A note related to the CS decomposition and the BK inequality for discrete determinantal processes

TL;DR

This paper establishes that the BK inequality holds for increasing events in discrete determinantal processes and explores their relationship with the CS decomposition, providing new insights into their structure.

Contribution

It introduces a proof of the BK inequality for discrete determinantal processes and links these processes to the CS decomposition, offering novel theoretical connections.

Findings

BK inequality holds for increasing events in discrete determinantal processes

Established a relationship between determinantal processes and the CS decomposition

Provided elementary insights into the structure of these processes

Abstract

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We give also some elementary, but nonetheless appealing relationship, between a discrete determinantal process and the well-known CS decomposition.

Equations

$$z^i=(z_1^i,\dots ,z_N^i{)}^t,i=1,\dots ,p$$

$$z_i=(z_i^1,\dots ,z_i^p),i=1,\dots ,N.$$

$$P\{\{i_1,\dots ,i_p\}=\varphi \}=\mid (\underset{i=1}{\overset{p}{\bigwedge }}{z}^i{)}_{\{i_1,\dots ,i_p\}}{\mid }^2=[\det ((z_{i_j}^k{)}_{k,j=1,\dots ,p}){]}^2$$

$$P\{\{i_1,\dots ,i_k\}\subset \varphi \}=\parallel \underset{j=1}{\overset{k}{\bigwedge }}{z}_{i_j}{\parallel }^2$$

$$\mid (\underset{i=1}{\overset{p}{\bigwedge }}{z}^i{)}_{\{i_1,\dots ,i_p\}}\mid =\mid (\underset{i=1}{\overset{p}{\bigwedge }}{\tilde{z}}^i{)}_{\{i_1,\dots ,i_p\}}\mid$$

$$\varphi ({\mathfrak{Z}}^{\perp })=\{1,\dots ,N\}\setminus \varphi (\mathfrak{Z}).$$

$$\mathfrak{A}\circ \mathfrak{B}=\{K\subset \mathcal{N}:\exists L\in \mathfrak{A},M\in \mathfrak{B},L,M\ne \varnothing ,L\cap M=\varnothing ,K\supset L\cup M\}.$$

$$P\{\psi \in \mathfrak{A}\circ \mathfrak{B}\}\le P\{\psi \in \mathfrak{A}\}\times P\{\psi \in \mathfrak{B}\}$$

$$\begin{array}{rl}& P\{A\not\subset \varphi \mid A_i\not\subset \varphi ,\forall i=1,\dots ,n\}\\ & \le P\{A\not\subset \varphi \mid A_i\not\subset \varphi ,\forall i=1,\dots ,n-1\}\end{array}$$

$$\begin{gathered} P\{i\notin {\psi }_k\mid i_j\notin {\psi }_k,\forall j=1,\dots ,n\} \\ =\frac{\left(\genfrac{}{}{0ex}{}{N-n-1}{k}\right)}{\left(\genfrac{}{}{0ex}{}{N-n}{k}\right)} \end{gathered}$$

$$u\to \sum _{i_1=0}^{n_1}...\sum _{i_M=0}^{n_M}\left(\genfrac{}{}{0ex}{}{N-u-(i_1+...+i_M)}{N-k-M}\right)$$

$$\begin{array}{rl}& c_1{c}_2\\ & Q=\left[\begin{array}{cc}{Q}_{11}& Q_{12}\\ Q_{21}& Q_{22}\end{array}\right]\begin{array}{c}{r}_1\\ r_2\end{array}\end{array},N=r_1+r_2=c_1+c_2,$$

$$\begin{bmatrix}U_{1}^{H}&\\ &V_{1}^{H}\\ \end{bmatrix}Q\begin{bmatrix}U_{2}&\\ &V_{2}\\ \end{bmatrix}=\left[\begin{array}[]{c|c}U_{1}^{H}Q_{11}U_{2}&U_{1}^{H}Q_{12}V_{2}\\ \hline\cr V_{1}^{H}Q_{21}U_{2}&V_{1}^{H}Q_{22}V_{2}\\ \end{array}\right]=\overset{c_{1}\quad c_{2}}{\left[\begin{array}[]{c|c}D_{11}&D_{12}\\ \hline\cr D_{21}&D_{22}\\ \end{array}\right]}\begin{matrix}r_{1}\\ r_{2}\\ \end{matrix}$$

$$D_{11}=\left[\begin{array}{ccc}I& & \\ & C& \\ & & 0_c\end{array}\right],D_{12}=\left[\begin{array}{ccc}{0}_s^H& & \\ & S& \\ & & I\end{array}\right],D_{21}=\left[\begin{array}{ccc}{0}_s& & \\ & S& \\ & & I\end{array}\right],$$

$$\begin{array}{rl}& pN-p\\ & Q=\left[\begin{array}{cc}{Q}_{11}& Q_{12}\\ Q_{21}& Q_{22}\end{array}\right]\begin{array}{c}n\\ N-n\end{array}\end{array}$$

