The goal of this section is to establish exact results for multi-point correlators of real and quaternionic ensembles, focusing mainly on the real case. We begin without the multiplicative perturbations in (1.16). The reason is mainly pedagogical; for \(\Omega = I_\) we can derive Pfaffian closed form expressions for (1.16) with only a few modifications of the approach of Sect. 2, while the case of general \(\Omega \) will require the use of zonal spherical functions which are discussed separately in Sect. 3.4.

Let \(\eta \) be a partition of length k. We say that \(\lambda = 2\eta = (2\eta _1, 2\eta _2,\ldots ,2\eta _)\) is a doubled partition derived from the partition \(\eta \) if it is constructed by doubling each part of the original partition. In what follows, we will say that a partition \(\lambda \) is an even partition if each of its parts is even, and consequently, there is a partition \(\eta \) such that \(\lambda = 2\eta \). If we repeat each part of \(\lambda \) twice, we then say it is a repeated partition, denoted \(\lambda ^2 = (\lambda _1, \lambda _1, \lambda _2, \lambda _2,\ldots )\). One may observe that \((2\lambda )' = (\lambda ')^2\). See Figs. 4 and 5.

Fig. 4

\(2\lambda = ( 8, 4, 2 )\)

Fig. 5

Doubled and repeated partitions for \(\lambda = (4,2,1)\). \(\lambda ^ = ( 4,4,2,2,1,1)\)

3.1 Real Ginibre Ensemble: \(\Omega =I_\)

We start by computing the multi-point correlator

$$\begin R^}_(\varvec) = }\left( \prod _^\det (G-z_)\right) . \end$$

(3.1)

After the same expansion of the determinants using the dual Cauchy identity, the relevant Schur function average can be computed with a result of Sommers and Khoruzhenko [62] and Forrester and Rains [26]:

$$\begin }(s_(G)) = 2^[N/2]^_, & \lambda =2\eta \\ 0, & \textrm\end\right. }, \end$$

(3.2)

where the generalized hypergeometric coefficient is

$$\begin [u]^_\eta = \prod ^N_ \frac . \end$$

(3.3)

Note the following property of generalized hypergeometric coefficients

$$\begin [u]^_ = (-\alpha )^ [-\alpha u]^_, \end$$

(3.4)

see Lemma A.1. These results allow one to proceed as in the complex case, starting with an expansion of (3.1) in terms of Schur functions via the dual Cauchy identity (2.5) and then applying (3.2). This step was essentially pointed out in [62]. The problem is then to evaluate the resulting sum over partitions which to our knowledge was never achieved. We show that this can be done with the following Pfaffian version of the Cauchy–Binet identity.

Lemma 3.1

[Ishikawa–Wakayama Pfaffian identity [35, 36]]. Let A be an \(N \times N\) anti-symmetric matrix and B be an \(M \times N\) matrix such that \(M \le N\) with M even. Then

$$\begin \begin \,}}( BAB^T )&= \,}}\bigg \ A_ ( B_ B_ - B_ B_ ) \bigg \}_ \nonumber \\&= \sum _\det \bigg \ \bigg \}_^\,}}\bigg \\bigg \}_. \end \end$$

This leads to a general summation identity over repeated partitions which results in a Pfaffian instead of a determinant. We were motivated to prove the following lemma by work of Betea and Wheeler [10] who proved a particular case in relation to so-called refined Cauchy–Littlewood identities.

Lemma 3.2

Let \(\varvec \in }^\) and let \(f:} \rightarrow }\). Then

$$\begin \sum _ \mu , l(\mu ) \le 2k, \mu _1 \le N \\ \mu '\,\textrm \end}&s_\mu (\varvec) \prod _^k f(\mu _+2k-2j) \nonumber \\&= \frac\,}}\bigg \^ f(l)(x_i x_j)^l \bigg \}_} )} . \end$$

(3.5)

Proof

Inserting definition (2.3) of the Schur functions, the left-hand side of (3.5) is

$$\begin \frac)} \sum _\det \bigg \\bigg \}_^\prod _^ f(\mu _+2(k-j)) \delta _, \mu _}.\nonumber \\ \end$$

