Theorem 1.1.
We look instead at the _random Gram matrix_ generated by the kernel vectors
associated to the random sequence $\Lambda$ as in (1), bypassing the
difficulty of not having a geometric characterization of Carleson measures for
the Hardy space in the polydisc.

Another ingredient for the proof of Theorem 1.1 is the $0-1$ law of random
sequences that can be partitioned into finitely many separated sequences with
respect to the pseudo-hyperbolic distance in $\mathbb{D}^{d}$. More
specifically, let

$\rho(z,w):=\max_{i=1,\dots,d}\left|\frac{z^{i}-w^{i}}{1-\overline{w^{i}}z^{i}}\right|,\qquad
z,w\in\mathbb{D}^{d}\,$

denote the pseudo-hyperbolic distance in the polydisc. We say that a sequence
$Z=(z_{n})_{n}$ in $\mathbb{D}^{d}$ is separated if

$\inf_{n\neq j}\rho(z_{n},z_{j})>0.$ (S)

Random separated sequences have been first studied by Cochran [12] who proved
that if $d=1$ then (2) is a necessary and sufficient condition for a random
sequence to be separated almost surely. In [13], the authors extended
Cochran’s result to the polydisc. Our next theorem extends this results to
finite unions of separated sequences.

######  Theorem 1.2.

Let $\Lambda$ be a random sequence in the polydisc. Then for all
$M\in\mathbb{N}$,

$\mathbb{P}(\Lambda\,\text{is the union of $M$ separated
sequences})=\begin{cases}1\quad\text{if}\,\,&\underset{m\in\mathbb{N}^{d}}{\sum}N_{m}^{1+M}2^{-M|m|}<\infty,\\\
0\quad\text{if}\,\,&\underset{m\in\mathbb{N}^{d}}{\sum}N_{m}^{1+M}2^{-M|m|}=\infty.\end{cases}$
(4)

It turns out that $\rho(z,w)$ coincides with the largest absolute value that
an analytic function on $\mathbb{D}^{d}$ bounded by $1$ that vanishes at $w$
can attain at $z$. The solution of the same extremal problem in $H^{2}_{d}$
gives raise to the distance

$\rho_{s}(z,w):=\sqrt{1-\frac{|\langle{s}_{z},{s}_{w}\rangle|^{2}}{\|s_{z}\|^{2}\|s_{w}\|^{2}}}\qquad
z,w\in\mathbb{D}^{d}.$

Since $\rho$ and $\rho_{s}$ are comparable, one can replace $\rho$ with
$\rho_{s}$ in (S), and this describes the same sequences. Since any Carleson
sequence for $H^{2}_{d}$ is the finite union of separated sequences with
respect $\rho_{s}$, [1, Proposition 9.11], the second half of Theorem 1.1 is
therefore deduced from the second half of Theorem 1.2.

The last part of this article is devote to the study of random Carleson
sequences for the Hardy space on the unit ball. Let
$\mathbb{B}_{d}:=\\{z\in\mathbb{C}^{d}\,|\,\sum_{i}|z^{i}|^{2}<1\\}$ denote
the $d$-dimensional unit ball on $\mathbb{C}^{d}$. The definition of a random
sequence $\Lambda=(\lambda_{n})_{n}$ in the unit ball is reminiscent of the
analogous construction on the unit disc: given a deterministic sequence of
radii $(r_{n})_{n}$ in $(0,1)$ and a sequence $(\xi_{n})_{n}$ of i.i.d. random
variables defined on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and
uniformly distributed on the unit sphere $\partial\mathbb{B}_{d}$, one defines

$\lambda_{n}(\omega):=r_{n}\xi_{n}(\omega),\qquad
n\in\mathbb{N},\omega\in\Omega.$

For all $m$ in $\mathbb{N}$, let

$N_{m}:=\\#\Lambda\cap\\{2^{-(m+1)}\leq 1-|z|<2^{-m}\\}\subset\mathbb{B}_{d}.$

The question of whether $\Lambda$ generates almost surely a Carleson measure
for some significant spaces of analytic functions on the unit ball has been
investigated in [13] and [16]. For all $0\leq a<d$, denote by $B^{a}_{d}$ the
reproducing kernel Hilbert space on $\mathbb{B}_{d}$ having kernel

$k^{(a)}_{w}(z):=\frac{1}{(1-\langle
z,w\rangle_{\mathbb{C}^{d}})^{d-a}},\qquad z,w\in\mathbb{B}_{d}.$

For $a=0$, $B^{a}_{d}$ is the Hardy space on the unit ball, while for $0<a<d$
one obtains a range of Besov-Sobolev spaces, including the Drury-Arveson space
($a=d-1$). For more information about Besov-Sobolev spaces see [22] and [14]
for the Drury Arveson space.

Regarding random Carleson sequences for $B^{a}_{d}$, for all $0<a<d$ the same
phenomenon observed in [10, Theorem 1.4] for the unit disc occurs on in the
unit ball: $\Lambda$ generates a Carleson measure for $B^{a}_{d}$ almost
surely if and only if it is a finite measure (see [13, Theorem 4.3]). We show
that this is not the case for $a=0$, as the $0-1$ law for Carleson sequences
for the Hardy space on the unit ball resembles the one for the unit disc and
the polydisc, refining a theorem of Massaneda [16, Theorem 3.2].

######  Theorem 1.3.

Let $\Lambda$ be a random sequence in the unit ball. Then

$\mathbb{P}(\Lambda\,\text{is a Carleson sequence for
}\,B^{0}_{d})=\begin{cases}1\,\,&\text{if}\,\,N_{m}=\mathcal{O}(2^{d(1-\varepsilon)m})\,\text{for
some}\,\,\varepsilon>0,\\\ 0\,\,&\text{otherwise}.\end{cases}$

The paper is structured as follows. Section 2 contains the proof of Theorem
1.2. An analogous result for the unit ball have been obtained by Massaneda in
[16, Theorem 3.4]. The probabilistic tools that we use are contained in
Section 2.1, and they differ from the ones used in Massaneda’s work.

As a by-product of our study of random sequences that can be written as the
union of finitely many separated sequences, we find in Section 3 the $0-1$ law
for a random sequence in the unit disc to be interpolating for the Bloch
space.

Section 4 contains the proof of the first half of Theorem 1.1. The main tool
used, Theorem 4.3, comes from the theory of random matrices, and it allows us
to estimate the probability that some diagonal blocks of the random Gramian
$G$ are big in norm. Section 4.3 contains some additional remarks on random
Carleson sequences for Dirichlet-type kernels in the polydisc. Finally, in
Section 5 we prove Theorem 1.3.

