inequality for determinantal probability measures

This paper describes a second order perturbation analysis of the BK property in the space of Hermitean determinantal probability measures around the subspace of product measures, showing that the second order Taylor approximation of the BK inequality holds for increasing events.


Motivation
The van den Berg Kesten (BK) inequality concerns occurrence of two events on disjoint sets.It has numerous applications in percolation theory, see e.g.Grimmett's book on percolation [2].For increasing events and product measures, the BK inequality was proven by van den Berg and Kesten in [9]; see also variants of it in [6].Reimer [5] proved the generalization of the BK inequality to arbitrary, not necessarily increasing events and product measures.Quite recently, several variants and generalizations of the BK inequality have been proven; see [8], [7], and [3].
Determinantal probability measures and their continuum analogue, determinantal point processes, have found considerable interest in mathematics and physics, e.g. in the description of quantum systems of fermions and in random matrix theory.For an interesting introduction to the theory of determinantal probability measures, including more details on the history and references, see [4].One of the facts shown in that paper is that determinantal probability measures are positively associated; compare the remark below (1.12).In §9 of [4], Lyons asks whether determinantal probability measures have the BK property.This question is still not answered, but it motivated the present work.
In this paper, we are mostly interested in increasing events A and B. In this case, A2B can be characterized as follows.
(1.2) Reimer has proven the following theorem; see Theorem 1.2 in [5].More precisely, we cite here the equivalent version of the theorem given in [8], Proposition 1.3.A detailed review of Reimer's proof can be found in [1], Sections 4 and 5. Reimer's theorem plays the key role in his proof of the van den Berg -Kesten -Reimer inequality.Fact 1.1 (Reimer's butterfly theorem).For all events A, B ⊆ Ω, the following holds: |A ∩ B| ≥ |A2B|.(1.3)We deduce the following corollary of this theorem; it plays an important role in this paper.

Corollary 1.2 (A variant of Reimer's theorem
).For all i, j ∈ [n] with i = j and for all increasing events A, B ⊆ Ω, one has We remark that this variant is not valid for arbitrary events A and B. A counterexample to this and some related counterexamples are given in Remark 2.2, below.
we denote by M I,J = (M ij ) i∈I,j∈J the submatrix with index sets I and J.For convenience of the reader, we have collected some basic facts and notation on positive definite matrices in Appendix A.1.We introduce the following sets of matrices: (1.9) (1.10) These probability measures P G are called Hermitian determinantal probability measures.
Although Fact 1.5 is well-known, we include a proof in Appendix A.2 to make the paper self-contained. Let denote the set of diagonal matrices in G n .Note that for any diagonal matrix D ∈ D, under P D , the event that there is a particle at position i ∈ [n] occurs independently of all particles at other locations.For increasing events A, B ⊆ Ω, we define (1.12) Whenever A and B are measurable with respect to deterministic disjoint sets of locations, one has BK A,B ≥ 0. This is called negative associations of determinantal probability measures and shown in Theorem 6.5 in [4].The classical BK inequality can be phrased as BK A,B (D) ≥ 0 for all D ∈ D. To our knowledge, it is not known whether BK A,B (G) ≥ 0 for all G ∈ G n .We prove here the following weaker statement: Theorem 1.6.For all increasing events A, B ⊆ Ω, the second order Taylor approximation of BK A,B at D is non-negative near D.More precisely, let G : (−1, 1) → G n be a C 2 path with G(0) ∈ D.Then, the second order Taylor polynomial of BK A,B •G at 0 is non-negative in a neighborhood of 0.
The proof of this theorem is based on the following theorem, which also might be interesting on its own.Theorem 1.7.For all increasing events A, B ⊆ Ω, the second order Taylor approximation of G n G → Reim A,B (P G ) at D is non-negative near D.More precisely, let G : (−1, 1) → G n be a C 2 path with G(0) ∈ D.Then, the second order Taylor polynomial of (−1, 1) t → Reim A,B (P G(t) ) at t 0 = 0 is non-negative in a neighborhood of 0.
We remark that Reim A,B (P G ) may take negative values.This holds even for G arbitrarily close to 1   2 Id ∈ D. For a counterexample, see Remark 3.14, below.However, in some numerical and computer algebraic searches, we did not find any counterexample to the conjecture BK A,B (G) ≥ 0 for any increasing events A and B and any G ∈ G n .
Overview of the proofs and related techniques in the literature.The Corollaries 1.2 and 1.4 of Reimer's butterfly theorem, Fact 1.1, are proven in Section 2. The key idea is to collapse the two locations i and j to a single one.
The Taylor expansions in Theorems 1.6 and 1.7 are proven in Section 3. Reimer's butterfly theorem and Corollary 1.2 are used in these Taylor expansions for the treatment of the 0-th order term and of the second order term, respectively.Parts of the techniques used in Section 3 have also been used by Lyons in [4] and van den Berg and Jonasson in [8], with different goals, perspectives, and notations.More precisely, conditioning of determinantal probability measures is described in §6 of [4], lifting of P G to P M (G) with a projection M (G) also appears in §8 of [4], and our partitioning of Ω × Ω (ω, η) according to different values of ξ = ω + η has some similarity with the method of proof used in Section 2.2 of [8].However, we have tried to make the paper as self-contained as possible.

