1. Introduction
There is an extensive mathematical literature, in several theoretical and applied areas, related to determinantal point processes; we cite, to mention a few recent applied works, [Reference Baccelli and O’Reilly2, Reference Gillenwater, Fox, Kulesza and Taskar7, Reference Gillenwater, Kulesza, Mariet and Vassilvtiskii8, Reference Launay, Galerne and Desolneux16, Reference Møller and O’Reilly21, Reference Østerbø and Grøndalen22]. A good overview of the main conceptual basis and properties can be found in [Reference Lyons18] and in the bibliography therein.
From the theoretical point of view, determinantal point processes could be defined (in a Bourbaki-like spirit) in the general locally compact Polish spaces setting, as point processes associated with some locally square integrable, Hermitian, positive semidefinite, locally trace-class operators, and thereafter specialized for particular cases, namely to discrete determinantal processes. Regarding the latter, the approach of [Reference Lyons18], which consists of constructing such processes, first in the most elementary discrete context and then gradually extending them to the general situation, provides, in our opinion, many advantages. It also turns out that some results for the most general processes are proved only [Reference Goldman9, Reference Lyons18], or more simply [Reference Lyons19], indirectly from the corresponding results of the basic processes.
The basic elementary determinantal point process can be described via the exterior product concept, as follows. Fix
$1< p < N$
and let
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
,
$ 1<p<N $
, be a set of orthonormal vectors in
$ \mathbb{C}^{N} $
. We write
$z^{i}=(z^{i}_{1},\dots,z^{i}_{N})^{t}$
,
$i=1,\dots,p$
, and
$z_{i}=(z^{1}_{i},\dots,z^{p}_{i})$
,
$i=1,\dots,N$
.
The associated determinantal process
$ \phi(\mathfrak{Z}) $
is a point process, view as a random subset of
$\mathcal{N}=\{1,\dots,N\}$
of cardinality
$ \vert \phi(\mathfrak{Z})\vert = p $
, characterized [Reference Lyons17, Reference Lyons18] by the formula
$\mathbb{P}\{\{i_{1},\dots,i_{p}\} = \phi\}= \big|\big(\bigwedge_{i=1}^{p}z^{i}\big)_{\{i_{1},\dots,i_{p}\}}\big|^{2}= \big[\!\det\big(\big(z_{i_{j}}^{k}\big)_{k,j=1,\dots,p}\big)\big]^{2}$
for all subsets
$\{i_{1},\dots,i_{p}\} \subset \mathcal{N}$
. Note also that this formula implies
$\mathbb{P}\{\{i_{1},\dots,i_{k}\} \subset \phi\}=\big\| \bigwedge_{j=1}^{k}z_{i_{j}}\big\|^{2}$
for all
$1\leq k \leq p$
.
Let
$ E=E(\mathfrak{Z})\subset \mathbb{C}^{N}$
be the vector space spanned by
$ \mathfrak{Z} $
. For all sets of linearly independent vectors
$v^{i}\in E$
,
$i=1,\dots,p$
, we have
$\bigwedge_{i=1}^{p}v^{i}=a\bigwedge_{i=1}^{p}z^{i}$
with
$a\neq 0$
; thus, in particular, if
$ \tilde{\mathfrak{Z}}=\{\tilde{z}^{1},\ldots,\tilde{z}^{p}\} $
is another orthonormal basis of
$ E=E(\mathfrak{Z}) $
then
$\big|\big(\bigwedge_{i=1}^{p}z^{i}\big)_{\{i_{1},\dots,i_{p}\}}\big|=\big|\big(\bigwedge_{i=1}^{p}\tilde{z}^{i}\big)_{\{i_{1},\dots,i_{p}\}}\big|$
for every
$\{i_{1},\dots,i_{p}\}\subset \mathcal{N}$
, and consequently
$ \phi(\mathfrak{Z})=\phi(\tilde{\mathfrak{Z}}) $
.
Note also that if
$ \mathfrak{Z}^{\perp}=\{z^{p+1},\ldots,z^{N}\} $
is an orthonormal basis of the orthogonal complement
$ E(\mathfrak{Z})^{\perp} $
of
$E(\mathfrak{Z})$
in
$\mathbb{C}^{N}$
then obviously
$\phi(\mathfrak{Z}^{\perp})=\{1,\dots,N\}\setminus \phi(\mathfrak{Z})$
. A remarkable example of a non-trivial basic determinantal process is given by uniform spanning tree measure on a finite connected graph G. Roughly speaking, if G is fixed and arbitrarily edge-oriented, and M is the vertex–edge incidence matrix (the columns being indexed by vertices), then the determinantal process associated with the vector space spanned by all the column vectors but one provides a uniform probability on spanning trees. This result. due to [Reference Burton and Pemantle5], is called the Transfer Current Theorem. For more details, with clever short proofs, see [Reference Lyons18, Section 2.6, p. 8]. Some extensions of this result are given in [Reference Benjamini, Lyons, Peres and Schramm4] with a series of open questions and conjectures, among them Conjecture 4.6, related to the van den Berg–Kesten (BK) inequality,
Recall that an event
$\mathfrak{A}\subset 2^{\mathcal{N}}$
,
$\mathcal{N}=\{1,\dots,N\}$
, is called increasing if, whenever
$A\in \mathfrak{A}$
and
$n \in \mathcal{N}$
, we also have
$A\cup \{n\}\in \mathfrak{A}$
. For a pair
$\mathfrak{A}, \mathfrak{B} \subset 2^{\mathcal{N}}$
of increasing events, the disjoint intersection
$\mathfrak{A}\circ\mathfrak{B}$
is then defined [Reference van den Berg and Kesten26] by
$\mathfrak{A}\circ\mathfrak{B}=\{ K \subset \mathcal{N} \,:\, \mathrm{there\ exist} \ L\in \mathfrak{A}, M \in \mathfrak{B}, L,M \neq \emptyset\ \mathrm{such\ that} \ L \cap M = \emptyset, K \supset L\cup M \}$
. A point process
$\psi$
on
$\mathcal{N}$
is said to have the BK property if
\begin{equation}\mathbb{P}\{ \psi \in \mathfrak{A}\circ\mathfrak{B}\}\leq\mathbb{P}\{\psi \in \mathfrak{A}\}\times\mathbb{P}\{\psi \in \mathfrak{B}\}\end{equation}
for every pair of increasing events. In [Reference van den Berg and Kesten26] it was proved that (1) is satisfied when
$\psi$
is related to a product probability on
$2^{\mathcal{N}}$
. In the basic determinantal process setting, Conjecture 4.6, which states that the same is true for the spanning trees determinantal point processes, is still unsolved. The question of whether general determinantal processes have the BK property was raised in [Reference Lyons17].
The purpose of this note is twofold. First, we introduce a new method to investigate discrete determinantal processes using the CS decomposition (CSD) of a partitioned unitary matrix, which is a useful non-trivial tool in numerical linear algebra; a precise statement of CSD is given in Section 2. We show that the CSD gives a pertinent description of conditioning and provides (at least in our opinion) a suggestive perspective for future investigations; see, for example, the result given by Proposition 2, which seems to us to be new, and the results of [Reference Goldman10]. Furthermore, this should be an appropriate framework for computational needs.
Second, we study the BK inequality. We prove that the BK inequality (1) is satisfied for all discrete determinantal processes when the increasing events
$\mathfrak{A}$
and
$\mathfrak{B}$
are generated by simple points: Theorem 3 in Sections 4 and 5. We also conjecture the following.
Conjecture 1. For all
$n\geq 2$
,
\begin{equation} \mathbb{P}\lbrace A \not \subset \phi \mid A_{i} \not \subset \phi \ {for\ all}\ i=1,\dots,n\rbrace \leq \mathbb{P}\lbrace A \not \subset \phi \mid A_{i} \not \subset \phi \ {for\ all}\ i=1,\dots,n-1\rbrace\end{equation}
for every choice of
$A, A_{i}$
,
$i=1,\dots,n$
, of disjoint subsets of
$\{1,\dots, N\}$
such that
$\mathbb{P}\lbrace A_{i} \not \subset \phi \ \mathrm{for\ all}\ i=1,\dots,n\rbrace > 0 $
.