$$\left[\begin{array}{c}{U}_1{D}_{11}\\ V_1{D}_{21}\end{array}\right],\left[\begin{array}{c}{U}_1{D}_{12}\\ V_1{D}_{22}\end{array}\right]$$

$${\mathbb{R}}_J^N=\{x=(x_k)\in {\mathbb{R}}^N;x_k=0ifk\notin J\}.$$

$$\cos \{E,{\mathbb{R}}_J^N\}=\prod _{i=1}^n\cos {\theta }_i,$$

$$\sin \{E,{\mathbb{R}}_J^N\}=\prod _{i=1}^n\sin {\theta }_i.$$

$$P\{|J\cap \varphi |=n\}=\prod _{i=1}^n{\cos }^2{\theta }_i$$

$$P\{|J\cap \varphi |=k\}=\sum _{1\le i_1<\cdots <i_k\le n}\prod _{j=1}^k{\cos }^2{\theta }_{i_j}\times \prod _{j\notin \{i_1,\dots ,i_k\}}{\sin }^2{\theta }_j.$$

$$P\{|J\cap {\varphi }^c|=n\}=\prod _{i=1}^n{\sin }^2{\theta }_i$$

$$P\{|J\cap {\varphi }^c|=k\}=\sum _{1\le i_1<\cdots <i_k\le n}\prod _{j=1}^k{\sin }^2{\theta }_{i_j}\times \prod _{j\notin \{i_1,\dots ,i_k\}}{\cos }^2{\theta }_j.$$

$$P\{K\subset \varphi (\mathfrak{W})\}\le P\{K\subset \varphi \},$$

$$P\{K\subset \varphi \}\le P\{K\subset \varphi (\mathfrak{V}\cup \mathfrak{W})\}.$$

$$\displaystyle\bigg{|}\langle z_{1},z_{n}\ \rangle+\sum_{k=1}^{n-2}(-1)^{k}\sum_{2\leq i_{1}<\dots<i_{k}\leq n-1}\langle z_{1}\wedge(\bigwedge_{j=1}^{k}z_{i_{j})},z_{n}\wedge(\bigwedge_{j=1}^{k}z_{i_{j}})\ \rangle\bigg{|}^{2}$$

$$\begin{array}{rl}\begin{array}{c}2\\ ⋮\\ n-1\\ 1\\ n\end{array}& [\begin{array}{cc}\cos {\theta }_1{u}_1^1\dots \cos {\theta }_{n-2}{u}_1^{n-2}& 0\dots 0\\ ⋮& ⋮\\ \cos {\theta }_1{u}_{n-2}^1\dots \cos {\theta }_{n-2}{u}_{n-2}^{n-2}& 0\dots 0\\ \sin {\theta }_1{V}_1^1\dots \sin {\theta }_{n-2}{V}_1^{n-2}& W_1^1\dots W_1^{p-n+2}\\ \sin {\theta }_1{V}_2^1\dots \sin {\theta }_{n-2}{V}_2^{n-2}& W_2^1\dots W_2^{p-n+2}\end{array}\\ & \\ & \begin{array}{cc}\sin {\theta }_1{u}_1^1\dots \sin {\theta }_{n-2}{u}_1^{n-2}& 0\dots 0\\ ⋮& ⋮\\ \sin {\theta }_1{u}_{n-2}^1\dots \sin {\theta }_{n-2}{u}_{n-2}^{n-2}& 0\dots 0\\ -\cos {\theta }_1{V}_1^1\cdots -\cos {\theta }_{n-2}{V}_1^{n-2}& {\tilde{W}}_1^1\dots {\tilde{W}}_1^{N+2-n-p}\\ -\cos {\theta }_1{V}_2^1\cdots -\cos {\theta }_{n-2}{V}_2^{n-2}& {\tilde{W}}_2^1\dots {\tilde{W}}_2^{N+2-n-p}\end{array}]\end{array}$$

$$\begin{split}&P\{x_{1}\in\phi|\{x_{2},\dots,x_{n}\}\subset\phi^{c}\}=\\ &P\{x_{1}\in\phi,x_{n}\in\phi^{c}|\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}\times\frac{P\{\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}}{P\{\{x_{2},\dots,x_{n}\}\subset\phi^{c}\}}\\ &=\big{[}P\{x_{1}\in\phi|\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}\\ &\qquad-P\{x_{1}\in\phi,x_{n}\in\phi|\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}\big{]}\\ &\quad\times\frac{P\{\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}}{P\{\{x_{2},\dots,x_{n}\}\subset\phi^{c}\}}\\ &=\big{[}\parallel V_{1}\parallel^{2}+\parallel W_{1}\parallel^{2}-\parallel(V_{1},W_{1})\wedge(V_{2},W_{2})\parallel^{2}\big{]}\times\frac{1}{\parallel\tilde{W}_{2}\parallel^{2}}.\end{split}$$