Proof of the variant of Reimer's theorem
Throughout this section, we fix i, j ∈ [n] with i = j as in Corollary 1.2.We abbreviate j c := [n] \ {j} and (ij) c := [n] \ {i, j}.The operation 2 is adapted to the index set j c rather than [n]; we write 2 j c in this case.We consider the restriction map : Ω i =j → Ω j c , ω → ω = (ω k ) k∈j c .Note that this map is a bijection.For an event A ⊆ Ω i =j , we write A = {ω : ω ∈ A}.The following lemma allows us to deduce Corollary 1.2 from Reimer's butterfly theorem.Lemma 2.1.For increasing events A, B ⊆ Ω, one has: 2) of A2B for increasing events A and B, we can take S ∈ A and T ∈ B with S ∧ T = 0 and S ∨ T ≤ ω.In order to show we define S ∈ Ω j c by Sk = S k for k ∈ (ij) c and Si = S i ∨ S j , and similarly Tk = T k for k ∈ (ij) c and Ti = T i ∨ T j .By definition of 2 j c , it suffices to show the following: Perturbation of the BK inequality for determinantal measures To prove claim (a), take k ∈ j c .In the case k ∈ (ij) c , we have Furthermore, we need to show τ j ≥ S j .We distinguish two cases.Case 1: Si = 0: Then, S j ≤ S i ∨ S j = Si = 0 implies S j = 0 and hence τ j ≥ S j .Case 2: Si = 1: (2.7) Thus, τ ≥ S is proven.
Using that S ∈ A and A is increasing, we conclude τ ∈ A and hence τ ∈ A ∩ Ω i =j .This yields claim (b) as follows: The claim (c) is proven just as claim (b); one only replaces A, S, and S by B, T , and T , respectively.

Summarizing, we have proven the claim (2.2).
Proof of Corollary 1.2.In the following estimate, we use in the first and last step that : Ω i =j → Ω j c is a bijection.Furthermore, in the first inequality we use Reimer's butterfly theorem Fact 1.1.Finally, in the second inequality, we apply Lemma 2.1.This yields the claim as follows: We remark also that the inequality need not be true for all increasing events A, B ⊆ Ω.Take for instance n = 2, i = 1, (2.10) Proof of Corollary 1.4.Let P , i, j, and A, B be as in the assumption of the corollary. Then, is the same for all ω ∈ Ω.We conclude

Proof of second order Taylor approximations
In Subsection 3.1, we derive a representation of P G in terms of a triangular matrix W (G), defined in (3.24), below.This representation allows a simpler second order Taylor expansion than the original form.This Taylor expansion is derived in Subsection 3.2 and then applied in Subsection 3.3 to derive Theorem 1.6.