If Conjecture 1 holds then it can be shown that the BK inequality (1) is satisfied for increasing events
$\mathfrak{A}$
and
$\mathfrak{B}$
generated by disjoint sets: Theorem 2 in Section 3. When the sets above are reduced to being simple points then the inequality (2) is a well-known result. For general sets, note that
$\mathbb{P}\lbrace A \not \subset \phi \mid A_{1} \not \subset \phi \rbrace \leq \mathbb{P}\lbrace A \not \subset \phi \rbrace$
,
$A\cap A_{1}=\emptyset $
, is the classical correlation inequality [Reference Lyons17], and that (2) was obtained in [Reference Goldman10] for
$n=2,3 $
with precise values of the conditional probabilities.
Remark 1. Note that for the process
$\psi$
related to a product probability on
$2^{\mathcal{N}}$
, the counting random variables
$\vert \psi \cap A_{i} \vert$
(the sets
$A_{i}$
,
$i=1,\dots,n$
, being disjoint) are independent and thus the inequality (2) becomes trivial. However, the situation is less obvious if the process
$\psi$
is conditioned to have exactly k points,
$1<k<N$
. In the particular case when the conditioned process
$\psi_{k}$
assigns equal probability to all subsets
$\{i_{1},\dots,i_{k}\} \subset \mathcal{N}$
, i.e. if
$\mathbb{P}\lbrace \psi_{k} = \{i_{1},\dots,i_{k}\}\rbrace = 1/ \binom{N}{k}$
, it was proved in [Reference van den Berg and Jonasson25] that
$\psi_{k}$
has the BK property. As regards the inequality (2), we have, with the choice
$\mathbb{P}\lbrace i \not \in \psi_{k}, i_{j} \not \in \psi_{k} \ \mathrm{for\ all}\ j=1,\dots,n\rbrace>0$
,
$\mathbb{P}\lbrace i \not \in \psi_{k} \mid i_{j} \not \in \psi_{k} \ \mathrm{for\ all}\ j=1,\dots,n\rbrace = {\binom{N - n - 1}{k}}/{\binom{N - n }{k}}$
and, consequently, the inequality (2) is equivalent, for simple points, to the well-known log-concave inequality
$\binom{N - n - 1}{N-k}\times\binom{N - n + 1}{N-k} \leq \binom{N - n}{N-k}^{2}$
, and thus is fulfilled. Likewise, for general sets, the correlation inequality
$\mathbb{P}\lbrace A \not \subset \psi_{k} \mid A_{1} \not \subset \psi_{k} \rbrace \leq \mathbb{P}\lbrace A \not \subset \psi_{k} \rbrace$
,
$A\cap A_{1}=\emptyset $
,
$\vert A \vert = n$
,
$\vert A_{1}\vert = m$
, with (the non-trivial case)
$n + m \leq k$
, follows from the BK property and is equivalent to the log-concave inequality
$\binom{N}{N-k}\times\binom{N - n - m}{N-k} \leq \binom{N-n}{N-k}\times \binom{N - m}{N-k} $
. For
$n\geq 2$
it is easy to see that the validity of the inequality (2) depends on whether or not functions of the form
\begin{equation*}u\rightarrow \sum_{i_{1}=0}^{n_{1}}\cdots\sum_{i_{M}=0}^{n_{M}}\! \binom{N - u - (i_{1}+\cdots+i_{M})}{N-k-M} \end{equation*}
are log-concave, a question which does not seem to me to have been really investigated. Finally, the occurrence of log-concave criteria for negative dependence properties is not quite a surprise; see, for example, [Reference Pemantle24].
2. The CS decomposition and the basic determinantal point process
Following [Reference Paige and Wei23], the general CSD for a matrix Q from the unitary group U(N) specifies that, for any
$2\times 2$
partitioning
\begin{align*} & \qquad \quad c_{1} \ \ \quad \quad c_{2} \\ & Q = \begin{bmatrix} Q_{11} & \quad Q_{12} \\ Q_{21} & \quad Q_{22} \\ \end{bmatrix} \begin{matrix} r_{1} \\ r_{2} \\ \end{matrix}\end{align*}
with
$N=r_{1} + r_{2}=c_{1} + c_{2}$
, there exist unitary matrices
$U_{1}$
,
$U_{2}$
,
$V_{1}$
,
$V_{2}$
such that (here, all unnamed blocks of the matrices are always zero, and the superscript H represents the conjugate transpose)
\begin{align} & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, \, {c_1}\quad \quad \, \, {c_2} \\ & \left[ {\matrix{
{U_1^H} & {} \cr
{} & {\quad V_1^H} \cr
} } \right]Q\left[ {\matrix{
{{U_2}} & {} \cr
{} & {\quad {V_2}} \cr
} } \right] = \left[ {\matrix{
{U_1^H{Q_{11}}{U_2}} & {\quad U_1^H{Q_{12}}{V_2}} \cr
{V_1^H{Q_{21}}{U_2}} & {\quad V_1^H{Q_{22}}{V_2}} \cr
} } \right] = \left[ {\matrix{
{{D_{11}}} & {\quad {D_{12}}} \cr
{{D_{21}}} & {\quad {D_{22}}} \cr
} } \right]\matrix{
{{r_1}} \cr
{{r_2}} \cr
} ,\end{align}
where the matrices
\begin{equation*}D_{11}=\begin{bmatrix}I & &\\& \quad C & \\& & \quad \textbf{0}_{c}\\\end{bmatrix},\ D_{12}=\begin{bmatrix}\textbf{0}_{s}^{H} & &\\& \quad S & \\& & \quad I\\\end{bmatrix},\ D_{21}=\begin{bmatrix}\textbf{0}_{s} & &\\& \quad S & \\& & \quad I\\\end{bmatrix},\ D_{22}=\begin{bmatrix}I & &\\& \quad -C & \\& & \quad \textbf{0}_{c}^{H}\\\end{bmatrix}\end{equation*}
are diagonal with
$C\equiv \mathrm{diag}(\!\cos\theta_{1},\dots,\cos\theta_{s})$
,
$S \equiv \mathrm{diag}(\!\sin\theta_{1},\dots,\sin\theta_{s})$
,
$1>\cos\theta_{1}>\dots > \cos\theta_{s}>0$
. In some cases the matrices of zeros
$\textbf{0}_{s}$
and
$\textbf{0}_{c}$
, as well as the unit matrices I, could be nonexistent. See [Reference Paige and Wei23, Theorem 1] and the discussion that follows it for the full statement, and below for a detailed description given from Jordan’s geometrical point of view.
The CS decomposition is a deep result which has a long history going back to the work of Camille Jordan in 1875 on angles between subspaces in
$\mathbb{R}^{n}$
[Reference Jordan15]. Nowadays it is a popular tool in numerical linear algebra, useful for solving various questions such as, for example, constrained least squares problems, computing principal angles between subspaces, the generalized singular value decomposition, quantum computing, and more [Reference Bai3, Reference Gawlik, Nakatsukasa and Sutton6, Reference Golub and Van Loan11, Reference Goubault de Brugière, Baboulin, Valiron and Allouche12, Reference Paige and Wei23].
Now, let
$ E\subset \mathbb{C}^{N}$
be a vector space of dimension
$ 1 < p< N $
,
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
an orthonormal basis of E, and
$ \mathfrak{Z}^{\perp}=\{z^{p+1},\ldots,z^{N}\} $
an orthonormal basis of the orthogonal complement
$ E(\mathfrak{Z})^{\perp} $
. Fix
$1\leq n\leq p$
and consider the CSD of the partitioned unitary matrix
$Q=(z^{1},\ldots,z^{p}, z^{p+1},\ldots,z^{N}) $
:
\begin{align*} & \qquad\ p \quad \quad N-p \\Q & = \begin{bmatrix} Q_{11} & \quad Q_{12} \\ Q_{21} & \quad Q_{22} \\ \end{bmatrix} \begin{matrix} n \\ N-n \\ \end{matrix}\end{align*}
It follows from (3) that the column vectors of these two matrices,
\begin{equation*} \begin{bmatrix} U_{1}D_{11} \\ V_{1}D_{21} \\ \end{bmatrix} , \qquad \begin{bmatrix} U_{1}D_{12} \\ V_{1}D_{22} \\ \end{bmatrix}\end{equation*}
are respectively orthonormal bases of E and
$ E(\mathfrak{Z})^{\perp} $
.