(3.6)

We introduce indices \(l_ = \mu _+2k-j\) for \(j=1,\ldots ,2k\). Then \(\\}_^\) satisfies the weakly decreasing property with \(\mu _ \ge 0\) and \(\mu _ \le N\) if and only if we have the constraints

$$\begin 0 \le l_< l_< \ldots < l_ \le N+2k-1. \end$$

Furthermore, \(\mu _=\mu _\) if and only if \(l_+1=l_\) for each \(j=1,\ldots ,k\). Then we can write the sum in (3.6) as

$$\begin \sum _< \ldots < l_ \le N+2k-1}\det \bigg \\bigg \}_^\prod _^ f(l_) \delta _, l_+1}. \end$$

(3.7)

Now we express the product over j as the Pfaffian of an anti-symmetric tridiagonal matrix, using the fact that for a generic sequence \(\_^\), we have

$$\begin \,}}\begin 0 & \quad a_ & \quad & \quad & \quad \\ -a_ & \quad 0 & \quad a_ & \quad & \quad & \\ & -a_ & \quad 0 & \quad \ddots & \quad \\ & & \quad & \quad \ddots & \quad a_ & \quad \\ & & \quad & \quad -a_ & \quad 0 & \quad a_ \\ & & \quad & \quad & \quad -a_ & \quad 0 \end = \prod _^a_. \end$$

This identity follows inductively from a Laplace expansion along the first row. Hence, we have

$$\begin \begin \prod _^k f(l_) \delta _,l_+1}&= \,}}\bigg \ f(l_) \delta _,l_+1} \bigg \}_ \\&= \,}}\bigg \) \delta _,l_+1} \bigg \}_, \end \end$$

where \(\delta _\delta _,l_+1} = \delta _,l_+1}\) follows from the strictly decreasing property of \(\\}_^\). Hence, after reordering indices, we can write (3.7) as

$$\begin \begin \sum _< \ldots< l_ \le N+2k} \,}}\bigg \-1) \delta _+1,l_} \bigg \}_\det \bigg \-1}\bigg \}_^ \end,\qquad \quad \end$$

(3.8)

to which Lemma 3.1 is applicable. For this, we identify the matrix elements in the lemma as

$$\begin \begin A_&= f(i-1)\delta _, \qquad 1 \le i < j \le N+2k\\\qquad B_&= x_^, \hspace i=1,\ldots ,2k, \quad j=1,\ldots ,N+2k \end, \end$$

so that (3.8) becomes

$$\begin \begin&\,}}\bigg \ f(l-1) \delta _ (x_i^ x_j^ - x_i^ x_j^) \bigg \}_ \\&= \,}}\bigg \^ f(l)(x_i x_j)^l \bigg \}_, \end \end$$

which completes the proof. \(\square \)

Theorem 3.3

We have

$$\begin R^}_(\varvec) = \prod ^_ (N + 2j-2)!\,\frac\,}}\bigg \(z_i, z_j) \bigg \}_ })}, \end$$

where

$$\begin B_(z,w) = \sum _^\frac}. \end$$

Proof

By the dual Cauchy identity (2.5) and average (3.2), we can write the correlator (3.1) as

$$\begin R^}_(\varvec) = \sum _ \mu : l(\mu ) \le 2k, \mu _ \le N\\ \mu ' \textrm \end}s_}(\varvec)2^[N/2]^_, \end$$

(3.9)

where \(\eta ' = \mu '/2\) and we used (2.7). Using Lemma A.1, equation (A.3) with \(u=N\) and \(\alpha =1/2\), we can express the coefficient in (3.9) as

$$\begin 2^[N/2]^_= \prod _^\frac}+2(k-j))!}, \end$$

where \(\tilde\) is the complement partition of \(\eta \), recall definition (2.6). Since \(\mu '\) is even, we deduce that \(\mu \) is repeated and \(\tilde_ = \tilde_\). Then replacing \(\tilde\) with \(\mu \), the sum in (3.9) becomes

$$\begin R^}_(\varvec) = \prod _^(N+2j-2)!\sum _ \mu : l(\mu ) \le 2k, \mu _ \le N\\ \mu ' \textrm \end} s_(\varvec) \prod _^\frac+2(k-j))!} . \end$$

Applying Lemma 3.2 with \(f(l) = \frac\) completes the proof. \(\square \)

3.2 Truncated Orthogonal Ensemble: \(\Omega =I_\)

We now prove an analogous Pfaffian formula for truncations of random orthogonal matrices.