$$\begin{array}{rl}& P\{\{x_2,\dots ,x_n\}\subset {\varphi }^c\}\times P\{\{x_2,\dots ,x_{n-1}\}\subset {\varphi }^c\}\\ & \times \left[P\{x_1\in \varphi |\{x_2,\dots ,x_n\}\subset {\varphi }^c\}-P\{x_1\in \varphi |\{x_2,\dots ,x_{n-1}\}\subset {\varphi }^c\}\right]\\ & =\prod _{i=2}^{n-2}{\sin }^4{\theta }_i\left[(\parallel V_1{\parallel }^2+\parallel W_1{\parallel }^2)(1-\parallel {\tilde{W}}_2{\parallel }^2)-\parallel (V_1,W_1)\wedge (V_2,W_2){\parallel }^2\right]\\ & =\prod _{i=1}^{n-2}{\sin }^4{\theta }_i|⟨(V_1,W_1),(V_2,W_2)⟩{|}^2.\end{array}$$

$$⟨z_1,z_n⟩+\sum _{k=1}^{n-2}(-1{)}^k\sum _{2\le i_1<\cdots <i_k\le n-1}⟨z_1\wedge (\underset{j=1}{\overset{k}{\bigwedge }}{z}_{i_j)},z_n\wedge (\underset{j=1}{\overset{k}{\bigwedge }}{z}_{i_j})⟩=A+B$$

Full text

1 Introduction.

There exists an extensive mathematical literature, in several theoretical and applied areas, related to determinantal point processes. Let us cite, to mention a few recent applied works, [2], [9], [10], [18], [23], [24]. A good overview of the main conceptual basis and properties can be found in [20] and in the bibliography therein.

From the theoretical point of view, determinantal point processes could be defined (in Bourbaki-like-spirit) in the general locally compact Polish spaces setting, as point processes associated to some locally square integrable, Hermitian, positive semidefinite, locally trace-class operators and thereafter specialize them for particular cases namely to discrete determinantal processes. Regarding the latter, the approach of [20] which consists to construct such processes, first in the most elementary discrete context and then gradually extend them to the general situation, provides, in our opinion, many advantages. It turns out also that some results for the most general processes are proved only [11], [20], or more simply [21], 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 $\mathfrak{Z}=\{z^{1},...,z^{p}\}$, $1<p<N$, be a set of orthonormal vectors in $\mathbb{C}^{N}$. Denote

[TABLE]

and

[TABLE]

The associated determinantal process $\phi(\mathfrak{Z})$ is a point process, view as a random subset of $\mathcal{N}=\{1,\dots,N\}$ of cardinality $|\phi(\mathfrak{Z})|=p$, characterized [19], [20] by the formulas

[TABLE]

for all subsets $\{i_{1},\dots,i_{p}\}\subset\mathcal{N}$. Note also that (1) implies

[TABLE]

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 $\tilde{\mathfrak{Z}}=\{\tilde{z}^{1},...,\tilde{z}^{p}\}$ is another orthonormal basis of $E=E(\mathfrak{Z})$ then

[TABLE]

for every $\{i_{1},\dots,i_{p}\}\subset\mathcal{N}$ and consequently $\phi(\mathfrak{Z})=\phi(\tilde{\mathfrak{Z}})$.

Remark also that if $\mathfrak{Z}^{\perp}=\{z^{p+1},...,z^{N}\}$ is an orthonormal basis of the orthogonal complement $E(\mathfrak{Z})^{\perp}$ of $E(\mathfrak{Z})$ in $\mathbb{C}^{N}$ then obviously

[TABLE]

A remarquable 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 fixed, is arbitrary edge-oriented and M is the vertex-edge incidence matrix (the columns been indexed by vertexes) then the determinantal process associated to the vector space spanned by column vectors but one, provides a uniform probability on spanning trees. This result due [7] is called the Transfer Current Theorem. For more details, with clever short proofs, see [20] 2.6. p.8. Some extensions of this result are given in [4] with a serie of open questions and conjectures. Among them, Conjecture 4.6. related to BK-type inequality after J. van den Berg and H.Kesten.

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 have also $A\cup\{n\}\in\mathfrak{A}$. 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 [6] by

[TABLE]

A point process $\psi$ on $\mathcal{N}$ is said to have the van den Berg - Kesten property (in short the BK property) if

[TABLE]

for every pair of increasing events. In [6] J. van den Berg and H.Kesten proved that inequality (4) is satisfied when $\psi$ is related to a product probability on $2^{\mathcal{N}}$. In the basic determinantal process setting the Conjecture 4.6. which states that the same is true for the spanning tree determinantal point processes is still unsolved. The question of whether general determinantal processes have the BK property was raised in [19].