Now we will detail the different cases given by these column vectors, which need to be distinguished. The description given here is somewhat lengthy but, in our opinion, useful for both theoretical and computational purposes. We denote by e(k),
$k=1,\dots,N $
, the null vector of the space
$\mathbb{C}^{k}$
. Note also the slight change with regard to angles appearing in CSD (3) which allows values 0 and
$\pi /2$
in order to recover all Jordan’s principal angles.
Case I:
$ n<p $
and
$p+n < N$
. There exist
-
a sequence
$ u^{1},\dots,u^{n} $
of orthonormal vectors in
$\mathbb{C}^{n}$
; -
three sequences of mutually orthonormal vectors in
$\mathbb{R}^{N-n}$
,
$ \mathfrak{V}=\{V^{1},\dots,V^{n}\} $
,
$\mathfrak{W}= \{W^{1},\dots,W^{p-n}\} $
, and
$ \tilde{\mathfrak{W}}=\{\tilde{W}^{1},\dots,\tilde{W}^{N-p-n}\} $
; -
Jordan angles
$0\leq \theta_{1}\leq \dots \leq \theta_{n}\leq \pi/2 $
such that, noting that
\begin{align*} z^{i} & = \begin{bmatrix} u^{i}\cos\theta_{i} \\ V^{i}\sin\theta_{i} \end{bmatrix} , \quad i=1,\dots,n ; \\ z^{i} & = \begin{bmatrix} e(n) \\ W^{i} \end{bmatrix} , \quad i=n+1,\dots,p ; \\ z^{p+i} & = \begin{bmatrix} u^{i}\sin\theta_{i} \\ -V^{i}\cos\theta_{i}) \end{bmatrix} , \quad i=1,\dots,n ; \\ z^{p+n+i} & = \begin{bmatrix} e(n) \\ \tilde{W}^{i} \end{bmatrix} , \quad i=1,\dots,N-p-n,\end{align*}
the sequence
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
is an orthonormal basis of E and the sequence
$ \mathfrak{Z}=\{z^{p+1},\ldots,z^{N}\} $
is an orthonormal basis of the orthogonal complement
$ E^{\perp} $
.
Case II:
$ n < p $
and
$ p+n > N $
. There exist
-
a sequence
$ u^{1},\dots,u^{n} $
of orthogonal vectors in
$\mathbb{C}^{n}$
; -
two sequences of mutually orthogonal vectors in
$\mathbb{C}^{N-n}$
,
$ \mathfrak{V}=\{V^{1},\dots,V^{N-p}\} $
and
$ \mathfrak{W}=\{W^{1},\dots,W^{p-n}\} $
; -
Jordan angles
$0 =\theta_{1}=\dots=\theta_{n+p-N}\leq \dots \leq \theta_{n}\leq \pi/2 $
such that, noting that
\begin{align*} z^{i} & = \begin{bmatrix} u^{i} \\ e(N-n) \end{bmatrix} , \quad i=1,\dots,n+p-N, \\ z^{i} & = \begin{bmatrix} u^{i}\cos\theta_{i} \\ V^{i-n-p +N}\sin\theta_{i}) \end{bmatrix} , \quad i=n+p-N +1,\dots,n ; \\ z^{n+i} & = \begin{bmatrix} e(n) \\ W^{i} \end{bmatrix} , \quad i=1,\dots,p-n ; \\ z^{p+i} & = \begin{bmatrix} u^{n+p-N+i}\sin\theta_{n+p-N+i} \\ -V^{i}\cos\theta_{n+p-N+i} \end{bmatrix} , \quad i=1,\dots,N-p,\end{align*}
the set
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
is an orthonormal basis of E and the set
$ \mathfrak{Z}=\{z^{p+1},\ldots,z^{N}\} $
is an orthonormal basis of
$ E^{\perp} $
.
Case III:
$ n < p $
and
$ p+n = N. $
There exist
-
a sequence
$ u^{1},\dots,u^{n} $
of orthogonal vectors in
$\mathbb{C}^{n}$
; -
two sequences of mutually orthogonal vectors in
$\mathbb{C}^{N-n}$
,
$ \mathfrak{V}=\{V^{1},\dots,V^{n}\} $
and
$ \mathfrak{W}=\{W^{1},\dots,W^{p-n}\} $
; -
Jordan angles
$0 \leq \theta_{1}\leq \dots \leq \theta_{n}\leq \pi/2 $
such that, noting that
\begin{align*} z^{i} & = \begin{bmatrix} u_{i}\cos\theta_{i} \\ V^{i}\sin\theta_{i} \end{bmatrix} , \quad i=1,\dots,n ; \\ z^{n+i} & = \begin{bmatrix} e(n) \\ W^{i} \end{bmatrix} , \quad i=1,\dots,p-n ; \\ z^{p+i} & = \begin{bmatrix} u^{i}\sin\theta_{i} \\ -V^{i}\cos\theta_{i} \end{bmatrix} , \quad i=1,\dots,n,\end{align*}
the set
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
is an orthonormal basis of E and the set
$ \mathfrak{Z}=\{z^{p+1},\ldots,z^{N}\} $
is an orthonormal basis of
$ E^{\perp}$
.
Case IV:
$ n = p $
. With the notations of cases I–III:
-
For
$ 2p< N $
,
\begin{align*} z^{i} & = \begin{bmatrix} u^{i}\cos\theta_{i} \\ V^{i}\sin\theta_{i} \end{bmatrix} , \quad i=1,\dots,p ; \\ z^{p+i} & = \begin{bmatrix} u^{i}\sin\theta_{i} \\ -V^{i}\cos\theta_{i} \end{bmatrix} , \quad i=1,\dots,p ; \\ z^{2p+i} & = \begin{bmatrix} e(n) \\ W^{i} \end{bmatrix} , \quad i=1,\dots,N-2p.\end{align*}
-
For
$ 2p > N $
,
\begin{align*} z^{i} & = \begin{bmatrix} u^{i} \\ e(N-n) \end{bmatrix} , \quad i=1,\dots,2p-N ; \\ z^{i} & = \begin{bmatrix} u^{i}\cos\theta_{i} \\ V^{i-2p+N}\sin\theta_{i} \end{bmatrix} , \quad i=2p-N+1,\dots,p; \\ z^{i} & = \begin{bmatrix} u^{i+p-N}\sin\theta_{i+p-N} \\ -V^{i-p}\cos\theta_{i+p-N} \end{bmatrix} , \quad i=p+1,\dots,N.\end{align*}
-
For
$ 2p = N $
,
\begin{equation*} z^{i} = \begin{bmatrix} u^{i}\cos\theta_{i} \\ V^{i}\sin\theta_{i}, \end{bmatrix} , \qquad z^{p+i} = \begin{bmatrix} u^{i}\sin\theta_{i} \\ -V^{i}\cos\theta_{i} \end{bmatrix} , \quad i=1,\dots,p. \end{equation*}
By reordering the rows of Q, the procedure described above works for every subset
$J=\{x_{1},\dots,x_{n}\}\subset \{1,\dots,N\} $
,
$ 1\leq n\leq p $
, and gives related bases of the spaces E and
$ E^{\perp} $
. Note that in the Euclidean context, i.e. for
$E\subset \mathbb{R}^{N}$
and the CSD applied to orthogonal matrices, the angles appearing in the CS decomposition (related to J) are the principal Jordan angles between the space E and the basic subspace
$\mathbb{R}_{J}^{N}=\{x=(x_{k})\in\mathbb{R}^{N} \,:\, x_{k}=0 \ \mathrm{if} \ k\notin J\}$
.