Theorem 3.4

Let \(\varvec \in }^\) and T be a \(M \times M\) truncation of a \(N \times N\) Haar distributed orthogonal matrix. Then

$$\begin R^}_(\varvec) = D^_ \frac\,}}\bigg \^ \frac (z_ z_)^ \bigg \}_ })}, \end$$

where \(D^_\) is given in (1.25).

Proof

We proceed as in the real Ginibre case and expand the characteristic polynomial product in terms of Schur functions. The relevant average is now the one computed in [61] as

$$\begin }( s_\mu (T)) = \delta _ \frac^}_\eta } . \end$$

We then have

$$\begin&R^}_(\varvec) \nonumber = \sum _ \mu : l(\mu ) \le 2k, \mu _ \le M\\ \mu ' \textrm \end}s_}(\varvec) \delta _ \frac^}_}. \end$$

Rewriting the hypergeometric coefficients using Lemma A.1 with \(\alpha =1/2\), we obtain

$$\begin&R^}_(\varvec) = D_^ \sum _ \mu , l(\mu ) \le 2k, \mu _1 \le M \\ \mu ' \text \end} s_(\varvec) \prod ^_ \frac + 2(k - j) )!} + 2(k - j))!}. \end$$

Finally, we employ Lemma 3.2 with the choice

$$\begin f(l) = \frac \end$$

to obtain the result. \(\square \)

3.3 Probabilistic Interpretation: Real Ensembles

In the spirit of Sect. 2.4, we introduce the following interpretation of the real ensemble multi-point correlators.

From the proof of Theorem 3.3, we recall the representation

$$\begin R^}_(\varvec) = \sum _ \eta , l(\eta ) \le 2k, \mu _ \le N\\ \eta ' \textrm \end} s_(\varvec) \prod _^ \frac+2(k-j))!} . \end$$

Consider the set \(}_\) of all partitions \(\eta \) of length \(l(\eta ) \le k\) and assign each \(\eta \in }_\) the probability

$$\begin p(\eta ) = \displaystyle }_} s_(\varvec) \frac+2(k-j))!}, } & \eta ' \text , \\ 0, \ & \text \end\right. }, \end$$

(3.10)

where \(}_\) is a normalization factor. To have a probabilistic interpretation, we require \(\varvec \in }^_\) so that (3.10) is positive by Schur positivity. The normalization constant in (3.10) is obtained by following the steps in the proof of Theorem 3.3 with no restriction on \(\eta _\). This gives

$$\begin }_&= \sum _ \eta , l(\eta ) \le 2k \\ \eta ' \textrm \end} s_(\varvec) \prod ^k_ \frac+2(k-j)+1)} = \frac\,}}\bigg \ z_} \bigg \}_ })} \nonumber \\&= \left( \prod _^ \frac \right) \int _(k)}dU\,e^ \textrm(UZU^Z^})}, \end$$

(3.11)

where CSE(k) denotes the Circular Symplectic Ensemble of \(2k \times 2k\) self-dual unitary matrices U, i.e., unitary matrices satisfying \(U^D = J^ U^T J = U\), where

$$\begin J = \left( 0 & & I_k \\[1.5mm] -I_k & & 0 \end} \right) . \end$$

The measure dU in (3.11) is the restriction of the Haar measure on U(2k) to self-dual unitary matrices satisfying \(U^}=U\) and normalized as a probability measure. See, e.g., [24] for further details about the CSE. The integration formula (3.11) in terms of a Pfaffian has been the subject of some attention recently, see the works [39, 65, 66].