A representation for determinantal probability measures
Let us first explain what is done in this subsection and why.In the definition (1.6) of Reim A,B (P G ), the probability P G (ω) of individual outcomes ω ∈ Ω plays an essential role.However, these probabilities are difficult to compute directly using the defining property (1.10) of P G , which is about events {ω I ≡ 1}.Events consisting of a single outcome can be written in the form Λ I,n = {ω I ≡ 1, ω [n]\I ≡ 0}.If P G is supported on configurations consisting of precisely |I| particles, the events {ω I ≡ 1} and Λ I,n differ only by a null set.Consequently, the probability P G (ω) of ω ∈ Ω is a simple determinant in this case.In particular, this holds when G is an orthogonal projector of rank |I|.However, G n does not contain any orthogonal projector.But one can write P G as a marginal of another determinantal probability measure P M (G) on the set of configurations {0, 1} 2n with twice the number of locations and an orthogonal projector M (G) of rank n; this is the meaning of Lemma 3.3 in combination with Lemma 3.1.Instead of working with the projector M (G), one can also work with an orthonormal basis of the space it projects to, encoded as columns in a matrix Ψ(G) * .This yields a description of P M (G) in terms of a measure µ Ψ(G) , described in Definition 3.2(a); see also Lemma 3.3.
Choosing another basis (not necessarily orthonormal) of the same space only changes a normalizing constant in this measure, as is shown in Lemma 3.5.A convenient choice of such a basis, encoded in a matrix, is of the form Σ * = (σ, Id) * , where the identity matrix corresponds to the "second half" of locations which are dropped by taking the marginal P G of P M (G) .Details are given in Lemma 3.4.As a marginal of µ Σ , one obtains another finite measure ν σ on {0, 1} n .For quadratic matrices σ ∈ C n×n , it is introduced in Definition 3.2(b), below.Unlike P G , the measure ν σ is defined in terms of probabilities of individual outcomes.By construction, for an appropriate choice of σ, it turns out to be a multiple of P G ; see Lemma 3.6 below.The second order perturbation analysis of Reim A,B (ν σ ) gets more elementary for triangular matrices σ with ones in the diagonal.For this reason, we reduce the general case to this special case using a QR-decomposition; this yields Lemma 3.10 below.
To make all this precise, we proceed as follows.Recall the definitions of G n , G n , and D from (1.9) and (1.11).For any positive semidefinite Hermitian matrix A ≥ 0, √ A ≥ 0 denotes its unique positive semidefinite square root.We define M : denote the event that there are particles precisely at locations in I.
We introduce now two measures with a matrix as a parameter.They are both closely related to P G as is shown in Lemmas 3. (3.5) Thus, ν σ is supported on particle configurations at n locations with an arbitrary number of particles.
If rank Σ < n, µ Σ is the zero measure.Although the definitions of µ Σ and ν σ look somehow similar, these measures are quite different and should not be confused with each other.In the special case Σ = Ψ(G), the following lemma establishes a connection between µ Σ and P G .Let ι : {0, 1} 2n → {0, 1} n denote the projection to the first n coordinates.We denote by ι[µ] the image measure of any measure µ on {0, 1} 2n with respect to ι.For ω ∈ Ω, we set The measures µ Σ and ν σ are related as follows.
Lemma 3.4.For σ ∈ C n×n and Σ = (σ, Id) ∈ C n×2n , one has ι[µ Σ ] = ν σ .In addition, for ω ∈ Ω, the following holds: Since µ Σ is supported on configurations with precisely n particles, we get For matrices E ∈ C n×i and F ∈ C i×n with natural numbers i ≤ n, the well-known Cauchy-Binet formula states the following:  (3.12) Because the measures µ CΣ and µ Σ are both supported on configurations with precisely n particles, the claim follows.