An important statistical application of principal angles is the canonical correlation analysis (CCA) of [Reference Hotelling13]. In order to develop a unified algebraic formulation of concepts in multivariate analysis (like, e.g., CCA), [Reference Afriat1] thoroughly studied (see also [Reference Miao and Ben-Israel20]) the geometry of subspaces in
$\mathbb{R} ^{N}$
in terms of orthogonal and oblique projectors, and introduced, among others, the notation of so-called multiplicative cosine and sine:
$\cos\!\big\{E,\mathbb{R}_{J}^{N}\big\} = \prod_{i=1}^{n}\cos\theta_{i}$
,
$\sin\!\big\{E,\mathbb{R}_{J}^{N}\big\} = \prod_{i=1}^{n}\sin\theta_{i}$
.
The basis of E given by the CSD is a pertinent tool for the study of the associated determinantal process. For example, it immediately gives the following proposition.
Proposition 1. For a set
$ J=\{x_{1},\dots,x_{n}\} $
,
$ n \leq p $
, we have:
-
(a)
$\mathbb{P}\{\vert J\cap \phi\vert = n\}=\prod_{i=1}^{n}\cos^{2}\theta_{i}$
, and, for
$k=1,\dots, n-1$
, (4)
\begin{equation} \mathbb{P}\{\vert J\cap \phi\vert = k\}=\sum_{1\leq i_{1}<\dots < i_{k}\leq n}\prod_{j=1}^{k}\cos^{2}\theta_{i_{j}} \prod_{j\not \in\{i_{1},\dots , i_{k}\} }\sin^{2}\theta_{j}. \end{equation}
-
(b)
(5)and, for
\begin{equation} \mathbb{P}\{\vert J\cap \phi^\mathrm{c}\vert = n\}=\prod_{i=1}^{n}\sin^{2}\theta_{i} \end{equation}
$k=1,\dots, n-1 $
,
\begin{equation*} \mathbb{P}\{\vert J\cap \phi^\mathrm{c}\vert = k\}=\sum_{1\leq i_{1}<\dots < i_{k}\leq n}\prod_{j=1}^{k}\sin^{2}\theta_{i_{j}} \prod_{j\not \in\{i_{1},\dots , i_{k}\} }\cos^{2}\theta_{j}. \end{equation*}
-
(c) If
$n < p$
and
$ \mathbb{P}\{ J \subset\phi\} >0$
, then the conditioned process
$\{\phi \mid J\subset\phi\}\setminus J$
is determinantal such that
$\{\phi \mid J\subset\phi\}\setminus J= \phi(\mathfrak{W})$
. -
(d) If
$N-p > n$
and
$ \mathbb{P}\{ J \subset\phi^\mathrm{c}\} >0$
, then the conditioned process
$\{\phi \mid J\subset\phi^\mathrm{c}\}$
is determinantal such that
$\{\phi\vert \ J\subset\phi^\mathrm{c}\} = \phi(\mathfrak{V}\cup \mathfrak{W})$
. -
(e) If
$ \mathbb{P}\{ J \subset\phi\} >0$
then, for all
$ K \subset \{1,\dots,N\}\setminus J $
,
$\mathbb{P}\{K\subset \phi(\mathfrak{W})\} \leq \mathbb{P}\{K\subset \phi\}$
, and if
$ \mathbb{P}\{ J \subset\phi^\mathrm{c}\} >0$
then
$\mathbb{P}\{K\subset \phi\} \leq \mathbb{P}\{K\subset \phi(\mathfrak{V}\cup \mathfrak{W})\}$
.
Remark 2. The fact that the conditioned processes
$\{\phi \mid J\subset\phi\}\setminus J$
and
$\{\phi \mid J\subset\phi^\mathrm{c}\}$
are determinantal, as well as the inequalities in Proposition 1(e), are well-known results proved in [Reference Lyons17].
Remark 3. Regarding Proposition 1(a) and (b), it was proved more generally in [Reference Hough, Krishnapur, Peres and Virag14, Theorem 5] that, for general determinantal processes with trace-class (both discrete and continuous case) kernels, the number of points in the process has the distribution of a sum of independent Bernoulli random variables.
More elaborate information can be obtained from this point of view.
Proposition 2. Consider the discrete determinantal process
$ \phi = \phi(\mathfrak{Z}) $
associated with a set
$ \mathfrak{Z}=\{z^{1},\ldots,z^{p}\} $
,
$ 1<p<N $
, of orthonormal vectors in
$ \mathbb{C}^{N} $
. Fix points
$J=\{x_{1},\dots,x_{n}\}\subset \{1,\dots,N\} $
,
$ 1\leq n\leq p $
, such that
$\mathbb{P}\{\{x_{2},\dots,x_{n}\}\subset \phi^\mathrm{c}\} > 0 $
. With the choice (to simplify the notation)
$x_{i}=i $
,
$ i=1,\dots,n $
, we have
\begin{align} & \Bigg\vert\langle z_{1},z_{n} \rangle + \sum_{k=1}^{n-2}({-}1)^{k}\sum_{2\leq i_{1}<\dots < i_{k}\leq n-1} \Bigg\langle z_{1}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg), z_{n}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg) \Bigg\rangle \Bigg\vert^{2} \nonumber \\ & = \mathbb{P}\{\{x_{2},\dots, x_{n}\}\subset \phi^\mathrm{c}\} \times \mathbb{P}\{\{x_{2},\dots, x_{n-1}\}\subset \phi^\mathrm{c}\} \nonumber \\ & \quad \times \big[ \mathbb{P}\{x_{1}\in \phi \vert \{x_{2},\dots,x_{n}\}\subset \phi^\mathrm{c}\} -\mathbb{P}\{x_{1}\in \phi \vert \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\}\big].\end{align}
Proof. The left- and right-hand sides of (6) do not depend of the choice of the basis of E. Choose the basis given by the CS decomposition related to the set
$J=\{2,\dots,n-1\}$
, with the reordering (
$2,\dots,n-1,1,n)^{t}$
and
$N-p - n +2>0$
(the general situation, case I). The first n coordinates of these bases have the following form:
\begin{align*} &\begin{matrix}2\\[4pt] \vdots\\[4pt] n-1\\[4pt] 1\\[4pt] n\end{matrix}\left[\begin{matrix}\cos \theta_{1}u^{1}_{1} \dots \cos \theta_{n-2}u^{n-2}_{1} & \quad 0 \quad \dots \quad 0 \\[4pt] \vdots \qquad & \quad \vdots \\[4pt] \cos \theta_{1}u^{1}_{n-2} \dots \cos \theta_{n-2}u^{n-2}_{n-2} & \quad 0 \quad \dots \quad 0 \\[4pt]
\sin\theta_{1}V^{1}_{1} \dots \sin \theta_{n-2}V^{n-2}_{1} & \quad W^{1}_{1} \dots W^{p-n+2}_{1} \\[4pt] \sin\theta_{1}V^{1}_{2} \dots \sin \theta_{n-2}V^{n-2}_{2} & \quad W^{1}_{2} \dots W^{p-n+2}_{2}\end{matrix}\right.\\[4pt] &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\left.\begin{matrix}\sin \theta_{1}u^{1}_{1} \dots \sin \theta_{n-2}u^{n-2}_{1} & \quad 0 \quad \dots \quad 0 \\[4pt] \vdots \qquad & \quad \vdots \\[4pt] \sin \theta_{1}u^{1}_{n-2} \dots \sin \theta_{n-2}u^{n-2}_{n-2} & \quad 0 \quad \dots \quad 0 \\[4pt] -\cos\theta_{1}V^{1}_{1} \dots -\cos \theta_{n-2}V^{n-2}_{1} & \quad \tilde{W}^{1}_{1} \dots \tilde{W}^{N+2-n-p}_{1} \\[4pt] -\cos\theta_{1}V^{1}_{2} \dots -\cos \theta_{n-2}V^{n-2}_{2} & \quad \tilde{W}^{1}_{2} \dots \tilde{W}^{N+2-n-p}_{2}\end{matrix}\right] .\end{align*}
It follows from Proposition 1 that
-
(a)
$\mathbb{P}\{\{x_{2},\dots, x_{n-1}\}\subset \phi^\mathrm{c}\}=\prod_{i=1}^{n-2}\sin^{2}\theta_{i}$
. -
(b)
$\mathbb{P}\{\{x_{2},\dots, x_{n}\}\subset \phi^\mathrm{c}\}=\prod_{i=1}^{n-2} \sin^{2}\theta_{i} \|\tilde{W}_{2}\|^{2} $