Corollary 3.5

When \(\Omega = I_\), we have the following probabilistic interpretation for the multi-point correlator in (1.16):

$$\begin R^}_(\varvec) = }_\left( \prod _^(N+2j-2)!\right) }(\eta _ \le N), \end$$

where the probability on the right-hand side is defined with respect to distribution (3.10) and \(}_\) is given by (3.11).

In a similar manner, we provide a description for the orthogonal truncations, assuming \(\varvec \in (0,1)^\). We define the probability distribution as

$$\begin p(\eta ) = \displaystyle }_}s_(\varvec) \prod ^_ \frac + 2(k - j))!} + 2(k - j))!}, } & \eta ' \text , \\ 0, \ & \text \end\right. },\qquad \quad \end$$

(3.12)

where the normalization constant is

$$\begin&}_ = \sum _ \eta , l(\eta ) \le 2k \\ \eta ' \text \end} s_(\varvec) \prod ^_ \frac + 2(k - j))!} + 2(k - j))!}\nonumber \\&=\,[(N-M)!]^\frac\,}}\bigg \} \bigg \}_})} \nonumber \\&= \left( \prod _^ \frac \right) \int _(k)}dU\,\det ( I_ - UZU^Z^})^ ( N - M + 2k - 1 )}. \end$$

(3.13)

The above integration identity can be deduced by application of Lemma 3.2; we postpone the details to a separate publication.

Corollary 3.6

When \(\Omega = I_\) and \(\varvec \in (0,1)^\), we have the following probabilistic interpretation for the multi-point correlator in (1.21):

$$\begin&R^}_(\varvec)=}_D^_}(\eta _ \le N) \end$$

where the probability on the right-hand side is defined with respect to distribution (3.12), \(}_\) is given by (3.13), and \(D^_\) is given by (1.25).

Similar character expansions were obtained in [25] for averages of O(N) characteristic polynomials in the context of last passage percolation models.

3.4 Duality Formulas in Real Ensembles: General \(\Omega \)

In this subsection, we derive dual integral representations of the multi-point correlators of real and quaternionic ensembles. Compared to the previous subsection, we now include the multiplicative perturbations \(\Omega \). Starting with the real Ginibre ensemble, our goal is to evaluate

$$\begin R^}_(\varvec;\Omega ) = }\left( \prod _^\det (\Omega G-z_iI_)\right) . \end$$

Before we proceed, let us introduce a few concepts we shall require. The generalized hypergeometric coefficients (3.3) obey the following relations, see [24],

$$\begin [u]^_ = 2^ [u/2]^_ [(u+1)/2]^_ , \qquad [u]_^ = [u]_^ [u-1]_^ .\qquad \quad \end$$

(3.14)

Further, we introduce \(\alpha \)-deformations of (2.1)

$$\begin&d'_\eta (\alpha ) = \prod _ (\alpha a(i,j) + l(i,j) + \alpha ) , \\&h_\eta (\alpha ) = \prod _ (\alpha a(i,j) + l(i,j) + 1 ) . \end$$

In what follows, we denote \(d'_\eta (1) = d'_\eta \). Note that \(d'_\eta = h_\eta (1)\). These coefficients with different values of \(\alpha \) can be related as follows, see, e.g., [26]

$$\begin h_(2) d'_(2) = d'_ , \quad 2^h_\eta (1/2)d'_\eta (1/2) = d'_, \quad d'_(\alpha ) = \alpha ^h_\eta (\alpha ^) .\qquad \quad \end$$

(3.15)

Let us introduce Jack polynomials \(P_\eta ^(X)\) that can be defined as eigenfunctions of the differential operator

$$\begin \sum ^N_ \left( x_j \frac \right) ^2 + \frac \sum ^N_ x_j \frac + \frac \sum _ \frac \left( \frac - \frac \right) , \end$$

with the additional structure

$$\begin P_^(X) = m_(X)+\sum _a^_m_(X), \end$$

where \(m_\) are the monomial symmetric functions, \(a^_\) are coefficients that do not depend on X, and \(\mu < \eta \) is the dominance ordering on partitions, see [24, 52, 63] for further details. In the Schur polynomial case \(\alpha =1\), these coefficients are better known as Kostka numbers. Jack polynomials are evaluated at the identity, see, e.g., [26], as

$$\begin P_\eta ^(I_N) = \frac[N/\alpha ]_^} . \end$$

(3.16)