Perturbation of the BK inequality for determinantal measures
We define the real-analytic map Note that for G ∈ G n , the matrix χ(G) = (Id −G) −1/2 G 1/2 is positive definite; in particular all its diagonal entries are positive.Note further that for diagonal matrices G ∈ D, the matrix χ(G) is diagonal.
denote the QR-decomposition of χ(G), where Note that the maps Q and R are uniquely determined by the Gram-Schmidt-algorithm in terms of real-analytic operations.As a consequence, these two maps are real-analytic.
Note also that for diagonal matrices G ∈ D, the matrix R(G) is diagonal.We get Lemma 3.8.For all G ∈ G n , we have Proof.Combining Lemmas 3.6 and 3.7, we get the claim as follows: It is convenient to work with triangular matrices having all diagonal entries equal to 1 rather than using arbitrary positive diagonal entries.To describe the corresponding normalization, we introduce the following notation: For any diagonal matrix D ∈ C n×n with positive entries, we define ii .(3.21) Proof.Using the defining formula (3.5) of ν R and abbreviating J = I(ω), the claim is proven as follows: We apply this lemma to the real-analytic maps  Proof.This follows from Lemmas 3.8 and 3.9.
We introduce the following real analytic function, which plays the role of a normalizing constant: Lemma 3.11.For all increasing events A, B ⊆ Ω and all G ∈ G n , one has Proof.For ω ∈ Ω, we have the following, using the definition (3.19) of κ(D(G)): Using this together with Lemma 3.10 yields Summing this over ω ∈ A ∩ B and over ω ∈ A2B and taking the difference, the claim follows.
Perturbation of the BK inequality for determinantal measures

Perturbation analysis around Reimer's butterfly theorem
We now take any matrix norm • on C n×n .Recall that T 1 denotes the set of all upper triangular complex n × n matrices with all diagonal entries equal to 1. Consequently, for σ ∈ T 1 , σ − Id measures the size of the off-diagonals in σ.In the following, we use i,j∈ [n] i =j as a short notation for i∈[n] j∈[n]\{i} .
Lemma 3.12.For all events C ⊆ Ω we have the following for σ ∈ T 1 in the limit as σ → Id: Proof.The error terms in this proof are always understood in the limit T 1 σ → Id.We prove the following for ω ∈ Ω:  (3.32) We distinguish several cases: Case 1: K = I(ω).In this case, we have because σ is a triangular matrix with ones on the diagonal.Case 2: K \ I(ω) consists of a single element j ∈ K. Then I(ω) \ K consists also of a single element i ∈ I(ω), i = j.Consider an index τ in the sum (3.32).
Case 2a: τ j = i and τ k = k for all k ∈ K \ {j}.Here, we get because σ has ones on the diagonal.Case 2b: τ k = k for more than one k ∈ K.In this case, we have the bound ), because the product contains at least two non-diagonal factors, which are bounded by O( σ − Id ).
Consequently in case 2, we obtain Case 3: K \ I(ω) consists of at least two elements.Consider again an index τ in the sum (3.32).Just as in case 2b, we have τ k = k for more than one k ∈ K.The same argument as in case 2b yields again k∈K σ k,τ k = O( σ − Id 2 ).We conclude in this case: (3.41) We combine the results obtained so far to obtain a Taylor-expansion of the function Reim A,B , which was defined in formula (1.6): Corollary 3.13.For all increasing events A, B ⊆ Ω, one has the following for σ ∈ T 1 in the limit as σ → Id:  The second order Taylor polynomial of t → Reim A,B (ν W (G(t)) ) at t 0 = 0 is non-negative near t 0 = 0 as a consequence of Corollary 3.13.Now c(G(t)) > 0 for t in a neighborhood of 0. Combining these facts yields the claim of the theorem.
One can show that the function W : G n → T 1 maps small neighborhoods in G n of 1 2 Id ∈ G n onto small neighborhoods in T 1 of Id ∈ T 1 .In view of Lemma 3.11, this implies that Reim A,B (P G ) may take negative values for G ∈ G n arbitrarily close to 1  2 Id.Note that the third order Taylor polynomial at σ 0 = Id of expression (3.45), which equals 4(|σ 12 | 2 − Re(σ 12 σ 23 σ * 13 )), may take negative values for σ arbitrarily close to Id.
This illustrates the following fact: if the BK inequality holds for a family of matrices, the corresponding third order Taylor approximation may violate it even close to the point of expansion.This implies also that our method of proof cannot be generalized to third order Taylor polynomials in a straightforward way.