. -
(c)
$\mathbb{P}\{x_{1}\in \phi \vert \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\} = \|V_{1} \|^{2} + \| W_{1}\|^{2}$
. -
(d)
$\mathbb{P}\{x_{1}\in \phi \vert \{x_{2},\dots,x_{n}\}\subset \phi^\mathrm{c}\} = $
\begin{equation*} \begin{split} & \mathbb{P}\{x_{1}\in \phi, x_{n} \in \phi^\mathrm{c}\mid \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\}\times \frac{\mathbb{P}\{\{x_{2},\dots, x_{n-1}\}\subset \phi^\mathrm{c}\}}{\mathbb{P}\{\{x_{2},\dots, x_{n}\}\subset \phi^\mathrm{c}\}} \\ & =\big[ \mathbb{P}\{x_{1}\in \phi\mid \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\} - \mathbb{P}\{x_{1}\in \phi, x_{n} \in \phi\mid \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\}\big] \\ & \quad \times \frac{\mathbb{P}\{\{x_{2},\dots, x_{n-1}\}\subset \phi^\mathrm{c}\}}{\mathbb{P}\{\{x_{2},\dots, x_{n}\}\subset \phi^\mathrm{c}\}} \\ & = \big[\| V_{1}\|^{2} + \| W_{1}\|^{2} -\| (V_{1},W_{1})\wedge (V_{2},W_{2})\|^{2}\big] \times \frac{1}{\|\tilde{W}_{2}\|^{2}}. \end{split} \end{equation*}
From (a)–(d), an elementary computation gives the right-hand side of (6) (note that
$\| V_{2} \|^{2} + \| W_{2} \|^{2}+\| \tilde{W}_{2}\|^{2} = 1$
). Indeed, we get
\begin{align} & \mathbb{P}\{\{x_{2},\dots, x_{n}\}\subset \phi^\mathrm{c}\} \times \mathbb{P}\{\{x_{2},\dots, x_{n-1}\}\subset \phi^\mathrm{c}\} \nonumber \\ & \quad \times \big[ \mathbb{P}\{x_{1}\in \phi \mid \{x_{2},\dots,x_{n}\}\subset \phi^\mathrm{c}\} -\mathbb{P}\{x_{1}\in \phi \mid \{x_{2},\dots,x_{n-1}\}\subset \phi^\mathrm{c}\}\big] \nonumber \\ & = \prod_{i=2}^{n-2}\sin^{4}\theta_{i}\big[ (\| V_{1} \|^{2} + \| W_{1} \|^{2}) (1 - \| \tilde{W}_{2}\|^{2})- \|(V_{1},W_{1})\wedge (V_{2},W_{2})\|^{2}\big] \nonumber \\ & = \prod_{i=1}^{n-2}\sin^{4}\theta_{i}\vert \langle(V_{1},W_{1}),(V_{2},W_{2})\rangle\vert^{2}. \end{align}
To compute the left-hand side of (6), we write
$z_{1}^{0}=(\!\sin\theta_{1}V^{1}_{1}, \dots, \sin \theta_{n-2}V^{n-2}_{1})$
,
$z_{n}^{0}=(\!\sin\theta_{1}V^{1}_{2}, \dots, \sin \theta_{n-2}V^{n-2}_{2})$
, and
$\tilde{z}^{i}=(\!\cos \theta_{i}u^{i}_{1}, \dots, \cos \theta_{i}u^{i}_{n-2},0)^{t}$
. Observe that
\begin{equation} \langle z_{1},z_{n} \rangle + \sum_{k=1}^{n-2}({-}1)^{k}\sum_{2\leq i_{1}<\dots < i_{k}\leq n-1} \Bigg\langle z_{1}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg), z_{n}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg) \Bigg\rangle = A + B , \end{equation}
with
\begin{align} A & = \langle z_{1}^{0},z_{n}^{0} \rangle + \sum_{k=1}^{n-2}({-}1)^{k}\sum_{2\leq i_{1}<\dots < i_{k}\leq n-1} \Bigg\langle z_{1}^{0}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg), z_{n}^{0}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg) \Bigg\rangle ,\\ B & = \langle W_{1},W_{2}\rangle \Bigg(1 + \sum_{k=1}^{n-2}({-}1)^{k}\sum_{1\leq i_{1}<\dots < i_{k}\leq n-2} \bigg\| \bigwedge_{j=1}^{k}\tilde{z}^{i_{j}} \bigg\|^{2}\Bigg) . \nonumber \end{align}
Obviously,
$\| \bigwedge_{j=1}^{k}\tilde{z}^{i_{j}} \|^{2} = \prod_{j=1}^{k}\cos^{2}\theta_{i_{j}}$
, and thus
\begin{equation*} 1 + \sum_{k=1}^{n-2}({-}1)^{k}\sum_{1\leq i_{1}<\dots < i_{k}\leq n-2} \bigg\| \bigwedge_{j=1}^{k}\tilde{z}^{i_{j}} \bigg\|^{2} =\prod_{i=1}^{n-2}(1-\cos^{2}\theta_{i})=\prod_{i=1}^{n-2 }\sin^{2}\theta_{i} , \end{equation*}
and consequently
\begin{equation} B = \langle W_{1},W_{2}\rangle\prod_{i=1}^{n-2}\sin^{2}\theta_{i}. \end{equation}
In order to compute A we introduce
$\tilde{z}^{i,l}=(\!\cos \theta_{i}u^{i}_{1}, \dots, \cos \theta_{i}u^{i}_{n-2},\sin\theta_{i}V^{i}_{l})^{t}$
,
$l=1,2$
and
$i=1,\dots,n-2$
. A little thought shows that, for
$k\geq1$
,
\begin{align} & \sum_{2\leq i_{1}\lt \dots \lt i_{k}\leq n-1} \Bigg\langle z_{1}^{0}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg), z_{n}^{0}\wedge\Bigg(\bigwedge_{j=1}^{k} z_{i_{j}}\Bigg) \Bigg \rangle \\
& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = \sum_{1\leq i_{1}\lt \dots \lt i_{k+1}\leq n-2} \Bigg[ \Bigg\langle \bigwedge_{j=1}^{k+1} \tilde{z}^{i_{j},1}, \bigwedge_{j=1}^{k+1} \tilde{z}^{i_{j},2} \Bigg \rangle - \bigg\| \bigwedge_{j=1}^{k+1} \tilde{z}^{i_{j}}\bigg\|^{2}\Bigg]. \end{align}
Moreover, we have
\begin{align*} & \bigwedge_{j=1}^{k} \tilde{z}^{i_{j},l} = \bigwedge_{j=1}^{k} \big( \tilde{z}^{i_{j}}+(e(n-2),\sin\theta_{i_{j}}V^{i_{j}}_{l})^{t}\big) \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad= \bigwedge_{j=1}^{k} \tilde{z}^{i_{j}} + \sum_{j=1}^{k}({-}1)^{j+1}(e(n-2),\sin\theta_{i_{j}}V^{i_{j}}_{l})^{t})\wedge\Bigg(\bigwedge_{s=1, s\neq j}^{k} \tilde{z}^{i_{s}} \Bigg).\end{align*}
From the orthogonality properties of the relevant multivectors we obtain, from the last equation,
\begin{align} & \Bigg\langle \bigwedge_{j=1}^{k} \tilde{z}^{i_{j},1}, \bigwedge_{j=1}^{k} \tilde{z}^{i_{j},2} \Bigg \rangle - \bigg\| \bigwedge_{j=1}^{k} \tilde{z}^{i_{j}}\bigg\|^{2} \nonumber \\ & = \sum_{j=1}^{k}\Bigg\langle (e(n-2),\sin\theta_{i_{j}}V^{i_{j}}_{1})^{t}\wedge\Bigg(\bigwedge_{s=1, s\neq j}^{k} \tilde{z}^{i_{s}} \Bigg), (e(n-2),\sin\theta_{i_{j}}V^{i_{j}}_{2})^{t}\wedge\Bigg(\bigwedge_{s=1, s\neq j}^{k} \tilde{z}^{i_{s}} \Bigg) \Bigg \rangle \nonumber \\ & = \sum_{j=1}^{k}V^{i_{j}}_{1}V^{i_{j}}_{2}\sin^{2}\theta_{i_{j}} \bigg\| \bigwedge_{s=1, s\neq j}^{k} \tilde{z}^{i_{s}}\bigg\|^{2} = \sum_{j=1}^{k}V^{i_{j}}_{1}V^{i_{j}}_{2}\sin^{2}\theta_{i_{j}} \prod_{s=1, s\neq j}^{k}\cos^{2}\theta{i_{s}}. \end{align}
From (9), (11), and (12), an elementary computation gives
\begin{equation*} A=\sum_{i=1}^{n-2}V^{i}_{1}V^{i}_{2}\sin^{2}\theta_{i} \prod_{j=1,j\neq i}^{n-2}(1- \cos^{2}\theta_{j}) = \langle V_{1},V_{2}\rangle\prod_{i=1}^{n-2 }\sin^{2}\theta_{i} , \end{equation*}
and, with (10),
$A+B=\langle (V_{1},W_{1}),(V_{2},W_{2})\rangle\prod_{i=1}^{n-2}\sin^{2}\theta_{i}$
. This and (7) prove Proposition 2. Note that from the last equation we also get that (8) is identified as a scalar product.