###  1.1. Notation

If $f$ and $g$ are positive expression, we will write $f\lesssim g$ if there
exists $C>0$ such that $f\leq Cg$, where $C$ does not depend on the parameters
behind $f$ and $g$, or $\lesssim_{a}$ if the implicit constant $C$ depends on
$a$. We will simply write $f\simeq g$ if $f\lesssim g$ and $g\lesssim f$.
Finally when $f$ and $g$ are expressions for which we can consider the limit
of their quotient, with $f\sim g$ we mean that $\lim f/g=1$, while
$f\sim_{a}g$ means that $\lim f/g$ is equal to a constant that depends on $a$.

##  2\. Union of finitely many separated sequences

###  2.1. A Probabilistic Tool

Let $N$ and $n$ be two positive integers, and consider the problem of placing
$n$ points at random into $N$ boxes, where the boxes are chosen independently
for each point, and for all points each box has the same probability of being
chosen. We are interested in the random variable $\mu_{r}(n,N)$ that counts
the number of boxes in which there are exactly $r$ points. We want to
estimate, for $n,N\to\infty$, the number $\mathbb{P}(\mu_{r}(n,N)=1)$. The
tools that we are going to use come from [15]. Define

$\displaystyle\alpha$ $\displaystyle:=\frac{n}{N}$ $\displaystyle p_{r}$
$\displaystyle:=\frac{\alpha^{r}e^{-\alpha}}{r!}$
$\displaystyle\sigma_{r}^{2}$
$\displaystyle:=\frac{\alpha}{1-p_{r}}\left(1-p_{r}-\frac{(\alpha-r)^{2}}{\alpha}p_{r}\right)$
$\displaystyle\alpha_{r}$ $\displaystyle:=\frac{\alpha-rp_{r}}{1-p_{r}}.$

Consider i.i.d. random variables $\eta_{1},\dots,\eta_{N}$ having Poisson
distribution with parameter $\alpha$ and let
$\zeta_{N}=\eta_{1}+\dots+\eta_{N}$. Similarly, define $\eta_{i}^{(r)}$,
$i=1,\dots,N$ i.i.d. random variables with distribution

$\mathbb{P}(\eta_{i}^{(r)}=l)=\mathbb{P}(\eta_{i}=l|\eta_{i}\neq r)$

and by $\zeta_{N}^{(r)}$ the sum of the $\eta_{i}^{(r)}$.

######  Lemma 2.1 ([15, Lemma 1, p.60]).

$\mathbb{P}(\mu_{r}(n,N)=k)=\binom{N}{k}p_{r}^{k}(1-p_{r})^{N-k}\frac{\mathbb{P}(\zeta_{N-k}^{(r)}=n-kr)}{\mathbb{P}(\zeta_{N}=n)}.$

######  Theorem 2.2 ([15, Theorem 1, p.61]).

If $m\to\infty$ and $\alpha m\to\infty$, then for fixed $r\geq 2$,

$\mathbb{P}(\zeta_{m}^{(r)}=l)=\frac{1}{\sigma_{r}\sqrt{2\pi
m}}e^{-\frac{(l-m\alpha_{r})^{2}}{2m\sigma_{r}^{2}}}(1+o(1)),$

uniformly with respect to $\frac{l-m\alpha_{r}}{\sigma_{r}\sqrt{m}}$ in any
finite interval.

The precise statement of the theorem is the following. Fixed $r\geq 2$ and
$M>0$ and consider the domain of parameters

$\mathcal{D}(M,r):=\\{(m,\alpha,l)\in\mathbb{N}\times(0,\infty)\times\mathbb{N}:\bigg{|}\frac{l-m\alpha_{r}}{\sigma_{r}\sqrt{m}}\bigg{|}\leq
M\\}.$

Then, for every $\varepsilon>0$, there exists $C_{0}=C(\varepsilon,M,r)>0$
such that

$\bigg{|}\mathbb{P}(\zeta_{m}^{(r)}=l)\sigma_{r}\sqrt{2\pi
m}\,e^{\frac{(l-m\alpha_{r})^{2}}{2m\sigma_{r}^{2}}}-1\bigg{|}<\varepsilon,$

for all $(m,\alpha,l)\in\mathcal{D}(M,r)$ such that $m,\alpha m>C_{0}$.
As a Corollary, we prove the following variation of [15, Theorem 3, p.67].

######  Corollary 2.3.

Suppose that $\alpha\to 0$ and $Np_{r}\to 0$ for $n,N\to\infty$. Then for
fixed $r$,

$\lim_{n,N\to\infty}\frac{\mathbb{P}(\mu_{r}(n,N)=1)}{Np_{r}}=1.$

###### Proof.

From Lemma 2.1 we have that

$\mathbb{P}(\mu_{r}(n,N)=1)=Np_{r}(1-p_{r})^{N-1}\frac{\mathbb{P}(\zeta_{N-1}^{(r)}=n-r)}{\mathbb{P}(\zeta_{N}=n)}.$
(5)

We want to use Theorem 2.2 with $m=N-1$ and $l=n-r$ to estimate
$\mathbb{P}(\zeta_{N-1}^{(r)}=n-r)$. The only hypothesis we have to check is
if $\frac{l-m\alpha_{r}}{\sigma_{r}\sqrt{m}}$ remains bounded with our
choices, while the other hypothesis easily hold. Notice that since $\alpha\to
0$ we can assume without loss of generality that $\alpha<1$. We have

$\frac{l-m\alpha_{r}}{\sigma_{r}\sqrt{m}}=\frac{n-r-(N-1)\alpha_{r}}{\sigma_{r}\sqrt{(N-1)}}.$
(6)

Since $p_{r}\sim_{r}\alpha^{r}$ we have that $\sigma_{r}^{2}\sim_{r}\alpha$
and so $\sigma_{r}\sqrt{(N-1)}\sim_{r}\sqrt{n}$. Furthermore

$n-r-(N-1)\alpha_{r}=n-r-(N-1)\frac{\alpha-
rp_{r}}{1-p_{r}}=\frac{rNp_{r}-np_{r}-r+\alpha}{1-p_{r}}$

which remains bounded since $Np_{r}\to 0$ and $r$ is fixed. So we have
obtained that (6) goes to $0$ for $n,N\to\infty$ and so it remains bounded. We
can finally apply Theorem 2.2:

$\mathbb{P}(\zeta_{N-1}^{(r)}=n-r)=\frac{1}{\sigma_{r}\sqrt{2\pi(N-1)}}e^{-\frac{(n-r-(N-1)\alpha_{r})^{2}}{2(N-1)\sigma_{r}^{2}}}(1+o(1))\sim_{r}\frac{1}{\sqrt{2\pi
n}}.$

Since $\zeta_{N}$ follows a Poisson distribution, using Stirling formula we
have

$\mathbb{P}(\zeta_{N}=n)=\frac{n^{n}}{n!}e^{-n}\sim\frac{1}{\sqrt{2\pi n}}.$

Finally, we have that
$(1-p_{r})^{N-1}=[(1-p_{r})^{\frac{1}{p_{r}}}]^{p_{r}(N-1)}\sim_{r}1$. The
result now follows from (5). ∎

###  2.2.  Random Separated Sequences in the Polydisc

We shall now show how Corollary 2.3 can be applied to prove Theorem 1.2:

###### Proof of Theorem 1.2.

For all $m$ in $\mathbb{N}^{d}$, divide the $d$-dimensional torus
$\mathbb{T}^{d}$ into $2^{|m|}$ dyadic rectangles
$\\{R^{m}_{j}\,|\,j_{i}=1,\dots,2^{m_{i}},i=1,\dots,d\\}$ of side-lenghts
$2^{-m_{1}},\dots,2^{-m_{d}}$, i.e.