From the variant of Reimer's theorem to the BK inequality
In this subsection, we work with arbitrary finite sets K ⊂ N of locations rather than only with [k].For this reason, we adapt the notations (1.8/1.9/1.10) as follows: {ω I ≡ j} = {ω ∈ {0, 1} K : ω i = j for all i ∈ I} for I ⊆ K and j = 0, 1,  The notations Reim A,B and 2 are adapted to arbitrary finite index sets K ⊂ N in the obvious way; we use the notations Reim K A,B and 2 K , respectively.The restriction of a configuration ω ∈ {0, 1} K to {0, 1} I , I ⊆ K, is denoted by ω I = (ω i ) i∈I .For I ⊆ K, we define I c = K \ I.
We now introduce two functions C 1 I,K and C 0 I,K .The function C 1 I,K corresponds to conditioning on having particles on I c , whereas C 0 I,K corresponds to conditioning on having holes on I c .A precise formulation of this fact is given in Lemma 3.16, below.
Note that C j I,K , j = 0, 1, are real analytic functions.They map diagonal matrices to diagonal matrices.
Lemma 3.15.The maps C j I,K , j = 0, 1, are well-defined.For all G ∈ G K and J In addition, the following relation holds for all G ∈ G K : Next, we show (3.51).We abbreviate L = J ∪ I c and L  (3.57) The case j = 0 is reduced to the case j = 1 by exchanging particles and holes as follows.It suffices to verify the claim for the events {ω J ≡ 0}, J ⊆ I.    Proof.This follows immediately by applying Lemma 3.16 twice.

Fact 1 . 5 .
For every G ∈ G k , there exists a unique probability measure P G which satisfies P G (ω I ≡ 1) = det G I,I for all I ⊆ [k].

( 3 . 7 )
Proof.Let I ⊆ [n].If at the locations in [n] there are precisely particles in I and the total number of particles in [2n] is n, then there must be n − |I| particles in [2n] \ [n].

( 3 .
10) We use it in the special case E = σ [n],I and F = σ * I,[n] to obtain

Lemma 3 . 6 .
For all G ∈ G n one has P G = | det φ(G)| 2 ν χ(G) .Proof.By definition, the formula φ(G) −1 Ψ(G) = (χ(G), Id) holds for G ∈ G n .The following calculation uses this fact in the third equality, Lemma 3.3 in the first equality, Lemma 3.5 in the second equality, and Lemma 3.4 in the last equality.

W
(G) = Id holds for all G ∈ D.
52) means intuitively that C 1 I,K and C 0 I,K are exchanged when exchanging particles with holes.Proof of Lemma 3.15.G ∈ G K implies G I c ,I c ∈ G I c and G I,I ∈ G I .In particular, G I c ,I c and Id −G I c ,I c are invertible.For the matrix T = Id I,I −G I,I c (G I c ,I c ) −1 0 Id I c ,I c , (3.53) the following holds: det T L,L = 1 and (3.54), we get det G L,L = det(T GT * ) L,L = det(C 1 I,K (G)) J,J det G I c ,I c .(3.55)The claim (3.52) follows then from the definitions of C 1 I,K (Id −G) and C 0 I,K (G) together with (Id −G) I,I c = −G I,I c and (Id −G) I c ,I = −G I c ,I .Finally, we conclude C 0 I,K (G) = Id −C 1 I,K (Id −G) ∈ Id −G I = G I .Lemma 3.16.For all G ∈ G K , all I ⊆ K, and all j ∈ {0, 1}, one has P G (ω I ∈ •|ω I c ≡ j) = P C j I,K (G) .

( 3 .
56)Proof.Let G ∈ G K and I ⊆ K. First, we show (3.56) in the case j = 1.It suffices to prove the claim for the events {ω J ≡ 1}, J ⊆ I.Note that P G (ωI c ≡ 1) = det G I c ,I c = 0.We calculate using (3.51) in the second but last inequality:P G (ω J ≡ 1|ω I c ≡ 1) = P G (ω J∪I c ≡ 1) P G (ω I c ≡ 1) = det G J∪I c ,J∪I c det G I c ,I c = det(C 1 I,K (G)) J,J = P C 1 I,K (G) (ω J ≡ 1).
Perturbation of the BK inequality for determinantal measuresThis variant of Reimer's theorem groups all indices in pairs, while Corollary 1.2 uses only a single pair.Nevertheless, the proofs are quite similar.