For further results by using the CSD, and for some extensions of Proposition 2, see [Reference Goldman10].
3. The BK inequality for increasing events generated by disjoint sets
Let
$\mathfrak{A}, \mathfrak{B} \subset 2^{\mathcal{N}}$
,
$\mathcal{N}=\{1,\dots,N\}$
, be a pair of increasing events, and suppose (obviously) that
$\emptyset \notin \mathfrak{A}\cup \mathfrak{B}$
. The events being increasing, there exist two minimal sets
$ S_{1}= S(\mathfrak{A})=\{A_{i},i=1,\dots, n_{1}\}\subset \mathfrak{A}$
and
$ S_{2}=S(\mathfrak{B})=\{B_{i},i=1,\dots, n_{2}\}\subset \mathfrak{B} $
such that
$ A\in \mathfrak{A}$
if and only if there exists
$A_{i}$
such that
$A\supset A_{i} $
, and
$ B\in \mathfrak{B}$
if and only if there exists
$B_{i}$
such that
$B\supset B_{i}$
. The sets
$A_{i}$
and
$B_{i}$
are minimal in the sense that none of
$A\in \mathfrak{A} $
(resp.
$B\in \mathfrak{B} $
) is stricly included in
$A_{i}$
(resp. in
$B_{i}$
).
Consider now a basic determinantal process
$\phi$
on
$\mathcal{N}$
. In the particular case when
$A \cap B =\emptyset$
for all
$A\in S _{1}$
and
$B\in S_{2}$
, we at once have
$\mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\}= \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{B}\}$
, and thus the BK inequality (1) becomes
$\mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\}\leq\mathbb{P}\{\phi \in \mathfrak{A}\}\times\mathbb{P}\{\phi \in \mathfrak{B}\}$
, which is a negative association inequality. It was proved in [Reference Lyons17, Reference Lyons18] that determinantal processes have negative association, meaning that this inequality is fulfilled.
In the general situation it is helpful to reformulate the BK inequality (1) as follows.
Proposition 3. The inequality (1) is satisfied if and only if
\begin{equation} \mathbb{P}\{\phi \notin \mathfrak{A}\cup\mathfrak{B}\} \leq \mathbb{P}\{\phi \notin \mathfrak{A}\} \times \mathbb{P}\{\phi \notin \mathfrak{B}\} + \mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\} - \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{B}\}. \end{equation}
Proof. Observe that
\begin{equation*} \nonumber \begin{split} \mathbb{P}\{\phi \notin \mathfrak{A}\cup\mathfrak{B}\} & =1 - \mathbb{P}\{\phi \in \mathfrak{A}\cup\mathfrak{B}\} =1- \mathbb{P}\{\phi \in \mathfrak{A}\} - \mathbb{P}\{\phi \in \mathfrak{B}\} +\mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\}\\ & =\mathbb{P}\{\phi \notin \mathfrak{A}\} \times \mathbb{P}\{\phi \notin \mathfrak{B}\} - \mathbb{P}\{\phi \in \mathfrak{A}\} \times \mathbb{P}\{\phi \in \mathfrak{B}\} + \mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\}. \end{split} \end{equation*}
Thus,
$\mathbb{P}\{\phi \notin \mathfrak{A}\cup\mathfrak{B}\} - \mathbb{P}\{\phi \notin \mathfrak{A}\} \times \mathbb{P}\{\phi \notin \mathfrak{B}\} -\mathbb{P}\{\phi \in \mathfrak{A}\cap\mathfrak{B}\} + \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{B}\} \leq 0$
if and only if
$\mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{B}\} - \mathbb{P}\{\phi \in \mathfrak{A}\} \times \mathbb{P}\{\phi \in \mathfrak{B}\} \leq 0$
.
Suppose now that
$\mathfrak{A}=\mathfrak{B}$
. The inequality in (13) becomes
\begin{equation} \mathbb{P}\{\phi \notin \mathfrak{A}\} \leq \mathbb{P}\{\phi \notin \mathfrak{A}\}^{2} + \mathbb{P}\{\phi \in \mathfrak{A}\} - \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{A}\}.\end{equation}
If the sets of
$S(\mathfrak{A})=\{A_{1},\dots,A_{n}\}$
are disjoint, that is if
$A_{i}\cap A_{j}=\emptyset $
for all
$ i\neq j $
, then
$ \{\phi \in \mathfrak{A}\backslash(\mathfrak{A}\circ\mathfrak{A})\}=\bigcup_{i=1}^{n}\lbrace A_{i}\subset \phi,A_{j}\not \subset\phi, \text{for all}\ j \neq i\rbrace$
. Therefore,
\begin{align*} \mathbb{P}\{\phi \in \mathfrak{A}\} - \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{A}\} & = \mathbb{P}\{\mathfrak{A}\backslash(\mathfrak{A}\circ\mathfrak{A})\} \\ & = \sum_{i=1}^{n}\mathbb{P}\lbrace A_{i}\subset \phi, A_{j}\not \subset \phi, \text{for all}\ j \neq i\rbrace \\ & = \sum_{i=1}^{n}\big[ \mathbb{P}\lbrace A_{j}\not \subset \phi, \text{for all}\ j \neq i\rbrace - \mathbb{P}\lbrace A_{i}\not \subset\phi, \text{for all}\ i=1,\dots,n\rbrace \big] \\ & = \sum_{i=1}^{n}\mathbb{P}\lbrace A_{j}\not \subset \phi, \text{for all}\ j \neq i\rbrace - n\mathbb{P}\lbrace \phi \notin \mathfrak{A}\rbrace ,\end{align*}
and (14) takes the form
\begin{equation} (n+1)\mathbb{P}\{\phi \notin \mathfrak{A}\} \leq \mathbb{P}\{\phi \notin \mathfrak{A}\}^{2} + \sum_{i=1}^{n}\mathbb{P}\lbrace A_{j}\not \subset \phi, \text{for all}\ j \neq i\rbrace.\end{equation}
Now fix
$n_{0}\geq 2$
, and suppose that Conjecture 1 is fulfilled for all
$2\leq n\leq n_{0} $
.