Remark 3.7

In the case \(\alpha = \frac\), Jack polynomials are frequently evaluated on self-dual matrices that have doubly degenerate eigenvalues. It is convenient for the rest of the paper to define the Jack polynomial for \(\alpha = \frac\) to evaluate only the distinct eigenvalues, see [21]. Hence, if X has eigenvalues \(\varvec = (x_1,x_1,x_2,x_2,\ldots ,x_k,x_k)\), we define \(P^)}_(X) := P^)}_(x_1,x_2,\ldots ,x_k)\). If \(\varvec \in }^\), the generalized Cauchy identity gives

$$\begin \sum _ P^_\eta (\varvec) P^_(\varvec) = \sqrt + \varvec \otimes \varvec)}. \end$$

(3.17)

For more details, see [21].

We begin with the duality formula for the real Ginibre ensemble, completing the proof of Theorem 1.4.

Proof of Theorem 1.4

Proceeding as in the unperturbed case, we employ the dual Cauchy identity, which yields

$$\begin \begin R^}_(\varvec;\Omega )&= } \left[ \prod ^_ \det ( \Omega G - z_j I) \right] \\&= \det (Z)^\sum _ s_\mu (\varvec^) } \left[ s_(\Omega G) \right] . \end \end$$

Now, we recognize the Schur function average as that computed by Forrester and Rains in [26], namely

$$\begin }\left[ s_\mu (\Omega G) \right] = \delta _ h_\eta (2)P_\eta ^(\Omega \Omega ^T) . \end$$

(3.18)

Substituting this average into the sum over partitions, we find

$$\begin \begin R^}_(\varvec;\Omega )&= \det (Z)^ \sum _ s_\mu (\varvec^) \delta _ h_(2)P_^(\Omega \Omega ^T) \\&=\det (Z)^\sum _ s_(\varvec^) h_(2)P_^(\Omega \Omega ^T), \end \end$$

(3.19)

where to obtain the last line we have used that \((2\eta ')'=\eta ^2\). Next we recognize the summand as the integral of Lemma B.9, that is,

$$\begin s_(Z^)\,h_(2) = \frac} \int _}_(}) } dX e^ \,}}XX^\dag } P_^ (Z^ X Z^ X^\dag ), \end$$

where \(C_ = \int _}_(}) } dX e^ \,}}XX^\dag }\). Substituting this expression in (3.19) and interchanging it with the summation, we obtain

$$\begin \begin R^}_(\varvec;\Omega )&= \det (Z)^ \frac} \int _}_(}) } dX e^ \,}}XX^\dag }\\&\times \sum _ P_^ (Z^ X Z^ X^\dag ) P_^(\Omega \Omega ^T) . \end \end$$

Computing the sum using the Cauchy identity (3.17) proves (1.18). Setting \(\Omega = I_\) in (1.18), similar to the complex case, we recognize the Pfaffian of a block matrix leading directly to (1.19). Together with the Pfaffian evaluation derived in Sect. 3.1, this completes the proof of Theorem 1.4. \(\square \)

We now conclude the proof of Theorem 1.5 regarding truncated orthogonal random matrices.