$R^{m}_{j}:=\\{x=(x_{1},\dots,x_{d})\in[0,1)^{d}\,|\,(j_{i}-1)/2^{m_{i}}\leq
x_{i}<j_{i}/2^{m_{i}}\\},\quad j_{i}=1,\dots,2^{m_{i}}.$

Let’s also label $\\{\lambda_{m,1},\dots,\lambda_{m,N_{m}}\\}$ the random
points in $A_{m}$. For any $I\subseteq\\{1,\dots,N_{m}\\}$, let $\Omega(m,I)$
be the event that the arguments of the points $\\{\lambda_{{m,i}}\,|\,i\in
I\\}$ are in the same dyadic rectangle $R^{n}_{j_{0}}$. Define

$\Omega_{m,M}:=\bigcup_{|I|=M+1}\Omega(m,I)$

as the event that $M+1$ of the $N_{m}$ random arguments of points in $A_{m}$
fall into the same dyadic rectangle. Moreover, the $\theta_{i}$ are
independent and identically distributed, so the probability of each
$\Omega(m,I)$ is equal to $2^{-m(|I|-1)}$.

Suppose $\sum_{m\in\mathbb{N}^{d}}N_{m}^{1+M}2^{-|m|M}<\infty$ for some $M\geq
1$. Since for every $M\in\mathbb{N}$,

$(M+1)!\geq\frac{(M+2)^{M+1}}{e^{M+1}}$

then

$\binom{N_{m}}{M+1}\leq\frac{N_{m}^{M+1}}{(M+1)!}\leq\left(\frac{e}{M+2}\right)^{M+1}N_{m}^{M+1}.$

So we finally obtain

$\mathbb{P}(\Omega_{m,M})\leq\binom{N_{m}}{M+1}2^{-|m|M}\leq\left(\frac{e}{M+2}\right)^{M+1}N_{m}^{M+1}2^{-|m|M}.$
(7)

This, in principle, doesn’t exclude the possibility that many random arguments
are very close to each other with high probability, even if they belong to
different rectangles. To take care of such eventuality, one can just shift the
rectangles by $2^{-m_{i}-1}$ modulo $1$ in each direction and repeat the above
argument. Therefore (7) controls the probability that $M+1$ arguments of the
points in $A_{m}$ belongs to a rectangle (non necesarlity dyadic) of side-
leghts $2^{-m_{i}}$, $i=1,\dots,d$. Moreover, two points in $A_{m}$ whose
arguments are not in the same rectangle of such dimensions are at a uniform
mutual pseudo-hyperbolic distance, hence thanks to Borel-Cantelli’s Lemma,
$\Lambda$ can be almost surely partitioned into $M$ weakly separated
sequences.

Now suppose $\sum_{m\in\mathbb{N}^{d}}N_{m}^{M+1}2^{-|m|M}=\infty$. We will
show that $\Lambda$ is not almost surely the union $M$ weakly separated
sequences by showing that for all $l$ in $\mathbb{N}$, almost surely there are
infinitely many clusters of $M+1$ points in $\Lambda$ in the same pseudo-
hyperbolic ball of radius $2^{-l}$. Fix $l$ in $\mathbb{N}$, and given $m$ in
$\mathbb{N}^{d}$ divide $A_{m}$ into $2^{dl}$ regions by refining the dyadic
partition that defines $A_{m}$ $l$ times in each direction. Namely,

$A_{m}^{j}:=\left\\{z\in
A_{m}\,\bigg{|}\,2^{-m_{i}+1}+\frac{j-1}{2^{m_{i}-1+l}}\leq
1-|z^{i}|<2^{-m_{i}+1}+\frac{j}{2^{m_{i}-1+l}},i=1,\dots,d\right\\},\,j_{i}=1,\dots,2^{l}.$

Since $A_{m}$ contains $N_{m}$ points of $\Lambda$, then there exists one
$A^{j}_{m}$, say $J_{m}$, that contains at least $L_{m}\geq N_{m}/2^{dl}$
points of $\Lambda$. If $m\oplus l:=(m_{1}+l,\dots,m_{d}+l)$, then every point
in $\mathbb{D}^{d}$ whose radii is in $J_{m}$ and arguments are in the same
dyadic rectangle $R^{m\oplus l}_{k}$ are in a ball of pseudo-hyperbolic
distance comparable to $2^{-l}$. Therefore, we need to apply Corollary 2.3 to
$\mu_{M}(N,n)$, with $N=2^{|m|+dl}$ and $m=L_{m}$. By eventually removing some
radii from the sequence $(r_{m})_{m\in\mathbb{N}^{d}}$, we can assume without
loss of generality that $N_{m}2^{-|m|}\underset{|n|\to\infty}{\to}0$, while
$\sum_{m\in\mathbb{N}^{d}}N_{m}^{1+M}2^{-M|m|}$ is still divergent (clearly if
the associated random sequences is not the union of $M$ weakly separated
sequences almost surely, so it won’t be the one associated to the whole
sequenced $(r_{m})_{m\in\mathbb{N}^{d}}$). In this setting, using the
notations of Corollary 2.3

$\alpha=\alpha_{n}\leq
N_{m}2^{-|m|-dl}\underset{|m|\to\infty}{\to}0,\quad\qquad
p_{M}=p_{M,m}\underset{|m|\to\infty}{\sim}\alpha_{m}^{M}/M!\underset{|m|\to\infty}{\to}{0},$

thus

$\mathbb{P}(\mu_{M}(2^{|m|+dl},L_{n})=1)\underset{|m|\to\infty}{\sim_{M}}L_{n}^{1+M}2^{-M|m|}\geq
N_{m}^{1+M}2^{-M|m|-dl(1+M)}.$

Thanks to Borel-Cantelli Lemma and the divergence of the series
$\sum_{n\in\mathbb{N}^{d}}N_{m}^{1+M}2^{-M|m|}$, we obtain that almost surely
infinitely many of the regions $(A_{m})_{m\in\mathbb{N}^{d}}$ contain a
cluster of $M+1$ points in a pseudo-hyperbolic ball of radius comparable to
$2^{-l}$. Since this is true for all $l$, we conclude by taking an
intersection of countably many events of probability $1$.