Lemma 1. Under this hypothesis, for all
$A_{i}$
,
$i=1,\dots,n$
, disjoint subsets of
$\{1,\dots, N\}$
with
$2\leq n\leq n_{0} $
and such that
$\mathbb{P}\lbrace A_{i} \not \subset \phi, {for\ all}\ i=1,\dots,n\rbrace>0$
, we have
\begin{equation} \mathbb{P}\lbrace A_{i} \not \subset \phi, {for\ all}\ i=1,\dots,n\rbrace^{n-1} \leq \prod_{i=1}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, {for\ all}\ j\neq i\rbrace. \end{equation}
Proof. For
$n=2$
the inequality (16) is the well-known correlation inequality. For
$n > 2$
, applying (2) we get
$\prod _{k=2}^{n}\mathbb{P}\lbrace A_{k}\not \subset \phi \mid A_{j} \not \subset \phi, \text{for all}\ j \neq k \rbrace \leq \prod_{k=2}^{n}\mathbb{P}\lbrace A_{k} \not \subset \phi \mid A_{j} \not \subset \phi, \text{for all}\ j \neq 1, k \rbrace$
if and only if
\begin{equation*} \dfrac{\mathbb{P}\lbrace A_{i}\not \subset \phi , \text{for all}\ i=1,\dots,n \rbrace ^{n-1}}{\prod_{i=1}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, \text{for all}\ j \neq i\rbrace} \leq \dfrac{\mathbb{P}\lbrace A_{i} \not \subset \phi, \text{for all}\ i \neq 1 \rbrace ^{n-2}}{\prod_{k=2}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, \text{for all}\ j \neq 1,k\rbrace} , \end{equation*}
and thus Lemma 1 follows by induction.
We will need the following elementary lemma. Its proof being trivial, we omit it.
Lemma 2. For all
$ 0<a\leq 1 $
and
$n>0$
,
$(n+1) -a - na^{-{1}/{n}}\leq 0$
.
Theorem 1. Let
$\mathfrak{A}$
be an increasing event generated by disjoint sets
$ A_{1},\dots,A_{n}$
. Suppose that Conjecture 1 holds. Then
\begin{equation} \mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{A}\} \leq \mathbb{P}\{\phi \in \mathfrak{A}\}^{2}. \end{equation}
Proof. We have to prove (15). By Lemma 2 we obtain
$(n+1)\mathbb{P}\{\phi \notin \mathfrak{A}\} \leq \mathbb{P}\{\phi \notin \mathfrak{A}\}^{2} + n\mathbb{P}\{\phi \notin \mathfrak{A}\}^{({n-1})/{n}}$
. Lemma 1 implies that
$\mathbb{P}\{\phi \notin \mathfrak{A}\}^{n-1} =\mathbb{P}\lbrace A_{i} \not \subset \phi, \text{for all}\ i=1,\dots,n\rbrace^{n-1} \leq \prod_{i=1}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, \text{for all}\ j\neq i\rbrace$
, so it remains to apply the arithmetic–geometric mean inequality,
$n\prod_{i=1}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, \text{for all}\ j\neq i\rbrace^{\frac{1}{n}} \leq \sum_{i=1}^{n}\mathbb{P}\lbrace A_{j} \not \subset \phi, \text{for all}$
$j\neq i\rbrace$
, to obtain (15) as desired.
Remark 4. Consider an event
$\tilde{S}=\{ D_{1},\dots,D_{n_{0}}\}\subset 2^{\mathcal{N}}$
of disjoint sets such that
$\mathbb{P}\{D \not \subset \phi, $
$\text{for all}\ D \in \tilde{S} \}>0$
. Write
$\psi = \{\phi \mid D \not \subset \phi, \text{for all}\ D \in \tilde{S}\}$
. If Conjecture 1 holds, then it is obvious that the inequality (2) is also satisfied for the conditioned process
$\psi$
provided that the sets occuring in (2) are disjoint from those in
$\tilde{S}$
. Consequently, if
$\mathfrak{A}$
is an increasing event generated by disjoint sets
$ A_{1},\dots,A_{n} $
such that
$ A_{i}\cap D = \emptyset$
for all
$i=1,\dots,n$
and
$D\in \tilde{S}$
, then we obtain
\begin{equation} \mathbb{P}\{\psi \in \mathfrak{A}\circ\mathfrak{A}\} \leq \mathbb{P}\{\psi \in \mathfrak{A}\}^{2}. \end{equation}
Let
$S_{1}=\{A_{i},i=1,\dots, n_{1}\}$
,
$S_{2}=\{B_{i},i=1,\dots, n_{2}\}$
, and
$S=\{C_{i},i=1,\dots, n_{3}\}$
be events such that all sets in
$S_{1}\cup S_{2}\cup S \subset 2^{\mathcal{N}}$
are pairwise disjoint.
Theorem 2. Suppose that Conjecture 1 holds. Then, for increasing events
$\mathfrak{A}$
and
$\mathfrak{B}$
such that
$S(\mathfrak{A})=S_{1}\cup S$
and
$S(\mathfrak{B})=S_{2}\cup S$
, we have
\begin{equation} \mathbb{P}\{\psi \in \mathfrak{A}\circ\mathfrak{B}\} \leq \mathbb{P}\{\psi \in \mathfrak{A}\}\times \mathbb{P}\{\psi \in \mathfrak{B}\} , \end{equation}
where
$\psi =\{ \phi \mid D \not \subset \phi, \text{for all}\ D \in \tilde{S}\}$
and all sets in
$S_{1}\cup S_{2}\cup S \cup \tilde{S}\subset 2^{\mathcal{N}}$
are pairwise disjoint.
Proof. The proof proceeds by induction using Theorem 1 and, starting from (18), applying Lemma 3 step by step.
Lemma 3. Fix
$S_{1}$
,
$S_{2}$
, and S, and suppose that the BK inequality (19) is fulfilled for all conditioned processes
$\psi$
subjected to the conditions of Theorem 2. Fix
$A \subset \mathcal{N}$
,
$A\neq \emptyset$
, such that
$A\cap A' = \emptyset$
for all
$A' \in S_{1}\cup S_{2}\cup S $
. Denote by
$ \tilde{\mathfrak{A}}=\sigma \{A,\mathfrak{A}\}$
the increasing event generated by A and
$ \mathfrak{A}$
. Then, the BK inequality
\begin{equation} \mathbb{P}\{\psi \in \tilde{\mathfrak{A}}\circ\mathfrak{B}\} \leq \mathbb{P}\{\psi \in \tilde{\mathfrak{A}}\} \times \mathbb{P}\{\psi \in \mathfrak{B}\} \end{equation}
is satisfied for all conditioned processes
$\psi = \{\phi \mid D \not \subset \phi \ {for\ all}\ D \in \tilde{S}\}$
such that all sets of
$S(\tilde{\mathfrak{A}})\cup S_{2}\cup S \cup \tilde{S}\subset 2^{\mathcal{N}}$
are pairwise disjoint.
Proof. By (13), we may suppose that
$\mathbb{P}\{A' \not \subset \psi, \text{for all}\ A' \in S(\tilde{\mathfrak{A}})\cup S_{2}\cup S \cup \tilde{S}\}>0$
. We have
$\lbrace \psi \in \mathfrak{A}\cap\mathfrak{B} \setminus \mathfrak{A}\circ\mathfrak{B}\rbrace= \cup_{C\in S}\lbrace C \subset \psi, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S, A'\neq C\rbrace$
and
\begin{align*} & \lbrace \psi \in\tilde{\mathfrak{A}}\cap\mathfrak{B} \setminus \tilde{\mathfrak{A}}\circ\mathfrak{B}\rbrace \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = \cup_{C\in S}\lbrace C \subset \psi, A\not \subset \psi, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S, A'\neq C\rbrace. \end{align*}
Formulas (13) and (21) imply that the BK inequality (20) can be written as
\begin{align*} & \mathbb{P}\{ A\not \subset \psi, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S\} \\ & \leq \mathbb{P}\{A\not \subset \psi, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S\} \times \mathbb{P}\{ A' \not \subset \psi, \text{for all}\ A' \in S_{2}\cup S\} \\ & \quad + \sum_{C\in S}\mathbb{P}\lbrace C \subset \psi, A\not \subset \psi, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S , A'\neq C \rbrace \end{align*}
or, introducing the process
$ \psi_{0} = \{\phi \mid A \not\subset \phi, A' \not\subset \phi \ \text{for all}\ A' \in \tilde{S}\}$
, as
\begin{align} & \mathbb{P}\{ A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S\} \nonumber \\ & \leq \mathbb{P}\{A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{1}\cup S\} \times \mathbb{P}\{ A' \not \subset \psi, \text{for all}\ A' \in S_{2}\cup S\} \nonumber \\ & \quad + \sum_{C\in S}\mathbb{P}\lbrace C \subset \psi_{0}, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S, A'\neq C \rbrace. \end{align}
The stated hypotheses imply that
\begin{align} & \mathbb{P}\{ A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S\} \nonumber \\ & \leq \mathbb{P}\{A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{1}\cup S\} \times \mathbb{P}\{ A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{2}\cup S\} \nonumber \\ & \quad + \sum_{C\in S}\mathbb{P}\lbrace C \subset \psi_{0}, A' \not \subset \psi, \text{for all}\ A' \in S_{1}\cup S_{2}\cup S , A'\neq C \rbrace. \end{align}
It is easy to see that Conjecture 1 implies the inequality
$\mathbb{P}\{ A' \not \subset \psi_{0}, \text{for all}\ A' \in S_{2}\cup S\} \leq \mathbb{P}\{ A' \not \subset \psi, \text{for all}\ A' \in S_{2}\cup S\}$
, and by this and (23) we obtain (22), which finishes the proof of Lemma 3.