The purpose of this note is twofold. First, we introduce a new method to investigate discrete determinantal processes using the CS decomposition (CSD) of 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 the proposition 2 below which seems us new and the results of [12]. Furthermore, this should be an appropriate framework for computational needs.

Second, we study the BK inequality. We prove that the BK inequality (4) is satisfied for all discrete determinantal processes when the increasing events $\mathfrak{A}$ and $\mathfrak{B}$ are generated by simple points: Theorem 3 in section 4 and section 5. We conjecture also that:

Conjecture 1

For all $n\geq 2$ we have

[TABLE]

for every choice $A$, $A_{i}$, $i=1,\dots,n$, of disjoint subsets of $\{1,\dots,N\}$ such that $P\{A_{i}\not\subset\phi,\forall i=1,\dots,n\}>0$.

If the Conjecture 1 holds then it can be shown that the BK inequality (4) 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 (5) is a well-known result. For general sets, note that $P\{A\not\subset\phi\mid A_{1}\not\subset\phi,\}\leq P\{A\not\subset\phi\}$, $A\cap A_{1}=\emptyset$ is the classical correlation inequality [19] and that (5) was obtained in [12] for $n=2,3$ with the precise values of conditional probabilities.

Remark 1

Note that for the process $\psi$ related to a product probability on $2^{\mathcal{N}}$ the counting random variables $|\psi\cap A_{i}|$ (the sets $A_{i}$, $i=1,\dots,n$, being disjoint) are independent and thus the inequality (5) 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 $\{i_{1},\dots,i_{k}\}\subset\mathcal{N}$, that is if $P\{\psi_{k}=\{i_{1},\dots,i_{k}\}\}=1/\binom{N}{k}$, it was proved in [5] that $\psi_{k}$ has the BK property. As regards to the inequality (5) we have, with the choice $P\{i\not\in\psi_{k},i_{j}\not\in\psi_{k},\forall j=1,\dots,n\}>0$,

[TABLE]

and, consequently, the inequality (5) 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 $P\{A\not\subset\psi_{k}\mid A_{1}\not\subset\psi_{k}\}\leq P\{A\not\subset\psi_{k}\}$, $A\cap A_{1}=\emptyset$, $|A|=n$, $|A_{1}|=m$ with (the non-trivial case) $n+m\leq k$, follows from 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 (5) depends on whether or not the functions of the form

[TABLE]

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 [26] .

2 The CS decomposition and the basic determinantal point process

Following [25] the general CS decomposition (CSD) for a matrix $Q$ from the unitary group $U(N)$ specifies that for any $2\times 2$ partitionning

[TABLE]

there exist unitary matrices $U_{1}$, $U_{2}$, $V_{1}$, $V_{2}$ such that (here all unnamed blocks of the matrices below are always zero and the superscript $H$ represents the conjugate transpose)

[TABLE]

where the matrices

[TABLE]

$D_{22}=\begin{bmatrix}I&&\\ &-C&\\ &&\mathbf{0}_{c}^{H}\\ \end{bmatrix}$ are diagonal with $C\equiv diag(\cos\theta_{1},\dots,\cos\theta_{s}),$

$S\equiv diag(\sin\theta_{1},\dots,\sin\theta_{s})$, $1>\cos\theta_{1}>\dots>\cos\theta_{s}>0$.

In some cases the matrices of zeros $\mathbf{0}_{s}$ and $\mathbf{0}_{c}$ as well as the unit matrices $I$ could be nonexistent. See [25] THEOREM 1 (page $12$ and the discussion that follows) for the full statement, and below for a detailled 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 $\mathbb{R}^{n}$ [17]. Nowaday it is a popular tool in numerical linear algebra, useful for solving various questions as, for example, constrained least square problems, computing principal angles between subspaces, the generalized singular value decomposition, quantum computing and more else [3], [8], [13], [14], [25].

Now, let $E\subset\mathbb{C}^{N}$ be a vector space of dimension $1<p<N$, $\mathfrak{Z}=\{z^{1},...,z^{p}\}$ an orthonormal basis of $E$ and $\mathfrak{Z}^{\perp}=\{z^{p+1},...,z^{N}\}$ an orthonormal basis of the orhogonal complement $E(\mathfrak{Z})^{\perp}$. Fix $1\leq n\leq p$ and consider the CSD of the partitioned unitary matrix $Q=(z^{1},...,z^{p},z^{p+1},...,z^{N})$

[TABLE]

It follows from (7) that the columns vectors of these two matrices

[TABLE]

are respectively orthonormal basis of $E$ and $E(\mathfrak{Z})^{\perp}$.

Now we will detail different cases given by (9) which need to be distinguished. The description given here is somewhat lengthy but, in our opinion, useful for both theoretical and computational purpose. We note by $e(k)$, $k=1,\dots,N$, the nul vector of the space $\mathbb{C}^{k}$. Note also the slight change with regard to angles appearing in CSD (7) which allows values [math] and $\pi/2$ in order to recover all Jordan’s principal angles.

