# Characteristic Polynomials in Coupled Matrix Models

Nicolas Babinet and Taro Kimura Institut de Mathématiques de Bourgogne,
Université Bourgogne Franche-Comté

###### Abstract

We study correlation functions of the characteristic polynomials in coupled
matrix models based on the Schur polynomial expansion, which manifests their
determinantal structure.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Introduction
    2. 1.2 Summary of the results
  2. 2 Coupled matrix model
    1. 2.1 Determinantal formula
    2. 2.2 Christoffel–Darboux kernel
    3. 2.3 Operator formalism
    4. 2.4 Polynomial ensemble
  3. 3 Characteristic polynomial averages
    1. 3.1 Schur polynomial average
    2. 3.2 Characteristic polynomial
    3. 3.3 Characteristic polynomial inverse
  4. 4 Pair correlation functions
    1. 4.1 Characteristic polynomial
    2. 4.2 Characteristic polynomial inverse
    3. 4.3 Mixed pair correlation

##  1 Introduction and summary

###  1.1 Introduction

Random Matrix Theory (RMT) has been playing an important role over the decades
in the both of physics and mathematics communities [Meh04, For10, ABDF11,
EKR15]. Applying the analogy with Quantum Field Theory (QFT), the asymptotic
behavior appearing in the large size limit (large $N$ limit) is interpreted as
a classical behavior as the parameter $1/N$ plays a role of the Planck
constant. From this point of view, it is an important task to explore the
finite $N$ result to understand the quantum $1/N$ correction and also the non-
perturbative effect beyond the perturbative analysis. The purpose of this
paper is to show the finite $N$ exact result of a class of the correlation
functions in the generalized two-matrix model, what we simply call the coupled
matrix model, which contains various models coupled in the chain. See, e.g,
[IZ80, EM98, BEH02, BEH03b, BEH03a, BE03, BGS09] and also [Eyn05, Ber11,
Ora11] for the development in this direction. We will show that this model can
be analyzed using its determinantal structure, which is a key property to
obtain the finite $N$ exact result. In this paper, we in particular consider
the correlation function of the characteristic polynomials in the coupled
matrix model. It has been known in the context of RMT that the characteristic
polynomial plays a central role in the associated differential equation system
through the Riemann–Hilbert problem and the notion of quantum curve. In
addition, the characteristic polynomial is essentially related to various
other important observables in RMT, e.g., the resolvent, the eigenvalue
density function, etc. See, e.g., [Mor94, BH00, FS03, SF03, AV03, BDS03, BS06]
and also [BH11] for earlier results in this direction.

###  1.2 Summary of the results

We state the summary of this paper. In Section 2, we introduce the generalized
coupled matrix model defined as the following formal eigenvalue integral,

$\displaystyle
Z_{N}=\frac{1}{N!^{2}}\int\prod_{k=L,R}\differential{X}_{k}\mathrm{e}^{-\tr
V_{k}(X_{k})}\prod_{i<j}^{N}(x_{k,i}-x_{k,j})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})$ (1.1)

for arbitrary potential functions $V_{k}(x)$ and a two-variable function
$\omega(x,y)$. See Definition 2.1 for details. We then show that this
eigenvalue integral is reduced to the determinant of the norm matrix of the
corresponding two-variable integral. We also show that biorthogonal
polynomials, which diagonalize the norm matrix, simplify the formulas. We
mention in Section 2.4 that the analysis shown there is straightforwardly
applied to the coupled matrix generalization of the polynomial ensemble [KS14]
defined for a set of arbitrary functions, containing various known models,
e.g., the external source model [BH16]. See also [Bor98]. In Section 3, we
study the correlation function for the coupled matrix model. In Section 3.1,
we show the Schur polynomial average, which will be a building block of the
characteristic polynomials discussed throughout the paper. In Sections 3.2 and
3.3, we explore the correlation function of the characteristic polynomial and
its inverse, and show that they are concisely expressed as a determinant of
the biorthogonal polynomial and its dual. We remark that these results are
natural generalization of the earlier results on the one-matrix model case. In
Section 4, we consider the pair correlation function, which involves both the
characteristic polynomials coupled with $X_{L}$ and $X_{R}$. In this case, the
correlation functions are again expressed as a determinant, while the
corresponding matrix element is written using the Christoffel–Darboux (CD)
kernel and its dual.

### Acknowledgments

We would like to thank Bertrand Eynard for useful conversation. This work was
supported in part by “Investissements d’Avenir” program, Project ISITE-BFC
(No. ANR-15-IDEX-0003), EIPHI Graduate School (No. ANR-17-EURE-0002), and
Bourgogne-Franche-Comté region.

##  2 Coupled matrix model

In this paper, we explore the coupled matrix model defined as follows.

######  Definition 2.1 (Partition function).

Let $V_{k}(x)$ $(k=L,R)$ be a polynomial function and $\omega(x_{L},x_{R})$ be
a two-variable function. Let $(X_{k})_{k=L,R}=(x_{k,i})_{k=L,R,i=1,\ldots,N}$
be a set of formal eigenvalues. Then, we define the partition function of the
coupled matrix model,

$\displaystyle
Z_{N}=\frac{1}{N!^{2}}\int\prod_{k=L,R}\differential{X}_{k}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N}(X_{L})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\Delta_{N}(X_{R})\,,$ (2.1)

where we denote the Vandermonde determinant by

$\displaystyle\Delta_{N}(X)=\prod_{i<j}^{N}(x_{i}-x_{j})\,.$ (2.2)

######  Remark 2.2.

We formally consider the eigenvalues $(x_{k,i})$ as complex variables, and
thus their integration contour is taken to provide a converging integral,
which is not unique in general. In this paper, we do not discuss the contour
dependence on the partition function, so that we always consider the
eigenvalue integral as a formal integral.

Throughout the paper, we frequently use the following identity.

######  Lemma 2.3 (Andréief–Heine identity).

Let $(f_{i}(x))_{i=1,\ldots,N}$ and $(g_{i}(x))_{i=1,\ldots,}$ be the
sequences of integrable functions on the domain $D$. Denoting
$\differential{X}=\differential{x}_{1}\cdots\differential{x}_{N}$, the
following identity holds,

$\displaystyle\frac{1}{N!}\int_{D^{N}}\differential{X}\det_{1\leq i,j\leq
N}f_{i}(x_{j})\det_{1\leq i,j\leq N}g_{i}(x_{j})=\det_{1\leq i,j\leq
N}\quantity(\int_{D}\differential{x}f_{i}(x)g_{j}(x))\,,$ (2.3)

which is called the Andréief–Heine (AH) identity.