4. The BK inequality for increasing events generated by simple points
As mentioned in the introduction, the inequality (2) is satisfied when the occurring sets are reduced to being simple points. This follows easily, for example, from Proposition 1. Therefore, Theorem 2 implies the following result.
Theorem 3. Let
$\mathfrak{A}$
,
$\mathfrak{B}$
be increasing events generated by simple points. The BK inequality
$\mathbb{P}\{\phi \in \mathfrak{A}\circ\mathfrak{B}\} \leq \mathbb{P}\{\phi \in \mathfrak{A}\}\times \mathbb{P}\{\phi \in \mathfrak{B}\} $
is then satisfied for all determinantal discrete processes
$ \phi $
associated with sets of orthonormal vectors of
$ \mathbb{C}^{N} $
.
Remark 5. For sets reduced to being simple points, the key inequality (16) can be seen from the point of view given by the CSD. Indeed, consider the CSD in case I applied to
$J=\{x_{1},\dots,x_{n}\}$
and, accordingly, let
$ v^{j}=(v_{1}^{j},\dots,v_{n}^{j})^{t}$
,
$v_{i}^{j}=(\!\sin\theta_{j})u_{i}^{j}$
,
$i,j=1,\dots,n$
be the vectors such that
$\mathbb{P}\{\{x_{i_{1}},\dots,x_{i_{k}}\} \subset \phi^\mathrm{c}\} = \big\| \bigwedge_{j=1}^{k}v_{i_{j}}\big\| ^{2}$
for all
$\{x_{i_{1}},\dots,x_{i_{k}}\} \subset J$
. Write
$\tilde{v}_{i}=\bigwedge\limits_{j\neq i}v_{j}=(\tilde{v}_{i}^{1},\dots,\tilde{v}_{i}^{n}) \in \mathbb{C}^{n}$
,
$i=1,\dots,n$
, where
$\tilde{v}_{i}^{j}= \prod_{k\neq j}\sin\theta_{k}\times \tilde{u}_{i}^{j}$
and
$\tilde{u}_{i}^{j}$
is the
$(i,n-j+1)$
-minor of the unitary matrix
$U=(u^{j}_{i})_{i,j=1,\dots,n}$
. By (5), we obtain
\begin{align*} \mathbb{P}\{x_{i}\in \phi^\mathrm{c}, i=1,\dots,n\}^{n-1} & =\prod_{i=1}^{n}(\!\sin\theta_{i})^{2(n-1)} \\ & = \bigg\| \bigwedge_{i=1}^{n}v_{i}\bigg\| ^{2(n-1)} \\ & = \bigg\| \bigwedge_{i=1}^{n}\tilde{v}_{i}\bigg\|^{2} \\ & \leq \prod_{i=1}^{n}\| \tilde{v}_{i}\|^{2} = \prod_{i=1}^{n}\mathbb{P}\lbrace x_{j}\in \phi^\mathrm{c}, \text{for all}\ j \neq i\rbrace . \end{align*}
Remark 6. It was pointed out to us that for an increasing event
$\mathfrak{A}$
generated by simple points
$S= \{x_{1},\dots,x_{n}\}$
, the inequality (17), which can be read as
\begin{equation} \mathbb{P}\{\vert S\cap \phi\vert\geq 2\}\leq \mathbb{P}\{\vert S\cap \phi\vert\geq 1\}^{2} , \end{equation}
can also be obtained by a direct computation from (4) of Proposition 1 and, moreover, if we consider the product measure
$\mu=\otimes_{i=1}^{n}((\!\cos^{2}\theta_{i})\delta_{1} + (\!\sin^{2}\theta_{i})\delta_{0})$
on the product space
$E=\{0,1\}^{n}$
and increasing events
$\mathfrak{A}_{i} = \{a=(a_{j})\in E$
such that
$\sum_{j=1}^{n}a_{j} \geq i\}$
,
$i=0,\dots,n$
, then the formulas in (4) imply that
$P\{\vert S\cap \phi\vert\geq i\} = \mu (\mathfrak{A}_{i})$
. From [Reference van den Berg and Kesten26, Theorem 3.3], we get
\begin{equation} \mathbb{P}\{\vert S\cap \phi\vert\geq i+j\}\leq \mathbb{P}\{\vert S\cap \phi\vert\geq i\}\times \mathbb{P}\{\vert S\cap \phi\vert\geq j\}, \qquad 2\leq i+ j \leq n. \end{equation}
Furthermore, note that by Remark 3 the inequalities (24) and (25) are still valid for general determinantal processes (both discrete and continuous) taking for S a Borel set.
5. Extensions and concluding remarks
Theorem 3 can be easily extended in the setting of general discrete determinantal processes. From the construction given in [Reference Lyons18, Paragraph 2.2], which starts from the basic processes, it follows at once that Theorems 1 and 2 are valid (the generated sets
$S(\mathfrak{A})$
and
$S(\mathfrak{B})$
being finite or infinite) for determinantal point processes defined on denumerable sets
$\mathcal{E}$
and associated with closed subspaces of
$l^{2}(\mathcal{E})$
. Now, let
$\phi$
be such a process on
$\mathcal{E}$
. Fix
$\mathcal{F} \subset \mathcal{E}$
and consider the process
$\psi= \phi \cap \mathcal{F}$
.
Let
$\mathfrak{A}, \mathfrak{B}\subset 2^{\mathcal{F}}$
,
$\tilde{\mathfrak{A}},\tilde{\mathfrak{B}}\subset 2^{\mathcal{E}}$
be the increasing events generated respectively by
$S_{1}=S(\mathfrak{A})=S(\tilde{\mathfrak{A}}) \subset \mathcal{F}$
and
$S_{2}=S(\mathfrak{B})=S(\tilde{\mathfrak{B}}) \subset \mathcal{F}$
. The BK inequalities for
$\phi $
,
$\tilde{\mathfrak{A}}$
,
$\tilde{\mathfrak{B}}$
and
$\psi$
,
$\mathfrak{A}$
,
$\mathfrak{B}$
involve only the generating sets
$S_{1}$
and
$S_{2}$
. Consequently, Theorem 3 is valid for
$\psi$
as well. To finish, just note that, by [Reference Lyons18, Paragraph 2.2], discrete determinantal processes associated with positive contractions (the general case) are of the form
$\psi= \phi \cap \mathcal{F}$
.
By the transference principle [Reference Lyons18, Section 3.6], Theorem 3 could also be extended to the continuous case, but this is of little use because in the continuous setting the intensity measures related to determinantal processes of interest are of diffusive type, which implies that
$\mathbb{P}\{x \in \phi \}=0$
for points x (however, as mentioned in Remark 6, inequalities (24) and (25) still hold).
Acknowledgement
I thank the anonymous referees for their constructive comments, especially for a pertinent question about the validity of Conjecture 1 for the non-determinantal point processes, which led to looking at the process described in Remark 1.
Funding information
There are no funding bodies to thank relating to this creation of this article.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