1. I.-

The case $n<p$ and $p+n<N$.

There exist:

1. (a)

a sequence $u^{1},\dots,u^{n}$ of orthonormal vectors in $\mathbb{C}^{n}$, 2. (b)

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

$\mathfrak{\tilde{W}}=\{\tilde{W}^{1},\dots,\tilde{W}^{N-p-n}\}$, 3. (c)

Jordan angles $0\leq\theta_{1}\leq\dots\leq\theta_{n}\leq\pi/2$

such that noting

1. $z^{i}=\begin{bmatrix}u^{i}cos\theta_{i}\\ V^{i}sin\theta_{i}\end{bmatrix}i=1,\dots,n,\quad z^{i}=\begin{bmatrix}e(n)\\ W^{i}\end{bmatrix}\quad i=n+1,\dots,p,$
2. $z^{p+i}=\begin{bmatrix}u^{i}sin\theta_{i}\\ -V^{i}cos\theta_{i})\end{bmatrix}i=1,\dots,n,$
3. $z^{p+n+i}=\begin{bmatrix}e(n)\\ \tilde{W}^{i}\end{bmatrix}i=1,\dots,N-p-n,$

the sequence $\mathfrak{Z}=\{z^{1},...,z^{p}\}$ is an orthonormal basis of $E$ and the sequence $\mathfrak{Z}=\{z^{p+1},...,z^{N}\}$ is an orthonormal basis of the orthogonal complement $E^{\perp}$. 2. II.-

The case $n<p$ and $p+n>N$.

There exist:

1. (a)

a sequence $u^{1},\dots,u^{n}$ of orthogonal vectors in $\mathbb{C}^{n}$, 2. (b)

two sequences of mutually orthogonal vectors in $\mathbb{C}^{N-n}$

$\mathfrak{V}=\{V^{1},\dots,V^{N-p}\}$ and $\mathfrak{W}=\{W^{1},\dots,W^{p-n}\}$, 3. (c)

Jordan angles $0=\theta_{1}=\dots=\theta_{n+p-N}\leq\dots\leq\theta_{n}\leq\pi/2$

such that noting

1. $z^{i}=\begin{bmatrix}u^{i}\\ e(N-n)\end{bmatrix}i=1,\dots,n+p-N,$
2. $z^{i}=\begin{bmatrix}u^{i}cos\theta_{i}\\ V^{i-n-p+N}sin\theta_{i})\end{bmatrix}i=n+p-N+1,\dots,n,$

$z^{n+i}=\begin{bmatrix}e(n)\\ W^{i}\end{bmatrix}i=1,\dots,p-n,$ 3. $z^{p+i}=\begin{bmatrix}u^{n+p-N+i}sin\theta_{n+p-N+i}\\ -V^{i}cos\theta_{n+p-N+i}\end{bmatrix}i=1,\dots,N-p,$

the set $\mathfrak{Z}=\{z^{1},...,z^{p}\}$ is an orthonormal basis of $E$ and the set $\mathfrak{Z}=\{z^{p+1},...,z^{N}\}$ is an orthonormal basis of $E^{\perp}$. 3. III.-

The case $n<p$ and $p+n=N.$

There exist:

1. (a)

a sequence $u^{1},\dots,u^{n}$ of orthogonal vectors in $\mathbb{C}^{n}$, 2. (b)

two sequences of mutually orthogonal vectors in $\mathbb{C}^{N-n}$

$\mathfrak{V}=\{V^{1},\dots,V^{n}\}$ and $\mathfrak{W}=\{W^{1},\dots,W^{p-n}\}$, 3. (c)

Jordan angles $0\leq\theta_{1}\leq\dots\leq\theta_{n}\leq\pi/2$

such that noting

1. $z^{i}=\begin{bmatrix}u_{i}cos\theta_{i}\\ V^{i}sin\theta_{i}\end{bmatrix}i=1,\dots,n,\quad z^{n+i}=\begin{bmatrix}e(n)\\ W^{i}\end{bmatrix}i=1,\dots,p-n,$
2. $z^{p+i}=\begin{bmatrix}u^{i}sin\theta_{i}\\ -V^{i}cos\theta_{i}\end{bmatrix}i=1,\dots,n,$

the set $\mathfrak{Z}=\{z^{1},...,z^{p}\}$ is an orthonormal basis of de $E$ and the set $\mathfrak{Z}=\{z^{p+1},...,z^{N}\}$ is an orthonormal basis of $E^{\perp}.$ 4. IV.-

The case $n=p$.