Proof of Theorem 1.5

As the proof is similar to the proof of Theorem 1.4, we just give the main ideas. Having expanded the characteristic polynomial product (1.21) in terms of Schur functions, the relevant Schur function average generalizing (3.18) was computed in [61] as

$$\begin } \left[ s_\mu (\Omega T) \right] = \delta _ \frac h_(2)}_} P_^(\Omega \Omega ^T). \end$$

Substituting this average in the sum over partitions, we have

$$\begin R^}_(\varvec;\Omega )=\det (Z)^\sum _\frac(\varvec^) h_(2)} \left[ -N \right] ^_}P_^(\Omega \Omega ^T), \end$$

(3.20)

where we used the transposition property (3.4) of \([u]^_\). Next we recognize the summand as the integral of Lemma B.10, namely the coefficient \(\frac(\varvec^) h_(2)} \left[ -N \right] ^_}\) is equal to

$$\begin \begin \frac_}\int _}_(})} dX\det ( I_ + XX^\dag )^ P_\eta ^(Z^ X Z^ X^\dag ). \end \end$$

Substituting this integral in (3.20) and computing the sum over partitions with the help of the generalized Cauchy identity yield the result. The computation of the normalization constant \(S^_\) is given in Lemma B.11. \(\square \)

3.5 Quaternionic Ensembles

Consider the multi-point correlator

$$\begin R^}_(\varvec;\Omega ) = }\left( \prod ^_ \det ( \Omega G -z_ I_ ) \right) , \end$$

where the average is over a matrix G from the quaternionic Ginibre ensemble. As in previous cases, we abbreviate \(R^}_(\varvec) := R^}_(\varvec;I_)\). For background on the quaternionic Ginibre ensemble, we refer the reader to the review article [16]. As this ensemble has a natural symplectic symmetry, it is also referred to as Ginibre Symplectic Ensemble (GinSE). Here, we will adopt the same underlying Gaussian measure as in [26]. The character expansion approach here goes through similarly to the real case so we just present the main differences. The required Schur function average was obtained in [26] as

$$\begin }(s_(\Omega G)) = \delta _}h_(1/2)P^_(\Omega \Omega ^). \end$$

(3.21)

Since the above will be applied to the conjugate partition \(\mu '\), we need an analogue of Lemma 3.2 where we sum over even partitions rather than repeated partitions.

Lemma 3.8

Let \(\varvec \in }^\) and let \(f:} \rightarrow }\). Then

$$\begin&\sum _ \mu , l(\mu ) \le 2k, \mu _1 \le 2N \\ \mu \,\textrm \end} s_\mu (\varvec) \prod _^ f(\mu _+2k-j)\\&\qquad =\frac\,}}\bigg \ f(2l) f(2p-1) (x_i^ x_j^ - x_i^ x_j^) \bigg \}_} )}. \end$$

Proof

Following the same steps as for the proof of Lemma 3.2, here we need to evaluate

$$\begin \sum _\det \bigg \^+2k-j}\bigg \}_^\prod _^ f(\mu _+2k-j)\prod _^\mathbbm _ \,\textrm}\qquad \quad \end$$

(3.22)

and we begin with the same substitution \(l_ = \mu _+2k-j\) for \(j=1,\ldots ,2k\). Note that the partition \(\mu \) is even if and only if \(l_\) is odd and \(l_\) is even for \(j=1,\ldots ,k\). We write the product of indicator functions in (3.22) as the Pfaffian

$$\begin \prod _^\mathbbm _\,\textrm}\mathbbm _\,\textrm} = \textrm\bigg \_\,\textrm}\mathbbm _\,\textrm}\bigg \}_. \end$$

This identity follows from the fact that for generic sequences \(\\}_^\), \(\\}_^\), we have

$$\begin \prod _^a_b_ = \textrm\bigg \b_\bigg \}_, \end$$

which follows by induction on k and Laplace expansion. Therefore, after shifting indices by 1 and reordering, we see that (3.22) is equal to

$$\begin \sum _< \ldots< l_ \le 2N+2k}\det \bigg \^-1}f(l_-1)\bigg \}_^\textrm\bigg \_\,\textrm}\mathbbm _\,\textrm}\bigg \}_. \end$$

Now the proof is completed by applying Lemma 3.1 with matrix elements

$$\begin \begin A_&= \mathbbm _}\mathbbm _}, \hspace\qquad 1 \le i < j \le 2N+2k,\\ B_&= x_^f(j-1) , \qquad i=1,\ldots ,2k, \quad j=1,\ldots ,2N+2k. \end \end$$

\(\square \)

Applying the above lemma, we obtain Pfaffian formulas for multi-point correlators.