######  Proposition 2.4 (Hermitian matrix chain models).

Let $(M_{k})_{k=1,\ldots,\ell}$ be a set of $\ell$ Hermitian matrices of rank
$N$. The following matrix chain models are reduced to the coupled matrix model
of the form of (2.1):

$\displaystyle Z_{\text{pot}}$
$\displaystyle=\int\prod_{k=1,\ldots,\ell}\differential{M_{k}}\mathrm{e}^{-\tr
V_{k}(M_{k})}\prod_{k=1}^{\ell-1}\mathrm{e}^{\tr M_{k}M_{k+1}}\,,$ (2.4a)
$\displaystyle Z_{\text{Cauchy}}$
$\displaystyle=\int\prod_{k=1,\ldots,\ell}\differential{M_{k}}\mathrm{e}^{-\tr
V_{k}(M_{k})}\prod_{k=1}^{\ell-1}\det(M_{k}\otimes\mathbbm{1}_{N}+\mathbbm{1}_{N}\otimes
M_{k+1})^{-N}\,.$ (2.4b)

We call them the potential-interacting matrix chain and the Cauchy-interacting
matrix chain, respectively.

###### Proof.

Diagonalizing each Hermitian matrix using the unitary transform for
$k=1,\ldots,\ell$,

$\displaystyle M_{k}=U_{k}X_{k}U_{k}^{-1}\,,\qquad
X_{k}=\operatorname{diag}(x_{k,1},\ldots,x_{k,N})\,,\qquad
U_{k}\in\mathrm{U}(N)\,,$ (2.5)

the matrix measure is given by

$\displaystyle\differential{M}_{k}=\frac{\differential{U_{k}}\differential{X}_{k}}{N!(2\pi)^{N}}\Delta_{N}(X_{k})^{2}\,,\qquad\differential{X}_{k}=\prod_{i=1}^{N}\differential{x}_{k,i}\,,$
(2.6)

where we denote the Haar measure of each unitary matrix by
$\differential{U_{k}}$. We remark that the factors $(2\pi)^{N}$ and $N!$ are
interpreted as the volumes of the maximal Cartan torus
$\mathrm{U}(1)^{N}\subset\mathrm{U}(N)$, and the symmetric group
$\mathfrak{S}_{N}$, which is the Weyl group of the unitary group
$\mathrm{U}(N)$.

For the potential-interacting chain, we may use the Harich-
Chandra–Itzykson–Zuber formula [IZ80],

$\displaystyle\int_{\mathrm{U}(N)}\differential{U}\mathrm{e}^{\tr
UXU^{-1}Y}=\frac{c_{N}}{\Delta_{N}(X)\Delta_{N}(Y)}\det_{1\leq i,j\leq
N}\mathrm{e}^{x_{i}y_{j}}$ (2.7)

where the constant factor $c_{N}=\Gamma_{2}(N+1)=\prod_{j=0}^{N-1}j!$ is
chosen to be consistent with the normalization of the group integral,
$\int_{\mathrm{U}(N)}\differential{U}=1$. Then, we obtain

$\displaystyle Z_{\text{pot}}$
$\displaystyle=\frac{c_{N}^{\ell-1}}{N!^{\ell}}\int\prod_{k=1,\ldots,\ell}\frac{\differential{X}_{k}}{(2\pi)^{N}}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N}(X_{1})\quantity(\prod_{k=1}^{\ell-1}\det_{1\leq
i,j\leq N}\mathrm{e}^{x_{k,i}x_{k+1,j}})\Delta_{N}(X_{\ell})$
$\displaystyle=\frac{c_{N}^{\ell-1}}{N!^{2}}\int\prod_{k=1,\ell}\frac{\differential{X}_{k}}{(2\pi)^{N}}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N}(X_{1})\det_{1\leq i,j\leq
N}\quantity(\int\prod_{k=2,\ldots,\ell-1}\frac{\differential{x}_{k}}{2\pi}\mathrm{e}^{-V_{k}(x_{k})}\prod_{k=1}^{\ell-1}\mathrm{e}^{x_{k}x_{k+1}})\Delta_{N}(X_{\ell})\,,$
(2.8)

where we apply the AH identity (Lemma 2.3) for $(X_{k})_{k=2,\ldots,\ell-1}$.
Identifying $(X_{1},X_{\ell})=(X_{L},X_{R})$ and

$\displaystyle\omega(x_{1},x_{\ell})=\int\prod_{k=2,\ldots,\ell-1}\frac{\differential{x}_{k}}{2\pi}\mathrm{e}^{-V_{k}(x_{k})}\prod_{k=1}^{\ell-1}\mathrm{e}^{x_{k}x_{k+1}}\,,$
(2.9)

we arrive at the expression (2.1) up to an overall constant.

For the Cauchy-interacting chain, we remark the relation [BGS09]

$\displaystyle\det(M_{k}\otimes\mathbbm{1}_{N}+\mathbbm{1}_{N}\otimes
M_{k+1})^{-N}$ $\displaystyle\xrightarrow{\text{diagonalization}}\prod_{1\leq
i,j\leq N}\frac{1}{x_{k,i}+x_{k+1,j}}$
$\displaystyle=\frac{1}{\Delta_{N}(X_{k})\Delta_{N}(X_{k+1})}\det_{1\leq
i,j\leq N}\quantity(\frac{1}{x_{k,i}+x_{k+1,j}})\,.$ (2.10)

Therefore, we may write the Cauchy-interacting chain partition function as

$\displaystyle Z_{\text{Cauchy}}$
$\displaystyle=\frac{1}{N!^{\ell}}\int\prod_{k=1,\ldots,\ell}\frac{\differential{X}_{k}}{(2\pi)^{N}}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N}(X_{1})\prod_{k=1}^{\ell-1}\det_{1\leq i,j\leq
N}\quantity(\frac{1}{x_{k,i}+x_{k+1,j}})\Delta_{N}(X_{\ell})\,.$ (2.11)

Similarly, applying the AH identity for $(X_{k})_{k=2,\ldots,\ell-1}$, and
identifying $(X_{1},X_{\ell})=(X_{L},X_{R})$ with

$\displaystyle\omega(x_{1},x_{\ell})=\int\prod_{k=2,\ldots,\ell-1}\frac{\differential{x}_{k}}{2\pi}\mathrm{e}^{-V_{k}(x_{k})}\prod_{k=1}^{\ell-1}\frac{1}{x_{k}+x_{k+1}}\,,$
(2.12)

we arrive at the expression (2.1). This completes the proof. ∎

######  Remark 2.5.

We can in general obtain the coupled matrix model (2.1) from the matrix chain
if the nearest-neighbor interaction is given in the determinantal form