With the notations of points I-III:

1. (i)

$2p<N$

1. $z^{i}=\begin{bmatrix}u^{i}cos\theta_{i}\\ V^{i}sin\theta_{i}\end{bmatrix}i=1,\dots,p,\quad z^{p+i}=\begin{bmatrix}u^{i}sin\theta_{i}\\ -V^{i}cos\theta_{i}\end{bmatrix}i=1,\dots,p,$
2. $z^{2p+i}=\begin{bmatrix}e(n)\\ W^{i}\end{bmatrix}i=1,\dots,N-2p.$
3. (ii)

$2p>N$

1. $z^{i}=\begin{bmatrix}u^{i}\\ e(N-n)\end{bmatrix}i=1,\dots,2p-N,$
2. $z^{i}=\begin{bmatrix}u^{i}cos\theta_{i}\\ V^{i-2p+N}sin\theta_{i}\end{bmatrix}i=2p-N+1,\dots,p,$
3. $z^{i}=\begin{bmatrix}u^{i+p-N}sin\theta_{i+p-N}\\ -V^{i-p}cos\theta_{i+p-N}\end{bmatrix}i=p+1,\dots,N.$
4. (iii)

$2p=N$

1. $z^{i}=\begin{bmatrix}u^{i}cos\theta_{i}\\ V^{i}sin\theta_{i},\end{bmatrix}\quad z^{p+i}=\begin{bmatrix}u^{i}sin\theta_{i}\\ -V^{i}cos\theta_{i}\end{bmatrix}i=1,\dots,p.$

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 basis of the spaces $E$ and $E^{\perp}$. Note that in the Euclidean context, that is for $E\subset\mathbb{R}^{N}$ and the CSD applied to orthogonal matrices, the angles appearing in the CS decomposition (related to $J$) are principal Jordan angles between the space E and the basic subspace

[TABLE]

An important statistical application of principal angles is the canonical correlation analysis (CCA) of H.Hotelling [15]. In order to develop a unified algebraic formulation of concepts in multivariate analysis (as for example CCA), S.Afriat has thoroughly studied in [1] (see also [22]) the geometry of subspaces in $\mathbb{R}^{N}$ in terms of orthogonal and oblique projectors and has introduced, among others, the notation of so-called *multiplicative cosine and sine *

[TABLE]

[TABLE]

The basis of $E$ given by the CSD is a pertinent tool for the study of the associated determinantal process. For example it gives at once

Proposition 1

For a set $J=\{x_{1},\dots,x_{n}\}$, $n\leq p$ we have:

[TABLE]

and for $k=1,\dots,n-1$

[TABLE] 2. 2.

[TABLE]

and for $k=1,\dots,n-1$

[TABLE] 3. 3.

If $n<p$ and $P\{J\subset\phi\}>0$ then the conditioned process $\{\phi|\ J\subset\phi\}\setminus J$ is determinantal such that $\{\phi|\ J\subset\phi\}\setminus J=\phi(\mathfrak{W}).$ 4. 4.

If $N-p>n$ and $P\{J\subset\phi^{c}\}>0$ then the conditioned process $\{\phi|\ J\subset\phi^{c}\}$ is determinantal such that $\{\phi|\ J\subset\phi^{c}\}=\phi(\mathfrak{V}\cup\mathfrak{W}).$ 5. 5.

If $P\{J\subset\phi\}>0$ then for all $K\subset\{1,\dots,N\}\setminus J$ we have

[TABLE]

if $P\{J\subset\phi^{c}\}>0$ then

[TABLE]

Remark 2

The fact that the conditioned processes $\{\phi|\ J\subset\phi\}\setminus J$ and $\{\phi|\ J\subset\phi^{c}\}$ are determinantal, as well as inequalities (14) and (15), are well-known results proved by R.Lyons [19].

Remark 3

Regarding items 1 and 2 of Proposition 1, it was proved more generally in [16] 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.

A more elaborate information can be obtained from this point of view. For example:

Proposition 2

Consider the discrete determinantal process $\phi=\phi(\mathfrak{Z})$ associated to a set $\mathfrak{Z}=\{z^{1},...,z^{p}\}$, $1<p<N$, 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 $P\{\{x_{2},\dots,x_{n}\}\subset\phi^{c}\}>0$. With the choice (to simplify the notations) $x_{i}=i$, $i=1,\dots,n$, we have

[TABLE]

Proof:

The left and rigt sides of (2) 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 basis have the following form:

[TABLE]

It follows from Proposition 1 that

1. (a)

$P\{\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}=\prod_{i=1}^{n-2}\sin^{2}\theta_{i}$ 2. (b)

$P\{\{x_{2},\dots,x_{n}\}\subset\phi^{c}\}=\prod_{i=1}^{n-2}\sin^{2}\theta_{i}\times\parallel\tilde{W}_{2}\parallel^{2}$ 3. (c)

$P\{x_{1}\in\phi|\{x_{2},\dots,x_{n-1}\}\subset\phi^{c}\}=\parallel V_{1}\parallel^{2}+\parallel W_{1}\parallel^{2}$ 4. (d)

[TABLE]

From (a)-(d) an elementary computation gives the right side of (2) (note that $\parallel V_{2}\parallel^{2}+\parallel W_{2}\parallel^{2}+\parallel\tilde{W}_{2}\parallel^{2}=1$). Indeed we get