∎

In particular, this gives the 0-1 for $\Lambda$ to be the finite union of
weakly separated sequences almost surely, which for the sake of completeness
we formulate via the following three equivalent conditions:

######  Corollary 2.4.

Let $\Lambda$ be a random sequence in $\mathbb{D}^{d}$. Then the following are
equivalent:

(i):

$\Lambda$ can be partitioned almost surely into finitely many weakly separated
sequences,

(ii):

There exists a $M$ in $\mathbb{N}$ such that

$\underset{m\in\mathbb{N}^{d}}{\sum}N_{m}^{1+M}2^{-M|m|}<\infty,$

(iii):

There exists an $\varepsilon>0$ such that

$N_{m}\lesssim 2^{(1-\varepsilon)|m|},$

(iv):

There exists some $\beta>1$ such that

$\sum_{m\in\mathbb{N}^{d}}N_{m}^{\beta}2^{-|m|}<\infty.$

##  3\. $\gamma$-Carleson Measures in the Unit Disc and Interpolating
Sequences for the Bloch Space

Let $\gamma\in(0,1)$. A sequence of points $Z:=(z_{n})_{n}\subset\mathbb{D}$
is a _ $\gamma$-Carleson sequence_ if the measure
$\mu_{Z,\gamma}:=\sum_{n}(1-|z_{n}|^{2})^{\gamma}\leavevmode\nobreak\
\delta_{z_{n}}$ satisfies the one-box condition

$\mu_{\gamma}(S_{I})\lesssim_{Z}|I|^{\gamma}.$

In [10, Theorem 1.4] the authors found the $0-1$ law for a random sequence
$\Lambda$ in the unit disc to satisfy the $\gamma$ \- Carleson condition:

######  Theorem 3.1.

Let $\Lambda$ be a random sequence in $\mathbb{D}$ and let be
$\gamma\in(0,1)$. Then,

$\mathbb{P}(\Lambda\text{ is
$\gamma$-Carleson})=\begin{dcases}1\quad\text{if}\quad&\sum_{m\in\mathbb{N}}N_{m}2^{-\gamma
m}<\infty,\\\ 0\quad\text{if}\quad&\sum_{m\in\mathbb{N}}N_{n}2^{-\gamma
m}=\infty.\end{dcases}.$

As an application of Theorem 3.1 and Theorem 1.2, we give the $0-1$ law for
random interpolating sequences for the Bloch space on the unit disc.

A holomorphic function $f:\mathbb{D}\to\mathbb{C}$ belongs to the Bloch space
$\mathcal{B}$ if

$\|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

We are going to consider interpolating sequences for $\mathcal{B}$ as
described in [6], where $Z=(z_{n})_{n}$ is said to be interpolating for the
Bloch space if for every collection of values $(a_{n})_{n}$ such that

$\sup_{n\neq m}\frac{|a_{n}-a_{m}|}{\beta(a_{n},a_{m})}<\infty$

there exists a function $f\in\mathcal{B}$ such that $f(z_{n})=a_{n}$, where
$\beta$ is the hyperbolic distance in $\mathbb{D}$. This choice for the trace
space is motivated by the fact that if $f\in\mathcal{B}$ then
$|f(z)-f(w)|\leq\|f\|_{\mathcal{B}}\beta(z,w)$. In [6] Bøe and Nicolau
characterized such interpolating sequences:

######  Theorem 3.2.

A sequence of points $Z$ in the unit disc is interpolating for the Bloch space
if and only if it can be expressed as a union of at most two separated
sequences and there exist $0<\gamma<1$ and $C>0$ such that

$\\#\\{n\in\mathbb{N}:\rho(z,z_{n})<r\\}\leq\frac{C}{(1-r)^{\gamma}}.$ (8)

for all $z\in\mathbb{D}$.

As proved in [20] and noted also in [18], condition (8) is equivalent to $Z$
being $\gamma$\- Carleson for some $\gamma<1$.
The $0-1$ law for random interpolating sequences for the Bloch space reads as
follows:

######  Theorem 3.3.

Let $\Lambda$ be a random sequence in $\mathbb{D}$. Then

$\mathbb{P}(\Lambda\text{ is interpolating for
$\mathcal{B}$})=\begin{dcases}1&\text{if}\quad\sum_{m\in\mathbb{N}}N_{m}^{3}2^{-2m}<\infty,\\\
0&\text{if}\quad\sum_{n\in\mathbb{N}}N_{m}^{3}2^{-2m}=\infty.\end{dcases}$

###### Proof.

Suppose $\sum N_{m}^{3}2^{-2m}<\infty$. By Theorem 1.2 we know that $\Lambda$
is almost surely an union of $2$ separated sequences. Furthermore we know that

$N_{m}\lesssim 2^{\frac{2}{3}m}.$

Take $2/3<\gamma<1$, then

$\sum_{m\in\mathbb{N}}N_{n}2^{-\gamma
m}\leq\sum_{m\in\mathbb{N}}2^{-\left(\gamma-\frac{2}{3}\right)}<\infty,$

and so by Theorem 3.1 we have that $\Lambda$ is almost surely
$\gamma$-Carleson for $\gamma>2/3$. By Theorem 3.2, we can conclude that
$\Lambda$ is almost surely an interpolating sequence for the Bloch space.

Suppose now $\sum N_{m}^{3}2^{-2m}=\infty$. Then by Theorem 1.2 we know that

$\mathbb{P}(\Lambda\text{ is the union of $2$ separated sequences})=0,$

thus $\Lambda$ is almost surely not interpolating for $\mathcal{B}$. ∎

##  4\. Random Carleson Measures in the Polydisc

###  4.1. Preliminaries on Gramians

Let $H$ be a Hilbert space, and let $V=(v_{n})_{n\in\mathbb{N}}$ be a sequence
in $H$. The associated _restriction map_ $R_{V}\colon
H\to\mathbb{C}^{\mathbb{N}}$ is defined as

$R_{V}(h):=\left(\left\langle
h,v_{n}\right\rangle\right)_{n\in\mathbb{N}},\qquad h\in H.$

The sequence $V$ is said to be a _Bessel system_ if $R_{V}$ maps $H$
continuously into $\ell^{2}$. In particular,

$V\,\text{is a Bessel system}\iff T_{V}:=(R_{V})^{*}R_{V}\,\text{is
bounded}\iff G_{V}:=R_{V}(R_{V})^{*}\,\text{is bounded}.$

The operator $T_{V}\colon H\mapsto H$ is usually referred to as the _frame
operator_ associated to $V$, and acts as follow:

$T_{V}(h):=\sum_{n\in\mathbb{N}}\langle h,v_{n}\rangle\leavevmode\nobreak\
v_{n},\qquad h\in H.$

The operator $G_{V}\colon\ell^{2}\to\ell^{2}$ is the _Gramian_ of the sequence
$V$, and, with respect to the standard basis of $\ell^{2}$, it is represented
by the infinite matrix

$\left(\langle v_{n},v_{j}\rangle_{H}\right)_{n,j\in\mathbb{N}}$

Let $H=\mathcal{H}_{k}$ be a reproducing kernel Hilbert space with kernel $k$
on a set $X$, and let $Z=(z_{n})_{n\in\mathbb{N}}$ be a sequence in $X$. The
measure

