A note on the Pfaffian integration theorem

Alexei Borodin¹ and Eugene Kanzieper²

¹ Department of Mathematics, California Institute of Technology, CA 91125, USA
² Department of Applied Mathematics, H.I.T.–Holon Institute of Technology, Holon 58102, Israel

E-mail: borodin@caltech.edu and eugene.kanzieper@weizmann.ac.il

Received 18 July 2007, in final form 31 July 2007
Published 21 August 2007

1. Introduction

In the recent studies of spectral properties of Ginibre's orthogonal ensemble (Ginibre 1965) of real asymmetric random matrices^Note1, the following theorem was presented by Kanzieper and Akemann (2005) and Akemann and Kanzieper (2007):

Pfaffian integration theorem. Let π(dz) be an arbitrary measure on with finite moments. Define the function in terms of the vector composed of arbitrary polynomials q_j(z) of the jth order, and of an n × n antisymmetric matrix μ. Then

where e_ℓ(p₁,  ..., p_ℓ) are the elementary symmetric functions^Note2 written as polynomials of the power sums, and n × n matrix [b.upsi] is [b.upsi]  =  μ g with

This theorem has been a key ingredient of the recent calculation (Kanzieper and Akemann 2005, Akemann and Kanzieper 2007) of the probability p_n,k to find exactly k real eigenvalues in the spectra of n × n real asymmetric random matrices drawn from Ginibre's orthogonal ensemble. An earlier attempt to address the same problem is due to Edelman (1997).

Remark 1.1. The explicit form of e_ℓ(p₁,  ..., p_ℓ) is well known (see, e.g., Macdonald 1998):

The notation stands for the frequency representation of the partition [b.lambda] of the size |[b.lambda]|  =  ℓ. It implies that the part ℓ_j appears σ_j times so that ℓ  =  ∑^g_j = 1ℓ_jσ_j, where g is the number of nonzero parts of the partition.

An immediate corollary of equation (1.1) is the identity

Remark 1.2. The Pfaffian integration theorem can be viewed as a generalization of the Dyson integration theorem (Dyson 1970, Mahoux and Mehta 1991) for the case where the quaternion kernel represented by the 2 × 2 matrix^Note3

does not satisfy the projection property

The original proof (Akemann and Kanzieper 2007) of the theorem involved an intricate topological interpretation of the ordered Pfaffian expansion combined with the term-by-term integration that spanned dozens of pages. In the present contribution, we provide two alternative, concise proofs of slight variations of the Pfaffian integration theorem. They are formulated in the form of the two theorems and represent the main result of our note.

Theorem 1. Let (X, m) be a measure space, and the vectors and be composed of measurable functions from X to . Define the functions

where μ is an n × n antisymmetric matrix. Then^Note4

where the n × n matrix [b.upsi] is [b.upsi]  =  μ g with

Theorem 2. In the notation of theorem 1, assume that the matrix μ is invertible (hence, n is even). Then

Remark 1.3. The equivalence of theorems 1 and 2 is easily established. Indeed, multiplying both sides of equation (1.3) by τ^ℓ and summing up over ℓ from 0 to ∞ with the help of equation (1.2), one finds that the right-hand side turns into

given that μ is invertible. This proves the equivalence for invertible (nondegenerate) μ. Theorem 1 with degenerate μ of even size follows by the limit transition, and decreasing the size n by 1 is achieved by setting ^±_n–1 ≡ 0 and nullifying the last, nth, row and column in the matrix μ.

2. Fredholm Pfaffian proof of theorem 1

For , define the infinite sequence

supplemented by σ₀(n)  =  1, and consider the series

By definition (equation (A.3)) introduced by Rains (2000), the function is the Fredholm Pfaffian on the measure space (X, m)

(See appendix for the matrix notation used.) Here, [b.Phi]_n is the 2 × 2 matrix kernel

that can also be written as

where the vector is .