$\displaystyle\frac{1}{\Delta_{N}(X_{k})\Delta_{N}(X_{k+1})}\det_{1\leq
i,j\leq N}I(x_{k,i},x_{k+1,j})$ (2.13)

after the diagonalization. We also remark that the supermatrix model

$\displaystyle Z_{\text{susy}}$
$\displaystyle=\frac{1}{N!^{2}}\int\differential{X}\differential{Y}\mathrm{e}^{-\tr
V(X)+\tr V(Y)}\Delta_{N}(X)^{2}\Delta_{N}(Y)^{2}\prod_{1\leq i,j\leq
N}(x_{i}-y_{j})^{-2}$
$\displaystyle=\frac{1}{N!^{2}}\int\differential{X}\differential{Y}\mathrm{e}^{-\tr
V(X)+\tr V(Y)}\det_{1\leq i,j\leq N}\quantity(\frac{1}{x_{i}-y_{j}})^{2}$
(2.14)

has a closed form to the partition function (2.1), but it does not belong to
the coupled matrix model of our current interest.

###  2.1 Determinantal formula

We show that the partition function (2.1) is written in a determinantal form.
In order to show this, we introduce the notations.

######  Definition 2.6.

We define the inner product with respect to the potentials $V_{L,R}(x_{L,R})$,

$\displaystyle(\,f\mid\omega\mid
g\,)=\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}f(x_{L})\omega(x_{L},x_{R})g(x_{R})\,.$
(2.15)

For a set of arbitrary monic polynomials
$(p_{i}(x),q_{i}(x))_{i\in\mathbb{Z}_{\geq 0}}$, where $p_{i}(x)=x^{i}+\cdots$
and $q_{i}(x)=x^{i}+\cdots$, we define the norm matrix,

$\displaystyle\mathsf{N}_{i,j}=(\,p_{i}\mid\omega\mid q_{j}\,)\,.$ (2.16)

######  Proposition 2.7.

The coupled matrix model partition function (2.1) is given as a rank $N$
determinant of the norm matrix,

$\displaystyle Z_{N}=\det_{1\leq i,j\leq N}\mathsf{N}_{N-i,N-j}\,.$ (2.17)

###### Proof.

Noticing that the Vandermonde determinant is written as a determinant of
arbitrary monic polynomials,

$\displaystyle\Delta_{N}(X_{L})=\det_{1\leq i,j\leq
N}p_{N-j}(x_{L,i})\,,\qquad\Delta_{N}(X_{R})=\det_{1\leq i,j\leq
N}q_{N-j}(x_{R,i})\,,$ (2.18)

the partition function (2.1) is evaluated as a rank $N$ determinant,

$\displaystyle Z_{N}$
$\displaystyle=\frac{1}{N!^{2}}\int\prod_{k=L,R}\differential{X}_{k}\mathrm{e}^{-\tr
V_{k}(X_{k})}\det_{1\leq i,j\leq N}p_{N-j}(x_{L,i})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq N}q_{N-j}(x_{R,i})$
$\displaystyle=\det_{1\leq i,j\leq
N}\quantity[\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}p_{N-i}(x_{L})\omega(x_{L},x_{R})q_{N-j}(x_{R})]$
$\displaystyle=\det_{1\leq i,j\leq N}\mathsf{N}_{N-i,N-j}\,,$ (2.19)

where we apply the AH identity for $X_{L,R}$. This completes the proof. ∎

######  Remark 2.8 (Biorthogonal polynomial).

Specializing the monic polynomials to the biorthogonal polynomials,

$\displaystyle(\,P_{i}\mid\omega\mid Q_{j}\,)=h_{i}\delta_{i,j}\,,$ (2.20)

the norm matrix is diagonalized $\mathsf{N}_{i,j}=h_{i}\delta_{i,j}$, so that
the partition function is given by

$\displaystyle Z_{N}=\prod_{i=0}^{N-1}h_{i}\,.$ (2.21)

###  2.2 Christoffel–Darboux kernel

######  Definition 2.9 (Christoffel–Darboux kernel).

We define the Christoffel–Darboux (CD) kernel associated with the coupled
matrix model,

$\displaystyle K_{N}(x_{R},x_{L})$
$\displaystyle=\mathrm{e}^{-V_{L}(x_{L})-V_{R}(x_{R})}\sum_{i,j=0}^{N-1}q_{i}(x_{R})\left(\mathsf{N}^{-1}\right)_{i,j}p_{j}(x_{L})$
$\displaystyle=\mathrm{e}^{-V_{L}(x_{L})-V_{R}(x_{R})}\sum_{i=0}^{N-1}\frac{Q_{i}(x_{R})P_{i}(x_{L})}{h_{i}}=\sum_{i=0}^{N-1}\psi_{i}(x_{R})\phi_{i}(x_{L})\,.$
(2.22)

We denote the inverse of the norm matrix by
$\left(\mathsf{N}^{-1}\right)_{i,j}$, and define the biorthonormal functions,
that we call the wave functions, by

$\displaystyle\phi_{i}(x)=\frac{\mathrm{e}^{-V_{L}(x)}}{\sqrt{h_{i}}}p_{i}(x)\,,\qquad\psi_{i}(x)=\frac{\mathrm{e}^{-V_{R}(x)}}{\sqrt{h_{i}}}q_{i}(x)\,.$
(2.23)

######  Proposition 2.10.

The probability distribution associated with the partition function (2.1) is
written using the CD kernel,

$\displaystyle\mathsf{P}_{N}(X_{L,R})$
$\displaystyle=\frac{Z_{N}^{-1}}{N!^{2}}\prod_{k=L,R}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N}(X_{L})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\Delta_{N}(X_{R})$
$\displaystyle=\frac{1}{N!^{2}}\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq N}K_{N}(x_{R,i},x_{L,j})\,,$
(2.24)

which obeys the normalization condition

$\displaystyle\int\prod_{k=L,R}\differential{X}_{k}\mathsf{P}_{N}(X_{L,R})=1\,.$
(2.25)

######  Definition 2.11 (Expectation value).

We define the expectation value with respect to the probability distribution
function $\mathsf{P}_{N}(X_{L,R})$ as follows,

$\displaystyle\langle\,\mathcal{O}(X_{L,R})\,\rangle=\int\prod_{k=L,R}\differential{X}_{k}\mathsf{P}_{N}(X_{L,R})\mathcal{O}(X_{L,R})\,.$
(2.26)

###  2.3 Operator formalism

######  Definition 2.12.

We define an inner product symbol,

$\displaystyle\langle\,f\mid g\,\rangle$
$\displaystyle=\int\differential{x}f(x)g(x)\,,$ (2.27a)
$\displaystyle\langle\,f\mid\omega\mid g\,\rangle$
$\displaystyle=\int\differential{x}_{L,R}f(x_{L})\omega(x_{L},x_{R})g(x_{R})\,.$
(2.27b)

We remark that, compared with the previous notation (2.15), this definition
does not depend on the potential function.