$\mu_{Z}:=\sum_{n\in\mathbb{N}}\|k_{z_{n}}\|^{-2}\leavevmode\nobreak\
\delta_{z_{n}}$

is a Carleson measure if $\mathcal{H}_{k}$ embedds continuosly in
$L^{2}(X,\mu_{\Lambda})$. Therefore, $\Lambda$ generates a Carleson measure
for $\mathcal{H}_{k}$ if and only if the sequence of normalized kernels
$\left(k_{z_{n}}/\|k_{z_{n}}\|\right)_{n\in\mathbb{N}}$ forms a Bessel system
in $\mathcal{H}_{k}$, that is, if and only if the Gram matrix

$G_{\Lambda}:=\left(\left\langle\frac{k_{z_{n}}}{\|k_{z_{n}}\|},\frac{k_{z_{j}}}{\|k_{z_{j}}\|}\right\rangle\right)_{n,j\in\mathbb{N}}$

defines a bounded operator from $\ell^{2}$ to itself.

The following two Lemmas will be relevant for the proof of Theorem 1.1.

######  Lemma 4.1.

Let $\mathbb{N}=\bigcup_{j\in\mathbb{N}}I_{j}$, where each $I_{j}$ is finite,
and suppose that a sequence $V=(v_{n})_{n\in\mathbb{N}}$ in a Hilbert space is
such that

$\langle v_{n},v_{k}\rangle=0$

whenever $n\in I_{j}$, $k\in I_{l}$, and $I_{j}\cap I_{l}=\emptyset$. Suppose
that

$M:=\sup_{j\in\mathbb{N}}\\#\\{k\,|\,I_{k}\cap I_{j}\\}<\infty.$ (9)

Then

$\|G_{V}\|\leq M\leavevmode\nobreak\ \sup_{j}\|G_{V_{j}}\|.$ (10)

###### Proof.

Since

$\left(\sum_{j}G_{V_{j}}\right)-G_{V}$

is positive semi-definite, one has that

$\|G_{V}\|\leq\left\|\sum_{j}G_{V_{j}}\right\|.$

Thanks to Cotlar-Stein Lemma,

$\left\|\sum_{j}G_{V_{j}}\right\|\leq\sqrt{C_{0}\leavevmode\nobreak\
C_{\infty}},$

where

$C_{0}:=\sup_{j}\sum_{l}\sqrt{\|G_{V_{j}}G_{V_{l}}^{*}}\|,\quad
C_{\infty}:=\sup_{l}\sum_{j}\sqrt{\|G_{V_{j}}^{*}G_{V_{l}}\|}.$

Thanks to (9), fixed any $j$, $G_{V_{l}}$ and $G_{V_{j}}$ have not orthogonal
ranges at most $M$ times, thus (10) holds.

∎

In what follows, if $h\in\mathcal{H}$ is a vector of a Hilbert space we will
denote by $hh^{*}$ the rank one operator on $\mathcal{H}$ which acts naturally
as follows, $hh^{*}(x)=\langle x,h\rangle h,\,\,x\in\mathcal{H}$. Let
$Z=(z_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathbb{D}^{d}$, and let, for
all $n$, $S_{n}$ be the normalized Szegö kernel at $z_{n}$. The measure
$\mu_{Z}:=\sum_{n}S_{z_{n}}(z_{n})^{-1}\delta_{z_{n}}$ is a Carleson measure
for $H^{2}_{d}$ if and only if the frame operator $T$ associated to the
sequence $(S_{n})_{n\in\mathbb{N}}\subset H^{2}_{d}$ is bounded, where

$T(f):=\sum_{n\in\mathbb{N}}\langle f,S_{n}\rangle\leavevmode\nobreak\
S_{n}=\left(\sum_{n\in\mathbb{N}}S_{n}S_{n}^{*}\right)(f),\qquad f\in H^{2}.$

Define, for all $a\leq b$ in $\mathbb{N}$,

$T_{[a,b]}:=\sum_{|m|=a}^{b}\sum_{z_{j}\in A_{m}}S_{j}S_{j}^{*},$

as the frame operator of the kernel functions associated to points in the
annulii $\\{A_{m}\,|\,a\leq|m|\leq b\\}$. By writing each $S_{j}$ in their
coordinates with respect to the monomials basis of $H^{2}_{d}$

$S_{j}(z)=\left(\prod_{i=1}^{d}\sqrt{1-|z^{i}_{j}|^{2}}\right)\sum_{l\in\mathbb{N}^{d}}\overline{z_{j}}^{l}z^{l}\qquad
z\in\mathbb{D}^{d},$

we will be able to use some results on the highest eigenvalue of random
matrices that can be written as the sum of rank-one independent components.
Since such components must be _finite_ dimensional square matrices (see
Theorem 4.3 below), we need first to approximate the vectors
$(S_{j})_{j\in\mathbb{N}}$ using partial sums. Fix a $L$ in $\mathbb{N}$, and
let

$T_{[a,b]}^{L}:=\sum_{|m|=a}^{b}\sum_{z_{j}\in
A_{m}}P_{L}(S_{j})(P_{L}(S_{j}))^{*}$

be the frame operator of the collection
$\left\\{P_{L}(S_{j})\,\bigg{|}\,z_{j}\in\overset{b}{\underset{|m|=a}{\bigcup}}A_{m}\right\\}$,
where

$P_{L}(S_{j})(z)=\left(\prod_{i=1}^{d}\sqrt{1-|z^{i}_{j}|^{2}}\right)\sum_{l_{1},\dots,l_{d}\leq
L}\overline{z_{j}}^{l}z^{l}\qquad z\in\mathbb{D}^{d}.$

Fixed $a$ and $b$ and set $N_{[a,b]}:=\sum_{|m|=a}^{b}N_{m}$. We seek how
large must $L$ be for $\|T_{[a,b]}\|$ and $\|T_{[a,b]}^{L}\|$ to be close.