[TABLE]

To compute the left side of (2) denote $z_{1}^{0}=(\sin\theta_{1}V^{1}_{1},\dots,\sin\theta_{n-2}V^{n-2}_{1})$,

$z_{n}^{0}=(\sin\theta_{1}V^{1}_{2},\dots,\sin\theta_{n-2}V^{n-2}_{2})$ and $\tilde{z}^{i}=(\cos\theta_{i}u^{i}_{1},\dots,\cos\theta_{i}u^{i}_{n-2},0)^{t}$. Observe that

[TABLE]

with

[TABLE]

and

[TABLE]

Obviously $\parallel\bigwedge_{j=1}^{k}\tilde{z}^{i_{j}}\parallel^{2}=\prod_{j=1}^{k}\cos^{2}\theta_{i_{j}},$ thus

[TABLE]

and consequently

[TABLE]

In order to compute A introduce the notations

$\tilde{z}^{i,l}=(\cos\theta_{i}u^{i}_{1},\dots,\cos\theta_{i}u^{i}_{n-2},\sin\theta_{i}V^{i}_{l})^{t}$, $l=1,2$ and $i=1,\dots,n-2$.

A little tought provides that for $k\geq 1$

[TABLE]

Moreover we have

[TABLE]

From orthogonality properties of relevant multivectors we obtain from (24)

[TABLE]

From (21), (23) and (25) an elementary computation gives

[TABLE]

and with (22)

[TABLE]

This and (19) prove Proposition 2. Note that from (27) we get that (20) is identified as a scalar product. $\square$

For further results by using the CSD as well as for some extensions of Proposition 2 see [12].

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

$A\in\mathfrak{A}\quad\Leftrightarrow\exists A_{i}\quad such\quad that\quad A\supset A_{i}$, 2. 2.

$B\in\mathfrak{B}\quad\Leftrightarrow\exists B_{i}\quad such\quad that\quad B\supset B_{i}.$

The sets $A_{i}$, $B_{i}$ are minimal in the sense that none of $A\in\mathfrak{A}$ (resp. $B\in\mathfrak{B}$) is strictly included in $A_{i}$ (resp. in $B_{i}$).

Consider now a basic determinantal process $\phi$ on $\mathcal{N}$. In the particular case when

[TABLE]

we have at once $P\{\phi\in\mathfrak{A}\cap\mathfrak{B}\}=P\{\phi\in\mathfrak{A}\circ\mathfrak{B}\}$ ans thus the BK-inequality (4) becomes

[TABLE]

which is a negative association inequality. R.Lyons proved in [19], [20] that determinantal processes have negative association, meaning that (29) is fulfilled.

In the general situation it is helpful to reformulate BK inequality (4) as follows.

Proposition 3

The inequality (4) is satisfied if and only if

[TABLE]

Proof:

Observe that

[TABLE]

Thus

[TABLE]

if and only if

[TABLE]

$\square$

Suppose now that $\mathfrak{A}=\mathfrak{B}$. Formula (30) becomes

[TABLE]

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

[TABLE]

Therefore

[TABLE]

and formula (31) takes the following form

[TABLE]

Fix now $n_{0}\geq 2$ and suppose that the Conjecture 1 is fulfilled for all $2\leq n\leq n_{0}$.

Lemma 4

Under this hypothesis, for all $A_{i}$, $i=1,\dots,n$, disjoint subsets of $\{1,\dots,N\}$ with $2\leq n\leq n_{0}$, such that $P\{A_{i}\not\subset\phi,\forall i=1,\dots,n\}>0$ we have

[TABLE]

Proof:

For $n=2$ the inequality (33) is the well-known correlation inequality. For $n>2$ applying (5) we get

[TABLE]

and thus Lemma 4 follows by induction. $\square$

We will need the following elementary lemma. Its proof being trivial we omit it.

Lemma 5

For all $0<a\leq 1$ and $n>0$ we have

[TABLE]

Theorem 1

Let $\mathfrak{A}$ be an increasing event generated by disjoint sets $A_{1},\dots,A_{n}$. Suppose that the Conjecture 1 holds. We have

[TABLE]

Proof:

We have to prove (32). By Lemma 5 we obtain

[TABLE]

Lemma 4 implies that

[TABLE]

so it remains to apply the geometric-arithmetic mean inequality

[TABLE]

to obtain (32) as desired. $\square$

Remark 4

Consider an event $\tilde{S}=\{D_{1},\dots,D_{n_{0}}\}\subset 2^{\mathcal{N}}$ of disjoint sets such that $P\{D\not\subset\phi,\forall D\in\tilde{S}\}>0$. Denote $\psi=\{\phi|D\not\subset\phi,\forall D\in\tilde{S}\}$. If the Conjecture 1 holds then it is obvious that the inequality (5) is also satisfied for the conditioned process $\psi$ provided that the sets occuring in (5) 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

[TABLE]

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 the Conjecture 1 holds. Then for increasing events $\mathfrak{A}$, $\mathfrak{B}$ such that $S(\mathfrak{A})=S_{1}\cup S$ and $S(\mathfrak{B})=S_{2}\cup S$ we have

[TABLE]

where $\psi=\{\phi|D\not\subset\phi,\forall D\in\tilde{S}\}$ 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 the following lemma.