Then, the orthonormality of the wave functions $(\phi_{i},\psi_{i})$ defined
in (2.23) is expressed as

$\displaystyle\langle\,\phi_{i}\mid\omega\mid\psi_{j}\,\rangle=\int\differential{x}_{L,R}\phi_{i}(x_{L})\omega(x_{L},x_{R})\psi_{j}(x_{R})=\delta_{i,j}\,,$
(2.28)

where we write

$\displaystyle\phi_{i}(x)=\langle\,\phi_{i}\mid
x\,\rangle\,,\qquad\psi_{i}(x)=\langle\,x\mid\psi_{i}\,\rangle\,,\qquad\omega(x_{L},x_{R})=\langle\,x_{L}\mid\hat{\omega}\mid
x_{R}\,\rangle\,.$ (2.29)

together with the completeness condition

$\displaystyle 1=\int\differential{x}\ket{x}\bra{x}\,.$ (2.30)

In this operator formalism, the CD kernel is given by a matrix element of the
operator defined as

$\displaystyle K_{N}(x_{R},x_{L})$
$\displaystyle=\langle\,x_{R}\mid\hat{K}_{N}\mid
x_{L}\,\rangle\,,\qquad\hat{K}_{N}=\sum_{i=0}^{N-1}\ket{\psi_{i}}\bra{\phi_{i}}\,.$
(2.31)

Introducing infinite dimensional vectors

$\displaystyle\ket{\underline{\phi}}=\begin{pmatrix}&\ket{\phi_{0}}&\ket{\phi_{1}}&\ket{\phi_{2}}&\cdots&\end{pmatrix}^{\text{T}}\,,\qquad\ket{\underline{\psi}}=\begin{pmatrix}&\ket{\psi_{0}}&\ket{\psi_{1}}&\ket{\psi_{2}}&\cdots&\end{pmatrix}^{\text{T}}\,,$
(2.32)

together with the projection matrix

$\displaystyle\left(\Pi_{N}\right)_{i,j}=\begin{cases}1&(i=j\in[0,\ldots,N-1])\\\
0&(\text{otherwise})\end{cases}$ (2.33)

the CD kernel operator is written as

$\displaystyle\hat{K}_{N}=\ket{\underline{\psi}}\Pi_{N}\bra{\underline{\phi}}\,.$
(2.34)

In the limit $N\to\infty$, we have

$\displaystyle\lim_{N\to\infty}{K}_{N}(x_{R},x_{L})=\sum_{i=0}^{\infty}\psi_{i}(x_{R})\phi_{i}(x_{L})=\langle\,x_{R}\mid\omega^{-1}\mid
x_{L}\,\rangle=:\tilde{\omega}(x_{R},x_{L})\,,$ (2.35)

such that

$\displaystyle\int\differential{z}\omega(x,z)\tilde{\omega}(z,y)=\int\differential{z}\tilde{\omega}(x,z){\omega}(z,y)=\delta(x-y)\,.$
(2.36)

######  Proposition 2.13.

The CD kernel is self-reproducing

$\displaystyle\hat{K}_{N}\cdot\hat{\omega}\cdot\hat{K}_{N}=\hat{K}_{N}\,,\qquad\tr\left(\hat{\omega}\cdot\hat{K}_{N}\right)=N\,,$
(2.37)

and therefore the correlation functions are in general determinantal
(Eynard–Mehta’s theorem [EM98]).

###  2.4 Polynomial ensemble

We consider a generalization of the coupled matrix model, that we call the
coupled polynomial ensemble, which is a coupled version of the polynomial
ensemble introduced in Ref. [KS14]. We define the following generalized
coupled matrix model partition functions.

######  Definition 2.14.

Let $(f_{k,i})_{k=L,R,0=1,\ldots,N-1}$ be a set of arbitrary functions. We
define the polynomial ensemble partition functions as follows,

$\displaystyle Z_{N,f_{L}}$
$\displaystyle=\frac{1}{N!^{2}}\int\differential{X}_{L,R}\mathrm{e}^{-\tr
V_{R}(X_{R})}\det_{1\leq i,j\leq N}f_{L,N-i}(x_{L,j})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\Delta_{N}(X_{R})\,,$ (2.38a) $\displaystyle
Z_{N,f_{R}}$
$\displaystyle=\frac{1}{N!^{2}}\int\differential{X}_{L,R}\mathrm{e}^{-\tr
V_{L}(X_{L})}\Delta_{N}(X_{L})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq N}f_{R,N-i}(x_{R,j})\,.$ (2.38b)

######  Remark 2.15.

Specializing each function $(f_{k,i})_{k=L,R,i=0,\ldots,N-1}$ to be a monic
polynomial, these partition functions (2.38) are reduced to the original one
(2.1).

These partition functions show the determinantal structure as discussed
before. In order to discuss their properties, we introduce the notation.

######  Definition 2.16 (Mixed braket notation).

We define the following inner product symbol,

$\displaystyle(\,f\mid g\,\rangle$
$\displaystyle=\int\differential{x}_{L,R}\mathrm{e}^{-V_{L}(x_{L})}f(x_{L})g(x_{R})\,,$
(2.39a) $\displaystyle\langle\,f\mid g\,)$
$\displaystyle=\int\differential{x}_{L,R}\mathrm{e}^{-V_{R}(x_{R})}f(x_{L})g(x_{R})\,.$
(2.39b)

We obtain the following result.