######  Lemma 4.2.

For all $a<b$ in $\mathbb{N}$ and for any $L$ in $\mathbb{N}$

$\|T_{[a,b]}\|\leq\|T^{L}_{[a,b]}\|+CN_{[a,b]}(1-2^{-b})^{2L}.$ (11)

###### Proof.

Thanks to orthogonality,

$T_{[a,b]}=T_{[a,b]}^{L}+\tilde{T}_{[a,b]}^{L},$

where $\tilde{T}^{L}_{[a,b]}$ is the frame operator of the collection
$\left\\{(Id-P_{L})(S_{j})\,|\,z_{j}\in\bigcup_{|m|=a}^{b}A_{m}\right\\}$.
Thus $\|T_{[a,b]}\|$ and $\|T_{[a,b]}^{L}\|$ differs at most by
$\|\tilde{T}^{L}_{[a,b]}\|$. Notice that for all $f$ in $H^{2}$

$\displaystyle\left\|\sum_{|m|=a}^{b}\sum_{z_{j}\in A_{m}}\langle{f,(Id-
P_{L})(S_{j})}\rangle\leavevmode\nobreak\ (Id-P_{L})(S_{j})\right\|$
$\displaystyle\leq N_{[a,b]}\leavevmode\nobreak\ \|f\|\leavevmode\nobreak\
\sup\left\\{\|(Id-
P_{L})(S_{j})\|^{2}\,\bigg{|}\,z_{j}\in\bigcup_{|m|=a}^{b}A_{m}\right\\}$
$\displaystyle=N_{[a,b]}\leavevmode\nobreak\ \|f\|\leavevmode\nobreak\
\sup\left\\{1-\prod_{i=1}^{d}(1-|z^{i}_{j}|^{2L})\,\bigg{|}\,z_{j}\in\bigcup_{|m|=a}^{b}A_{m}\right\\}$
$\displaystyle\leq N_{[a,b]}\leavevmode\nobreak\ \|f\|\leavevmode\nobreak\
\left(1-\left(1-\left(1-2^{-b}\right)^{2L}\right)^{d}\right)$
$\displaystyle\lesssim_{d}N_{[a,b]}\leavevmode\nobreak\
\|f\|\leavevmode\nobreak\ (1-2^{-b})^{2L}.$

since for all $z_{j}\in A_{m}$, $|m|=a,\dots,b$, one has that

$|z_{j}^{i}|\leq 1-2^{-m_{i}}\leq 1-2^{-|m|}\leq 1-2^{-b},\qquad i=1,\dots d.$

∎

###  4.2. Random Szegö Gramians

We are now ready for the proof of Theorem 1.1. Thanks to Corollary 2.4, if
$N_{m}\neq\mathcal{O}(2^{(1-\varepsilon)|m|})$ for all $\varepsilon>0$ then
$\Lambda$ is almost surely not the union of finitely many separated sequences.
Hence, [1, Proposition 9.11], $\Lambda$ is not a Carleson sequence almost
surely.
Let $\Lambda$ be a random sequence in $\mathbb{D}^{d}$, and assume that
$N_{m}\lesssim 2^{(1-\varepsilon)|m|}$, for some $\varepsilon>0$. The idea is
to arrange a (deterministic) decomposition of the associated random Gramian
$G_{\Lambda}$, by writing it as the sum of a sequence of overlapping blocks
and the remaining off-blocks part. More precisely, set

$I_{j}:=[2^{j-1},2^{j}/\varepsilon]\cap\mathbb{N},\qquad j\in\mathbb{N}.$

Let $G_{\Lambda}=G_{1}+G_{2}$, where the entries of $G_{1}$ coincide with the
ones of $G$ on the overlapping blocks $(X_{j})_{j\in\mathbb{N}}$, $X_{j}$
being the Gram matrix of the collection

$\\{S_{n}\,|\,n\in A_{m},|m|\in I_{j}\\},$

and they are zero elsewhere. $G_{2}$ instead is zero on the overlapping blocks
and carries the entries of $G_{\Lambda}$ outside of such blocks. We prove
Theorem 1.1 by showing in two separate steps that $G_{1}$ and $G_{2}$ are
bounded almost surely:

#### Diagonal overlapping blocks estimates

The key result that we are going to use from the theory of random matrices can
be extracted from the matrix Chernoff’s inequality, [21, Theorem 1.1]:

######  Theorem 4.3.

Let $T$ be the frame operator of finitely many random independent vectors
$v_{1},\dots,v_{N}$ in $\mathbb{C}^{L}$, and let $\mu:=\|\mathbb{E}(T)\|$. If
$\|v_{j}\|\leq 1$ for all $j$ almost surely, then

$\mathbb{P}\left(\left\|T\right\|\geq(1+\delta)\mu\right)\leq
L\leavevmode\nobreak\
\left(\frac{e}{1+\delta}\right)^{\delta\mu}\qquad\delta\geq 0.$

A computation shows that by our choice of the sequence $(I_{j})_{j}$, the
number of overlaps of the blocks that compose $G_{1}$ is uniformly bounded by
$M_{\varepsilon}:=\log_{2}(1/\varepsilon)+1$. Therefore, thanks to Lemma 4.1
in order to show that $G_{1}$ is bounded almost surely it suffices to show
that

$\sup_{j}\|X_{j}\|<\infty$ (12)

almost surely. Notice that $X_{j}$ is the Gram matrix associated to the frame
operator $T_{[a_{j},b_{j}]}$, where $a_{j}=2^{j-1}$ and
$b_{j}=2^{j}/\varepsilon$. Consider $T_{[a_{j},b_{j}]}^{L_{j}}$, where
$L_{j}=2^{3b_{j}}$ so that

$N_{[a_{j},b_{j}]}(1-2^{-b_{j}})^{2L_{j}}\lesssim_{\varepsilon}2^{2(1-\varepsilon)b_{j}}(1-2^{-b_{j}})^{2^{3b_{j}}}\underset{b_{j}\to\infty}{\to}0.$
(13)

Thanks to Lemma 4.2, $\|X_{j}\|$ and $\|T^{L_{j}}_{[a_{j},b_{j}]}\|$ are
closer than an uniform constant. We have, with respect to the coordinates
given by the monomials in $H^{2}_{d}$,

$\begin{split}T^{L_{j}}_{[a_{j},b_{j}]}&=\sum_{|m|=a_{j}}^{b_{j}}\sum_{\lambda_{n}\in
A_{m}}P_{L_{j}}(S_{n})(P_{L_{j}}(S_{n}))^{*}\\\
&=\sum_{|m|=a_{j}}^{b_{j}}\sum_{\lambda_{n}\in
A_{m}}\left(\prod_{i=1}^{d}(1-(r^{i}_{n})^{2}\right)\left(r_{n}^{k+l}\leavevmode\nobreak\
e^{-i\theta_{n}(k-l)}\right)_{|l|,|k|=0}^{L_{j}},\end{split}$

where

$r_{n}^{k+l}\leavevmode\nobreak\
e^{-i\theta_{n}(k-l)}=\prod_{i=1}^{d}(r^{i}_{n})^{k_{i}+l_{i}}e^{-i\theta^{i}_{n}(k_{i}-l_{i})}.$

Hence the expectation of $T^{L_{j}}_{[a_{j},b_{j}]}$ is diagonal, and its norm
is

$\mu_{j}:=\left\|\mathbb{E}\left(T^{L_{j}}_{[a_{j},b_{j}]}\right)\right\|\simeq\sum_{|m|=a_{j}}^{b_{j}}N_{m}2^{-|m|}\lesssim\sum_{|m|=a_{j}}^{b_{j}}2^{-\varepsilon|m|}\lesssim
2^{-\frac{\varepsilon a_{j}}{2}}.$

Fix a positive number $A$, to be determined later. By applying Theorem 4.3,
$\delta_{j}:=\frac{A}{\mu_{j}}-1$, we obtain

$\begin{split}\mathbb{P}\left(\left\|T_{[a_{j},b_{j}]}^{L_{j}}\right\|\geq
A\right)&\leq L_{j}^{d}\left(\frac{\mu_{j}\leavevmode\nobreak\
e}{A}\right)^{A-\mu_{j}}\\\
&\underset{j\to\infty}{\sim}L_{j}^{d}\left(\frac{\mu_{j}\leavevmode\nobreak\
e}{A}\right)^{A}\\\ &\lesssim_{A}2^{3db_{j}-\frac{A\varepsilon a_{j}}{2}}\\\
&=2^{a_{j}\left(\frac{6d}{\varepsilon}-\frac{A\varepsilon}{2}\right)},\end{split}$

since $b_{j}=2a_{j}/\varepsilon$. It suffices then to pick
$A>\frac{12d}{\varepsilon^{2}}$, and Borel-Cantelli Lemma gives (12).