The matrix I – τ J[b.Phi]_n appearing under the sign of the Fredholm determinant can be represented as I  +  A B with

Following Tracy and Widom (1998), one observes the `needlessly fancy' general relation det[ I  +  A B]  =  det[ I  +  B A] that holds for arbitrary Hilbert–Schmidt operators A and B. They may act between different spaces as long as the products make sense. In the present context, det[ I  +  A B] is the Fredholm determinant det_X[ I  +  A B] whilst the determinant det[ I  +  B A] is that of the n × n matrix

where [b.upsi]  =  µ g with g defined by equation (1.4).

The above calculation allows us to write down the Fredholm Pfaffian in the form

Identity equation (1.2) concludes the proof.

3. Linear-algebraic proof of theorem 2

A linear-algebraic proof of theorem 2 is based on the works by Ishikawa and Wakayama (1995, 2000) and de Bruijn (1955) as formulated in sections 3.1 and 3.2. Section 3.3 contains a proof of theorem 2.

3.1.  Minor summation formulae and identities by Ishikawa and Wakayama

Let T be any M × N matrix, and let [m] denote the set {1, 2,  ..., m} for a positive integer . For n-element subsets I  =  {i₁ < ··· < i_n} ⊆ [M] and J  =  {j₁ < ··· < j_n} ⊆ [N] of row and column indices, let denote the submatrix of T obtained by picking up the rows and columns indexed by I and J. In this notation, the following three lemmas hold^Note5.

Lemma 1 (Ishikawa and Wakayama 1995). Let M ≤ N and assume M is even. For any M × N matrix A and any N × N antisymmetric matrix B, one has

Lemma 2 (Ishikawa and Wakayama 2000). Let A be an N × N invertible antisymmetric matrix. Then, for any I ⊆ [N], one has

where stands for the complementary of I, and |I| denotes the sum of the elements of I, |I|  =  ∑_i I i.

Lemma 3 (Ishikawa and Wakayama 2000). Let A and B be N × N antisymmetric matrices. Then

Here, n denotes the integer part of n.

In what follows, we will need the following corollary.

Corollary 1. Let A and B be N × N antisymmetric matrices, and A is invertible. Then

Proof. Using lemma 3 and lemma 2 (in this order), we write down

This concludes the proof.        

3.2.  de Bruijn integration formula

Lemma 4 (de Bruijn 1955). Let (X, m) be a measure space, and the vectors and be composed of measurable functions from X to . Then

where the 2ℓ × 2ℓ matrix g is^Note6

3.3.  Proof of theorem 2

In the notation of theorem 1, let us set A to be the 2ℓ × n matrix

and identify B with the n × n matrix μ,

Noting that

we make use of the lhs of lemma 1, to write down the expansion

Consequently, the lhs of equation (1.5) reduces to (note that n is even by the hypothesis)

Here, we have used lemma 3. By corollary 1, the rhs of equation (3.10) is equivalent to

This concludes the proof.

Acknowledgments

EK is grateful to G Akemann, a joint work with whom on the integrable structure of Ginibre's Orthogonal Ensemble has triggered this study. AB was partially supported by the NSF grants DMS-0402047 and DMS-0707163. EK acknowledges a partial support by the Israel Science Foundation through the grant no 286/04.

Appendix. The Fredholm Pfaffian

Let K(x, y) be a 2 × 2 matrix kernel

which is antisymmetric under the change of its arguments, K(x, y)  =  – K^T(y, x), and yet another 2 × 2 matrix kernel J(x, y) be

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Notes

Note1

 For a discussion of physical applications of Ginibre's random matrices, the reader is referred to the detailed paper by Akemann and Kanzieper (2007).

Note2

 Up to a factorial prefactor, the elementary symmetric functions e_ℓ(p₁,  ..., p_ℓ) coincide with the zonal polynomials appearing in the original formulation of the theorem.

Note3

 See appendix for the notation.

Note4

 In what follows, we assume that our measures are such that all integrals are finite.

Note5

 Lemma 2 is a reformulation of theorem 3.1 by Ishikawa and Wakayama (2000).

Note6

 Compare to equation (1.4).