######  Proposition 2.17.

The partition function of the polynomial ensemble is written as a rank $N$
determinant with a set of arbitrary monic polynomials
$(p_{i},q_{i})_{i=0,\ldots,N-1}$,

$\displaystyle Z_{N,f_{L}}$ $\displaystyle=\det_{1\leq i,j\leq
N}\langle\,f_{L,N-i}\mid\omega\mid q_{N-j}\,)\,,$ (2.40) $\displaystyle
Z_{N,f_{R}}$ $\displaystyle=\det_{1\leq i,j\leq N}(\,p_{N-i}\mid\omega\mid
f_{R,N-j}\,\rangle\,.$ (2.41)

###### Proof.

We obtain this formula by direct calculation. Recalling the Vandermonde
determinant is given as (2.18) with a set of arbitrary monic polynomials, we
have

$\displaystyle Z_{N,f_{L}}$
$\displaystyle=\frac{1}{N!^{2}}\int\differential{X}_{L,R}\mathrm{e}^{-\tr
V_{R}(X_{R})}\det_{1\leq i,j\leq N}f_{L,N-i}(x_{L,j})\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq N}q_{N-j}(x_{R,i})$
$\displaystyle=\det_{1\leq i,j\leq
N}\quantity(\int\differential{x}_{L,R}\mathrm{e}^{-V_{R}(x_{R})}f_{L,N-i}(x_{L})\omega(x_{L},x_{R})q_{N-j}(x_{R}))$
$\displaystyle=\det_{1\leq i,j\leq N}\langle\,f_{L,N-i}\mid\omega\mid
q_{N-j}\,)\,.$ (2.42)

We can obtain the other formula in the same way. ∎

######  Definition 2.18 (Biorthogonal functions).

We can then define two pairs of biorthogonal families
$(F_{L,i},Q_{j})_{i,j=0,\ldots,N-1}$ and $(P_{i},F_{R,j})_{i,j=0,\ldots,N-1}$
such that:

  * •

The functions $P_{i}$ and $Q_{j}$ are monic polynomials.

  * •

The functions $F_{L,i}$ (resp. $F_{R,i}$) are linearly spanned by the
functions $(f_{L,k})_{k=0,\cdots,i}$ (resp. $(f_{R,k})_{k=0,\cdots,i}$).

  * •

They satisfy the following scalar product properties:

$\displaystyle\langle\,F_{L,i}\mid\omega\mid Q_{j}\,)$
$\displaystyle=h_{L,i}\delta_{i,j}\qquad(i,j=0,\ldots,N-1),$ (2.43a)
$\displaystyle(\,P_{i}\mid\omega\mid F_{R,j}\,\rangle$
$\displaystyle=h_{R,i}\delta_{i,j}\qquad(i,j=0,\ldots,N-1).$ (2.43b)

######  Corollary 2.19.

The partition functions of the coupled polynomial ensemble (2.38) take the
following form in terms of the normalization constants
$(h_{k,i})_{k=L,R,i=0,\ldots,N-1}$,

$\displaystyle Z_{N,f_{k}}$
$\displaystyle=\prod_{i=0}^{N-1}h_{k,i}\qquad(k=L,R)\,.$ (2.44)

###### Proof.

Once recalling that the determinant is invariant under linear operations on
rows and columns, one can express it in terms of the biorthogonal functions
defined before,

$\displaystyle Z_{N,f_{L}}$ $\displaystyle=\det_{1\leq i,j\leq
N}\langle\,F_{L,i-1}\mid\omega\mid Q_{j-1}\,)\,,$ (2.45a) $\displaystyle
Z_{N,f_{R}}$ $\displaystyle=\det_{1\leq i,j\leq N}(\,P_{i-1}\mid\omega\mid
F_{R,j-1}\,\rangle\,.$ (2.45b)

which is exactly the desired expression. ∎

######  Definition 2.20 (Christoffel–Darboux kernel).

We define the CD kernels for the coupled polynomial ensemble as follows,

$\displaystyle K_{N,f_{L}}(x,y)$
$\displaystyle=\mathrm{e}^{-V_{R}(x)}\sum_{i=0}^{N-1}\frac{Q_{i}(x)F_{L,i}(y)}{h_{L,i}}\,,$
(2.46a) $\displaystyle K_{N,f_{R}}(x,y)$
$\displaystyle=\mathrm{e}^{-V_{L}(y)}\sum_{i=0}^{N-1}\frac{F_{R,i}(x)P_{i}(y)}{h_{R,i}}\,.$
(2.46b)

######  Remark 2.21.

As for the ordinary coupled matrix model (2.1), one can define the following
biorthonormal wave functions

$\displaystyle\psi_{L,i}(x)=\frac{1}{\sqrt{h_{L,i}}}\mathrm{e}^{-V_{R}(x)}Q_{i}(x)\,,\qquad\phi_{L,i}(x)=\frac{1}{\sqrt{h_{L,i}}}F_{L,i}(x)\,,$
(2.47)
$\displaystyle\psi_{R,i}(x)=\frac{1}{\sqrt{h_{R,i}}}\mathrm{e}^{-V_{L}(x)}P_{i}(x)\,,\qquad\phi_{R,i}(x)=\frac{1}{\sqrt{h_{R,i}}}F_{R,i}(x)\,,$
(2.48)

and the CD kernels take then a very compact form.

######  Proposition 2.22.

The probability distributions for the coupled polynomial ensemble can be
expressed as

$\displaystyle\mathsf{P}_{N,f_{L}}(X_{L,R})$
$\displaystyle=\frac{1}{N!^{2}}\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq
N}K_{N,f_{L}}(x_{R,i},x_{L,j})\,,$ (2.49)
$\displaystyle\mathsf{P}_{N,f_{R}}(X_{L,R})$
$\displaystyle=\frac{1}{N!^{2}}\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq
N}K_{N,f_{R}}(x_{R,i},x_{L,j})\,.$ (2.50)

######  Remark 2.23.

All the previous formulas lead to the familiar matrix model formalism.
Therefore the correlation functions of the Schur polynomial and the
characteristic polynomials shown in the following sections are
straightforwardly generalized to the coupled polynomial ensemble (except for
the pair correlation functions discussed in Section 4). We obtain a natural
generalization of the results for the characteristic polynomial average with
the source term [Kim14b, Kim14a, KM21] and also the one-matrix polynomial
ensemble [ASW20].

##  3 Characteristic polynomial averages

###  3.1 Schur polynomial average

We first compute the Schur polynomial average for the coupled matrix model,
which will be a building block of the correlation functions of the
characteristic polynomials [KM21]. See also [ST21] for a related work.