#### Off diagonal estimates

We are left with showing that $G_{2}$ is bounded almost surely, under the
assumption that $N_{m}\lesssim 2^{(1-\varepsilon)|m|}$. The advantage of
taking overlapping blocks in the first step of our proof is that the entries
that are left composing $G_{2}$ are far away from the diagonal, hence we can
exploit the decay of their expectation. We show that
$\mathbb{E}(\|G_{2}\|_{HS})$, the expectation of the Hilbert-Schmidt norm of
$G_{2}$, is finite, concluding the proof of Theorem 1.1. Thanks to [13, Remark
3.2], if $\lambda_{n}$ is in $A_{m}$ and $\lambda_{l}$ is in $A_{k}$, then

$\mathbb{E}(|\left\langle
S_{\lambda_{n}},S_{\lambda_{l}}\right\rangle|^{2})=\frac{(1-r^{2}_{n})(1-r^{2}_{j})}{1-r_{n}r_{j}}\simeq\prod_{i=1}^{d}\frac{1}{2^{m_{i}}+2^{k_{i}}}.$
(14)

Thus the expectation of the square of the Hilbert-Schmidt norm of the off-
diagonal block

$\left(\langle S_{n},S_{l}\rangle\right)_{\lambda_{n}\in A_{m},\lambda_{l}\in
A_{k}}$

of $G_{\Lambda}$ is controlled by
$\frac{N_{m}N_{l}}{\prod_{i}2^{m_{i}}+2^{l_{i}}}$. Hence

$\begin{split}\mathbb{E}(\|G_{2}\|_{HS}^{2})\simeq&\sum_{j=1}^{\infty}\sum_{|m|=2^{j-1}}^{2^{j}}\sum_{|l|\geq
2^{j}/\varepsilon}N_{m}N_{l}\prod_{i=1}^{d}\frac{1}{2^{m_{i}}+2^{l_{i}}}\\\
\lesssim&\sum_{j=1}^{\infty}\sum_{|m|=2^{j-1}}^{2^{j}}\sum_{|l|\geq
2^{j}/\varepsilon}\frac{2^{(1-\varepsilon)(|m|+|l|)}}{2^{|m|}+2^{|l|}}\\\
\leq&\sum_{j=1}^{\infty}\sum_{|m|=2^{j-1}}^{2^{j}}2^{(1-\varepsilon)|m|}\sum_{|l|\geq
2^{j}/\varepsilon}2^{-\varepsilon|l|}\\\
\lesssim&\sum_{j=1}^{\infty}\sum_{|m|=2^{j-1}}^{2^{j}}2^{(1-\varepsilon)|m|}\sum_{s\geq
2^{j}/\varepsilon}s^{d-1}2^{-\varepsilon s}\\\
\lesssim&\sum_{j=1}^{\infty}\sum_{|m|=2^{j-1}}^{2^{j}}2^{(1-\varepsilon)|m|-2^{j}(1-\varepsilon/2)}\\\
\lesssim&\sum_{j=1}^{\infty}\sum_{r=2^{j-1}}^{2^{j}}r^{d-1}2^{(1-\varepsilon)r-2^{j}(1-\varepsilon/2)}\\\
\leq&\sum_{j=1}^{\infty}2^{dj-\varepsilon 2^{j-1}}<\infty.\end{split}$

This concludes the proof of Theorem 1.1.

###  4.3. Random Dirichlet Gramians

Given a random sequence $\Lambda$ in the polydisc, one can ask whether it
generates a Carleson measure for a reproducing kernel Hilbert space other than
the Hardy space. As in the one variable setting, for all $0\leq a\leq 1$, let
$D^{a}_{d}$ be the associated Dirichlet-type space on $\mathbb{D}^{d}$, that
is, the reproducing kernel Hilbert space having kernel

$k^{(a)}_{w}(z):=\begin{cases}\prod_{i=1}^{d}\frac{1}{(1-\overline{w^{i}}z^{i})^{1-a}}\qquad&a\leq
0<1\\\
\prod_{i=1}^{d}\frac{1}{z^{i}\overline{w^{i}}}\log\frac{1}{1-\overline{w^{i}}z^{i}}\qquad&a=1\end{cases}$

Let $S^{(a)}_{w}(z):=k^{(a)}_{w}(z)/\left\|k^{(a)}_{w}\right\|$ denote the
associated normalized kernel. The case $a=0$ corresponds to the Hardy space,
while the case $a=1$ corresponds to the Dirichlet space on the polydisc. In
order to study radom sequence on the polydisc that are Carleson for the spaces
$D^{2}_{d}$, we first extend (14) to this setting:

######  Lemma 4.4.

Let $\Lambda=(\lambda_{n})_{n}$ be a random sequence in the polydisc. Then

$\mathbb{E}(|S^{(a)}(\lambda_{n},\lambda_{j})|^{2})\simeq\prod_{i=1}^{d}(1-r^{i}_{n})^{1-a}(1-r^{i}_{j})^{1-a}\begin{cases}\frac{1}{(1-r^{i}_{n}r^{i}_{j})^{1-2a}}\quad&0\leq
a<1/2\\\ \log\frac{1}{1-r^{i}_{n}r^{i}_{j}}\quad&a=1/2\\\
1\quad\quad&1/2<a\leq 1\end{cases}.$

###### Proof.

First, observe that it is enough to prove the Lemma for the case $d=1$, since
the $d$ coordinates of each random variable $\theta_{n}$ are independent, and
the expectation of the product of independent random variables factorizes.
Write then