Theorem 3.9

For \(\varvec \in }^\), we have

$$\begin R^}_(\varvec) = \left( \prod ^_ \Gamma (N+j/2+1/2)\right) \,\frac\,}}\bigg \}_(z,w)\bigg \}_})} \end$$

where the kernel is

$$\begin B^}_(z,w) = \sum _^\sum _^\frac\,(z^w^-w^z^). \end$$

(3.23)

Proof

For \(\Omega = I_\), the result (3.21) is

$$\begin }(s_(G)) = \delta _2^[2N]^_. \end$$

We apply Lemma A.1, equation (A.3) with \(\alpha =2\), so that

$$\begin 2^[2N]^_ = \prod _^\frac_+(2k-j)/2+1)}. \end$$

Then we have

$$\begin R^}_(\varvec) = \sum _ \mu : l(\mu ) \le 2k, \mu _ \le 2N\\ \mu \,\textrm \end}s_(\varvec)\prod _^\frac+2k-j)/2+1)}. \end$$

(3.24)

The proof is completed by applying Lemma 3.8 with

$$\begin f(j) = \frac. \end$$

\(\square \)

The Pfaffian expression above recovers a result of Akemann and Basile [3]. Their approach differs from ours through their use of the joint probability density function of eigenvalues of G. The kernel (3.23) was first discovered in the context of eigenvalue correlations of symplectic ensembles in [41]. Using the representation as a character sum in (3.24), one can write an analogous probabilistic interpretation that we obtained in the real case in Corollary 3.5; we omit the details.

Dualities of the type (1.19) and (1.23) also hold in the quaternionic case, the main difference being that the dual integrals are over complex symmetric matrices instead of anti-symmetric matrices. Let \(}_(})\) denote the space of \(2k \times 2k\) symmetric matrices with complex entries. On the space \(}_(})\), we put the Gaussian measure

$$\begin \mu _(dX) = \frac\pi ^}\,e^\textrm(XX^)}\,dX, \end$$

(3.25)

where dX is the Lebesgue measure on the upper triangular and diagonal elements of X.

Theorem 3.10

Let \(\varvec \in }^\). Then with \(}_\) denoting the average with respect to (3.25), we have

$$\begin R^}_(\varvec;\Omega ) = \det (Z)^}_(\textrm( I_ + \Omega \Omega ^\dag \otimes Z^ X Z^ X^\dag )^})\qquad \end$$

(3.26)

and for \(\Omega = I_\) (3.26) equals

$$\begin 4^\pi ^ \int _}_(})} dX e^ \,}}XX^\dag } \det \left( X & & Z\\ -Z & & X^\dag \end} \right) ^. \end$$

(3.27)

Proof

Applying the dual Cauchy identity, we have

$$\begin R^}_(\varvec;\Omega ) = \det (Z)^\sum _\le 2N}s_(Z^)h_(1/2)P_^(\Omega \Omega ^).\qquad \quad \end$$

(3.28)

Using integral identity (B.6), we recognize the summand as

$$\begin \begin s_(Z^)h_(1/2)=\frac}\int _}_(}) } dX e^\,}}XX^\dag } P_^ (Z^ X Z^ X^\dag ). \end \end$$

(3.29)

Inserting (3.29) into (3.28) and summing using the generalized Cauchy identity (3.17) complete the proof. \(\square \)

For a derivation of (3.27) using Grassmann calculus, see [53]. We now give the corresponding result for truncations of symplectic unitary matrices. The main difference with the Ginibre ensemble is that the duality comes from a different integral representation for the coefficients that arise in the character expansion (see (B.9)). Consider the following dual measure on the space \(}_(})\)

$$\begin \mu _(dX) = \frac^}\,\det ( I_ + XX^\dag )^\, dX, \end$$

(3.30)

where \(S_^ = \int _}_(}) } dX \det ( I_ + XX^\dag )^\).

Theorem 3.11

Let T be an \(M \times M\) truncation of a \(N \times N\) Haar distributed symplectic matrix. Then with \(}_\) denoting the average with respect to (