######  Definition 3.1 (Schur polynomial).

Let $\lambda$ be a partition, a non-increasing sequence of non-negative
integers,

$\displaystyle\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\ell}>\lambda_{\ell+1}=\cdots=0)\,,$
(3.1)

where $\ell=\ell(\lambda)$ is called the length of the partition. Denoting the
transposed partition by $\lambda^{\text{T}}$, we have
$\ell(\lambda)=\lambda_{1}^{\text{T}}$. Then, the Schur polynomial of $N$
variables, $X=(x_{i})_{i=1,\ldots,N}$, is defined as follows,

$\displaystyle s_{\lambda}(X)=\frac{1}{\Delta_{N}(X)}\det_{1\leq i,j\leq
N}x_{i}^{\lambda_{j}+N-i}\,.$ (3.2)

If $\ell(\lambda)>N$, we have $s_{\lambda}(X)=0$. We also remark
$s_{\emptyset}(X)=1$.

######  Lemma 3.2.

The Schur polynomial average with respect to the probability distribution
function $\mathsf{P}_{N}(X_{L,R})$ (2.24) is given as a rank $N$ determinant,

$\displaystyle\langle\,s_{\lambda}(X_{L})s_{\mu}(X_{R})\,\rangle$
$\displaystyle=\frac{1}{Z_{N}}\det_{1\leq i,j\leq
N}(\,x_{L}^{\lambda_{i}+N-i}\mid\omega\mid x_{R}^{\mu_{j}+N-j}\,)\,.$ (3.3)

###### Proof.

This can be shown by direct calculation,

$\displaystyle\langle\,s_{\lambda}(X_{L})s_{\mu}(X_{R})\,\rangle$
$\displaystyle=\int\prod_{k=L,R}\differential{X}_{k}\mathsf{P}_{N}(X_{L,R})s_{\lambda}(X_{L})s_{\mu}(X_{R})$
$\displaystyle=\frac{Z_{N}^{-1}}{N!^{2}}\int\prod_{k=L,R}\mathrm{e}^{-\tr
V_{k}(X_{k})}\det_{1\leq i,j\leq N}x_{L,i}^{\lambda_{j}+N-j}\det_{1\leq
i,j\leq N}\omega(x_{L,i},x_{R,j})\det_{1\leq i,j\leq N}x_{R,i}^{\mu_{j}+N-j}$
$\displaystyle=\frac{1}{Z_{N}}\det_{1\leq i,j\leq
N}\quantity(\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}x_{L}^{\lambda_{i}+N-i}\omega(x_{L},x_{R})x_{R}^{\mu_{j}+N-j})$
$\displaystyle=\frac{1}{Z_{N}}\det_{1\leq i,j\leq
N}(\,x_{L}^{\lambda_{i}+N-i}\mid\omega\mid x_{R}^{\mu_{j}+N-j}\,)\,.$ (3.4)

This completes the proof. ∎

######  Lemma 3.3 (Schur polynomial expansion).

Let $Z=\operatorname{diag}(z_{1},\ldots,z_{M})$. The characteristic polynomial
is expanded with the Schur polynomial as follows,

$\displaystyle\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)$
$\displaystyle=\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}s_{\lambda^{\vee}}(Z)s_{\lambda}(X)\,,$
(3.5a) $\displaystyle\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}$
$\displaystyle=\det_{M}Z^{-N}\sum_{\lambda|\ell(\lambda)\leq\operatorname{min}(M,N)}s_{\lambda}(Z^{-1})s_{\lambda}(X)\,,$
(3.5b)

where we define the dual partition

$\displaystyle\lambda^{\vee}=(\lambda_{1}^{\vee},\ldots,\lambda_{M}^{\vee})=(N-\lambda_{M}^{\text{T}},\ldots,N-\lambda_{1}^{\text{T}})\,,$
(3.6)

and the length of the partition denoted by $\ell(\lambda)=\lambda_{1}$.

###### Proof.

This follows from the Cauchy sum formula. See, e.g., [Mac15]. ∎

###  3.2 Characteristic polynomial

Based on the Schur polynomial expansion, we obtain the determinantal formula
for the characteristic polynomial average as follows.

######  Proposition 3.4 (Characteristic polynomial average).

The $M$-point correlation function of the characteristic polynomial is given
by a rank $M$ determinant of the associated biorthogonal polynomials,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}P_{N+M-\beta}(z_{\alpha})\,,$ (3.7a)
$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{R})}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}Q_{N+M-\beta}(z_{\alpha})\,.$ (3.7b)

###### Proof.

We may use Lemma 3.2 and Lemma 3.3 to show this formula. Considering the
characteristic polynomial coupled with the matrix $X_{L}$, we obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})}$
$\displaystyle=\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}s_{\lambda^{\vee}}(Z)\expectationvalue{s_{\lambda}(X_{L})}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)}\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}z_{\alpha}^{\lambda_{\beta}^{\vee}+M-\beta}\det_{1\leq i,j\leq
N}(x_{L}^{\lambda_{i}+N-i}\mid\omega\mid q_{N-j})$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{pmatrix}z_{\alpha}^{N+M-\beta}&z_{\alpha}^{N-j}\\\
(x_{L}^{N+M-\beta}\mid\omega\mid q_{N-j})&(x_{L}^{N-j}\mid\omega\mid
q_{N-j})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{pmatrix}p_{N+M-\beta}(z_{\alpha})&p_{N-j}(z_{\alpha})\\\
(p_{N+M-\beta}\mid\omega\mid q_{N-i})&(p_{N-j}\mid\omega\mid
q_{N-i})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{pmatrix}P_{N+M-\beta}(z_{\alpha})&P_{N-j}(z_{\alpha})\\\
(P_{N+M-\beta}\mid\omega\mid Q_{N-i})&(P_{N-j}\mid\omega\mid
Q_{N-i})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{pmatrix}P_{N+M-\beta}(z_{\alpha})&P_{N-j}(z_{\alpha})\\\
0&h_{N-i}\,\delta_{N-i,N-j}\end{pmatrix}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}P_{N+M-\beta}(z_{\alpha})\,,$ (3.8)

where we apply the rank $M$ co-factor expansion of the rank $N+M$ determinant.
The other formula (3.7b) is similarly obtained. ∎

###  3.3 Characteristic polynomial inverse

In order to write down the characteristic polynomial inverse average, we
define the Hilbert transform.

######  Definition 3.5 (Hilbert transform).

We define the Hilbert transform of the polynomial functions as follows,

$\displaystyle\widetilde{p}_{j}(z)=\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}\frac{\omega(x_{L},x_{R})q_{j}(x_{R})}{z-x_{L}}\,,$
(3.9a)
$\displaystyle\widetilde{q}_{j}(z)=\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}\frac{p_{j}(x_{L})\omega(x_{L},x_{R})}{z-x_{R}}\,.$
(3.9b)

We obtain the following formula.