$k^{(a)}_{w}(z)=\sum_{l=0}^{\infty}c_{l}(z\overline{w})^{l}\qquad
z,w\in\mathbb{D}$

where $c_{l}\simeq(1+l)^{-a}$. Hence

$\begin{split}&\mathbb{E}(|S^{(a)}(\lambda_{n},\lambda_{j})|^{2})\\\
\simeq&(1-r_{n})^{1-a}(1-r_{j})^{1-a}\sum_{l,r=0}^{\infty}c_{l}c_{r}(r_{n}r_{j})^{l+r}\mathbb{E}(e^{i(l-r)(\theta_{n}-\theta_{j})})\\\
=&(1-r_{n})^{1-a}(1-r_{j})^{1-a}\sum_{l=0}^{\infty}c_{l}^{2}(r_{n}r_{j})^{2l},\end{split}$

and the Lemma follows. ∎

Since all the kernels involved are invariant under rotations, $\Lambda$
generates for all $\omega$ in $\Omega$ a finite measure for $D^{a}_{d}$ if and
only if

$\sum_{n\in\mathbb{N}}\|k^{(a)}_{r_{n}}\|^{-2}=\sum_{m\in\mathbb{N}^{d}}N_{m}2^{(1-a)|m|}<\infty.$
(15)

Recall that for $d=1$ and $0<a\leq 1$, (15) is also sufficient for $\Lambda$
to generate a Carleson measure for $D^{a}$, see [10, Theorem 1.4]. Thanks to
Lemma 4.4, this can be seen to be true also in the multi-variable case, though
our argument covers only the case $1/2<a\leq 1$. Indeed, if (15) holds,thanks
to Lemma 4.4, $1/2<a\leq 1$, one obtains that

$\begin{split}&\mathbb{E}\left(\sum_{n\neq
j}|S^{(a)}(\lambda_{n},\lambda_{j})|^{2}\right)=\sum_{n\neq
j}\mathbb{E}\left(|S^{(a)}(\lambda_{n},\lambda_{j})|^{2}\right)\\\
\simeq&\sum_{n\neq
j}\|k^{(a)}_{r_{n}}\|^{-2}\|k^{(a)}_{r_{j}}\|^{-2}<\infty,\end{split}$

hence the Gram matrix of $\Lambda$ in $D^{a}_{d}$

$G_{\Lambda}^{a}:=\left(S^{(a)}(\lambda_{n},\lambda_{j})\right)_{n,j}$

is almost surely a Hilbert-Schmidt perturbation of the identity. In
particular,

######  Corollary 4.5.

Let $d\geq 1$, and let $\Lambda$ be a random sequence in $\mathbb{D}^{d}$. For
all $1/2<a\leq 1$,

$\mathbb{P}(\Lambda\,\text{is a Carleson sequence for
}\,D^{a}_{d})=\begin{cases}1\qquad&\text{if}\,\,\sum_{m\in\mathbb{N}^{d}}N_{m}2^{(1-a)|m|}<\infty\\\
0\qquad&\text{if}\,\,\sum_{m\in\mathbb{N}^{d}}N_{m}2^{(1-a)|m|}=\infty\end{cases}$
(16)

We conjecture that Corollary 4.5 holds also for $0<a\leq 1/2$. The random
matrix argument used in Theorem 1.1 for the case $a=0$ can’t be used to prove
(16) for $0<a\leq 1/2$, since it requires that the sequence
$(N_{m}2^{-|m|})_{m\in\mathbb{N}^{d}}$ decay exponentially. A geometric
sufficient condition for Carleson measures for $D^{a}_{d}$ in the
deterministic setting is only available for $d=2,3$: see [17] for the case
$0<a<1$ and [3] for the case $a=1$.

##  5\. Random Carleson Measures on the Unit Ball

The proof of Theorem 1.3 relies on some properties of positive semi-definite
matrices. An infinite matrix $A=(a_{nj})_{n,j}$, is positive semi-definite,
$A\geq 0$, if for any $N>0$ and $c_{1},\dots,c_{N}$ in $\mathbb{C}$

$\sum_{n,j=1}^{N}c_{n}\overline{c_{j}}a_{nj}\geq 0.$

For instance, any Gram matrix associated to a sequence of vectors
$(v_{n})_{n}$ in a Hilbert space is positive semi-definite. A noteworthy
result about positive semi-definite matrices is that if $A=(a_{nj})_{n,j}$ and
$B=(b_{nj})_{n,j}$ are positive semi-definite, then $A\odot
B:=(a_{nj}b_{nj})_{n,j}$ is positive semi-definite as well. As a corollary,
one proves the following:

######  Lemma 5.1.

Let $A\colon\ell^{2}\to\ell^{2}$ be a bounded infinite matrix, and let $H$ be
a positive semi-definite infinite matrix having all the entries on its main
diagonal equal to $1$. Then $\|A\odot H\|\leq\|A\|$.

###### Proof.

The norm of $A$ is the least $C>0$ such that

$C^{2}Id-A\geq 0.$ (17)

By Schur multiplying the left hand side of (17) by $H$, we obtain that
$C^{2}Id-A\odot H\geq 0$, hence $\|A\odot H\|\leq C$. ∎

This, together with [13, Theorem 4.3] and [16, Theorem 3.4], provides the
proof of Theorem 1.3:

###### Proof of Theorem 1.3.

If $N_{m}\neq\mathcal{O}(2^{d(1-\varepsilon)m})$ for all $\varepsilon>0$, then
thanks to [16, Theorem 3.4] it is not the union of finitely many separated
sequences almost surely with respect to the pseudo-hyperbolic metric

$\rho(z,w)^{2}:=1-\frac{(1-|z|^{2})(1-|w|^{2})}{|1-\langle
z,w\rangle_{\mathbb{C}^{d}}|^{2}}$

hence the Gram matrix

$G_{\Lambda}:=\left(\frac{(1-|z|^{2})^{\frac{d}{2}}(1-|w|^{2})^{\frac{d}{2}}}{\left(1-\langle
z,w\rangle_{\mathbb{C}^{d}}\right)^{d}}\right)_{n,j}$

is not bounded almost surely, and $\Lambda$ does not generate a Carleson
measure for the Hardy space. On the other hand, if there exists a positive
$\varepsilon$ such that $N_{m}\lesssim 2^{d(1-\varepsilon)m}$, then $\Lambda$
generates a finite measure for $B^{\nu}_{d}$ for some $0<\nu$. Hence, [13,
Theorem 4.3], $\Lambda$ is a Carleson sequence for $B^{\nu}_{d}$ almost
surely, and the Gram matrix

$G_{\Lambda}:=\left(\frac{(1-|z|^{2})^{\frac{d-\nu}{2}}(1-|w|^{2})^{\frac{d-\nu}{2}}}{\left(1-\langle
z,w\rangle_{\mathbb{C}^{d}}\right)^{d-\nu}}\right)_{n,j}$

is bounded almost surely. But since $G_{\Lambda}$ is the Schur product between
$G_{\Lambda}^{\nu}$ and $G_{\Lambda}^{d-\nu}$ then $G_{\Lambda}$ is bounded
almost surely thanks to Lemma 5.1, hence $\Lambda$ is Carleson for the Hardy
space almost surely.

∎

## Acknowledgements

The authors would like to express their gratitude to Roland Speicher and Marwa
Banna for directing us towards the literature on random matrices that contain
Theorem 4.3.

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