Lemma 6

Fix $S_{1}$, $S_{2}$, $S$ and and suppose that BK inequality (41) is fulfilled for all conditioned processes $\psi$ subjected to the conditions of Theorem 2. Fix $A\subset\mathcal{N}$, $A\neq\emptyset$ such that $A\cap A^{\prime}=\emptyset$ for all $A^{\prime}\in S_{1}\cup S_{2}\cup S$. Denote by $\mathfrak{\tilde{A}}=\sigma\{A,\mathfrak{A}\}$ the increasing event generated by $A$ and $\mathfrak{A}$. Then, the BK inequality

[TABLE]

is satisfied for all conditioned processes $\psi=\{\phi|D\not\subset\phi\quad\forall D\in\tilde{S}\}$ such that all sets of $S(\mathfrak{\tilde{A}})\cup S_{2}\cup S\cup\tilde{S}\subset 2^{\mathcal{N}}$ are pairwise disjoint.

Proof:

By (30) we may suppose that

[TABLE]

We have

[TABLE]

and

[TABLE]

Formulas (30) and (44) imply that the BK inequality (42) can be written as follows

[TABLE]

or, introducing the process $\psi_{0}=\{\phi\mid A\not\subset\phi,A^{\prime}\not\subset\phi\quad\forall A^{\prime}\in\tilde{S}\}$, as

[TABLE]

The stated hypotheses imply that

[TABLE]

It is easy to see that Conjecture 1 implies the inequality

[TABLE]

and thus by (47) and (48) we obtain (46) which finish the proof of Lemma 6. $\square$

Starting from (40) and applying step by step the Lemma 6 Theorem 2 follows. $\square$

4 The BK inequality for increasing events $\mathfrak{A}$, $\mathfrak{B}$ generated by simple points

As mentioned in the Introduction the inequality (5) is satisfied when the occuring sets are reduced to being simple points. This follows easily, for example, from Proposition 1. Therefore Theorem 2 implies

Theorem 3

Let $\mathfrak{A}$, $\mathfrak{B}$ be increasing events generated by simple points. The BK inequality

[TABLE]

is then satisfied for all determinantal discrete processes $\phi$ associated to sets of orthonormal vectors of $\mathbb{C}^{N}$.

Remark 5

For sets reduced to being simple points the key inequality (33) can be seen from the point of view given by the CSD. Indeed, consider the CSD in the case I applied to $J=\{x_{1},\dots,x_{n}\}$ and, according, 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

[TABLE]

for all $\{x_{i_{1}},\dots,x_{i_{k}}\}\subset J$. Denote

[TABLE]

where

[TABLE]

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 (12) we obtain:

[TABLE]

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 (36) which can be read as

[TABLE]

can be obtained also by a direct computation from formula (11) of Proposition 1 and, moreover, if one 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

[TABLE]

$i=0,\dots,n$, then the formulas (11) imply that $P\{|S\cap\phi|\geq i\}=\mu(\mathfrak{A}_{i})$. From the original J. van den Berg and H.Kesten Theorem (3.3) of [6] we get that

[TABLE]

Furthermore, note that by Remark 3 the inequalities (54) and (55) 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 paragraphs 2.2. of [20], which start from the basic processes, it follows at once that Theorems 1-2 are valid (the generated sets $S(\mathfrak{A})$ and $S(\mathfrak{B})$ being finite or infinite) for determinantal point processes defined on denumerables sets $\mathcal{E}$ and associated to closed subspaces of $l^{2}(\mathcal{E})$. Now, let $\phi$ be a such process on $\mathcal{E}$. Fix $\mathcal{F}\subset\mathcal{E}$ and consider the proces

[TABLE]

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 generating sets $S_{1}$, $S_{2}$. Consequently Theorem 3 is valid for $\psi$ as well. To finish just note that, by paragraph 2.2. of [20], discrete determinantal processes associated to positive contractions (the general case) are of the form (56).

By the transference principle ([20] 3.6.) Theorem 3 could also be extended to the continuous case but this is of little use due the fact that in the continuous setting the intensity measures related to determinantal processes of interest are of diffusive type which implies that $P\{x\in\phi\}=0$ for points $x$ (however, as mentioned in Remark 6, inequalities (54) and (55) still hold).

Acknowledgement

I thank the anonymous referees for their constructive comments, especially for a pertinent question about the validity of the Conjecture 1 for the non-determinantal point processes, which led to look at the process described in Remark 1.