######  Proposition 3.6 (Characteristic polynomial inverse average).

Let $Z=\operatorname{diag}(z_{1},\ldots,z_{M})$. The $M$-point correlation
function of the characteristic polynomial inverse is given by a rank $M$
determinant of the dual biorthogonal polynomials. Depending on the relation
between $N$ and $M$, we have the following formulas.

  1. 1.

$M\leq N$

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\widetilde{P}_{N-\beta}(z_{\alpha})\,,$ (3.10a)
$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\widetilde{Q}_{N-\beta}(z_{\alpha})\,.$ (3.10b)

  2. 2.

$M\geq N$

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}\widetilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\,,$ (3.10c)
$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}\widetilde{q}_{N-i}(z_{\alpha})\\\
q_{a-1}(z_{\alpha})\end{pmatrix}\,.$ (3.10d)

###### Proof.

We first consider the case $M\leq N$. In this case, the Schur polynomial
average for $\ell(\lambda)\leq M$ is obtained from Lemma 3.2 as

$\displaystyle\expectationvalue{s_{\lambda}(X_{L})}$
$\displaystyle=\frac{1}{Z_{N}}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{pmatrix}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
q_{N-\beta})&(x_{L}^{N-a}\mid\omega\mid q_{N-\beta})\\\\[5.0pt]
(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
q_{N-b})&(x_{L}^{N-a}\mid\omega\mid q_{N-b})\\\ \end{pmatrix}$
$\displaystyle=\frac{1}{Z_{N}}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{pmatrix}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
Q_{N-\beta})&0\\\\[5.0pt] (x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
Q_{N-b})&h_{N-a}\,\delta_{N-a,N-b}\\\ \end{pmatrix}$
$\displaystyle=\frac{Z_{N-M}}{Z_{N}}\det_{1\leq\alpha,\beta\leq
M}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid Q_{N-\beta})\,.$ (3.11)

Then, applying the Schur polynomial expansion as given in Lemma 3.3, the
characteristic polynomial inverse average is given as follows,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}}$
$\displaystyle=\det_{M}Z^{-N}\sum_{\ell(\lambda)\leq
M}s_{\lambda}(Z^{-1})\expectationvalue{s_{\lambda}(X_{L})}$
$\displaystyle=\frac{Z_{N-M}}{Z_{N}}\frac{1}{\Delta_{M}(Z)}\sum_{0\leq\lambda_{M}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{1\leq\alpha,\beta\leq
M}\quantity(z_{\alpha}^{-\lambda_{\beta}+\beta-(N+1)})\det_{1\leq\alpha,\beta\leq
M}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid Q_{N-\beta})$
$\displaystyle=\frac{Z_{N-M}}{Z_{N}}\frac{1}{\Delta_{M}(Z)}\frac{1}{M!}\sum_{\begin{subarray}{c}0\leq
r_{1},\cdots,r_{M}\leq\infty\\\ r_{\alpha}\neq
r_{\beta}\end{subarray}}\det_{1\leq\alpha,\beta\leq
M}\quantity(z_{\alpha}^{M-N-r_{\beta}-1})\det_{1\leq\alpha,\beta\leq
M}(x_{L}^{N-M+r_{\alpha}}\mid\omega\mid Q_{N-\beta})$
$\displaystyle=\frac{Z_{N-M}}{Z_{N}}\frac{1}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\sum_{r=0}^{\infty}z_{\alpha}^{M-N-r-1}(x_{L}^{N-M+r}\mid\omega\mid
Q_{N-\beta}))\,,$ (3.12)

where we have applied an analog of the AH identity for non-colliding discrete
variables, $(r_{\alpha})_{\alpha=1,\ldots,M}$ ($r_{\alpha}\neq r_{\beta}$).
Noticing

$\displaystyle\sum_{r=0}^{\infty}z^{-r-1}x^{r}=\frac{1}{z-x}\,,$ (3.13)

and

$\displaystyle\frac{x^{N-M}}{z-x}=\frac{z^{N-M}}{z-x}-\frac{z^{N-M}-x^{N-M}}{z-x}=\frac{z^{N-M}}{z-x}-O(x^{N-M-1})\,,$
(3.14)

we obtain

$\displaystyle\sum_{r=0}^{\infty}z_{\alpha}^{M-N-r-1}(x_{L}^{N-M+r}\mid\omega\mid
Q_{N-\beta})$
$\displaystyle=z_{\alpha}^{M-N}\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}\frac{x_{L}^{N-M}}{z_{\alpha}-x_{L}}\omega(x_{L},x_{R})Q_{N-\beta}(x_{R})$
$\displaystyle=\int\prod_{k=L,R}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}\frac{\omega(x_{L},x_{R})Q_{N-\beta}(x_{R})}{z_{\alpha}-x_{L}}$
$\displaystyle=\widetilde{P}_{N-\beta}(z_{\alpha})\,.$ (3.15)

We have used the biorthogonality $(\,x_{L}^{a}\mid\omega\mid Q_{N-\beta}\,)=0$
for $\beta=1,\ldots,M$ and $a=0,\ldots,N-M-1$ to obtain the last expression.
This completes the derivation of the formula (3.10a). We can similarly obtain
the formula (3.10b).

We then consider the case $M\geq N$. In this case, the $M$-variable Schur
polynomial with the condition $\ell(\lambda)\leq N$ for
$Z=\operatorname{diag}(z_{1},\ldots,z_{M})$ is given by

$\displaystyle\frac{s_{\lambda}(Z^{-1})}{\det Z^{N}}$
$\displaystyle=\det_{\begin{subarray}{c}i=1,\ldots,N\\\ \alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{pmatrix}$
$\displaystyle=\det_{\begin{subarray}{c}i=1,\ldots,N\\\ \alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\,.$ (3.16)

Hence, applying the Schur polynomial expansion, we obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\sum_{0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\det_{1\leq i,j\leq
N}(x_{L}^{\mu_{i}+N-i}\mid\omega\mid q_{N-j})$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}\displaystyle\sum_{r=0}^{\infty}z_{\alpha}^{-r-1}(x_{L}^{r}\mid\omega\mid
q_{N-i})\\\ p_{a-1}(z_{\alpha})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}\displaystyle\sum_{r=0}^{\infty}z_{\alpha}^{-r-1}(x_{L}^{r}\mid\omega\mid
q_{N-i})\\\ p_{a-1}(z_{\alpha})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{N}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}\widetilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\,.$ (3.17)

This is the determinantal formula shown in (3.10c). We can similarly obtain
the other formula (3.10d). This completes the proof. ∎

##  4 Pair correlation functions

In this Section, we consider the correlation function of both of the
characteristic polynomials coupled to the matrices $X_{L,R}$, that we call the
pair correlation function.

###  4.1 Characteristic polynomial

We have the following result regarding the pair correlation of the
characteristic polynomials.

######  Proposition 4.1 (Pair correlation of characteristic polynomials).

Let $Z=\operatorname{diag}(z_{1},\ldots,z_{M})$ and
$W=\operatorname{diag}(w_{1},\ldots,w_{M})$. The correlation function of $M$
pairs of the characteristic polynomials is given by a rank $M$ determinant of
the CD kernel,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})\det(w_{\alpha}-X_{R})}$
$\displaystyle=\frac{\mathrm{e}^{\tr V_{L}(Z)+\tr
V_{R}(W)}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{Z_{N+M}}{Z_{N}}\det_{1\leq\alpha,\beta\leq
M}K_{N+M}(w_{\alpha},z_{\beta})\,.$ (4.1)

###### Proof.

We use Lemma 3.2 and Lemma 3.3 as before. In addition, we apply the co-factor
expansion twice to obtain the following,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})\det(w_{\alpha}-X_{R})}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq N+M\end{subarray}}\begin{pmatrix}0&w_{\alpha}^{N+M-j}\\\
z_{\beta}^{N+M-i}&(x_{L}^{N+M-i}\mid\omega\mid x_{R}^{N+M-j})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq N+M\end{subarray}}\begin{pmatrix}0&q_{N+M-j}(w_{\alpha})\\\
p_{N+M-i}(z_{\beta})&\mathsf{N}_{N+M-i,N+M-j}\end{pmatrix}$
$\displaystyle=\frac{Z_{N+M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\sum_{k,k^{\prime}=0}^{N+M-1}q_{k}(w_{\alpha})(\mathsf{N}^{-1})_{k,k^{\prime}}p_{k^{\prime}}(z_{\beta}))$
$\displaystyle=\frac{\mathrm{e}^{\tr V_{L}(Z)+\tr
V_{R}(W)}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{Z_{N+M}}{Z_{N}}\det_{1\leq\alpha,\beta\leq
M}K_{N+M}(w_{\alpha},z_{\beta})\,,$ (4.2)

We have applied the definition of the CD kernel of degree $N+M$ (2.22) to
obtain the last expression. ∎

######  Remark 4.2.

This result can be also obtained using the self-reproducing property of the CD
kernel as follows. Noticing

$\displaystyle\Delta_{N}(X)\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)=\frac{\Delta_{N+M}(X;Z)}{\Delta_{M}(Z)}\,,$
(4.3)

the pair correlation is given by

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})\det(w_{\alpha}-X_{R})}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{1}{N!^{2}}\int\prod_{k=L,R}\differential{X}_{k}\mathrm{e}^{-\tr
V_{k}(X_{k})}\Delta_{N+M}(X_{L};Z)\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\Delta_{N+M}(X_{R};W)$
$\displaystyle=\frac{\mathrm{e}^{\tr V_{L}(Z)+\tr
V_{R}(W)}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{Z_{N+M}/Z_{N}}{N!^{2}}$
$\displaystyle\hskip
10.00002pt\times\int\prod_{k=L,R}\differential{X}_{k}\mathrm{e}^{-\tr
V_{k}(X_{k})}\det_{1\leq i,j\leq
N}\omega(x_{L,i},x_{R,j})\det_{\begin{subarray}{c}1\leq i,j\leq N\\\
1\leq\alpha,\beta\leq
M\end{subarray}}\begin{pmatrix}K_{N+M}(x_{R,i},x_{L,j})&K_{N+M}(x_{R,i},z_{\beta})\\\
K_{N+M}(w_{\alpha},x_{L,j})&K_{N+M}(w_{\alpha},z_{\beta})\end{pmatrix}$
$\displaystyle=\frac{\mathrm{e}^{\tr V_{L}(Z)+\tr
V_{R}(W)}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{Z_{N+M}}{Z_{N}}\det_{1\leq\alpha,\beta\leq
M}K_{N+M}(w_{\alpha},z_{\beta})\,.$ (4.4)

###  4.2 Characteristic polynomial inverse

We then consider the pair correlation of the characteristic polynomial
inverses. In order to write down the formula in this case, we define the dual
CD kernel as follows.

######  Definition 4.3 (Dual Christoffel–Darboux kernel).

For the dual wave functions defined through the Hilbert transform,

$\displaystyle\widetilde{\phi}_{i}(z)$
$\displaystyle=\mathrm{e}^{V_{L}(z)}\int\differential{x}_{L,R}\mathrm{e}^{-V_{L}(x_{L})}\frac{\omega(x_{L},x_{R})\psi_{i}(x_{R})}{z-x_{L}}\,,$
(4.5a) $\displaystyle\widetilde{\psi}_{i}(z)$
$\displaystyle=\mathrm{e}^{V_{R}(z)}\int\differential{x}_{L,R}\mathrm{e}^{-V_{R}(x_{R})}\frac{\phi_{i}(x_{L})\omega(x_{L},x_{R})}{z-x_{R}}\,,$
(4.5b)

we define the dual Christoffel–Darboux kernel of degree $N$ as follows,

$\displaystyle\widetilde{K}_{N}(w,z)=\sum_{i=N}^{\infty}\widetilde{\psi}_{i}(w)\widetilde{\phi}_{i}(z)\,.$
(4.6)

######  Proposition 4.4 (Pair correlation of characteristic polynomial
inverses).

Let $Z=\operatorname{diag}(z_{1},\ldots,z_{M})$ and
$W=\operatorname{diag}(w_{1},\ldots,w_{M})$. The correlation function of $M$
pairs of the characteristic polynomial inverses is given by a rank $M$
determinant of the dual CD kernel depending on the relation between $N$ and
$M$ as follows.

  1. 1.

$M\leq N$

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})^{-1}}=\frac{\mathrm{e}^{-\operatorname{tr}V_{L}(Z)}\mathrm{e}^{-\operatorname{tr}V_{R}(W)}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{Z_{N-M}}{Z_{N}}\det_{1\leq\alpha,\beta\leq
M}\widetilde{K}_{N-M}(w_{\beta},z_{\alpha})$ (4.7a)

  2. 2.

$M\geq N$

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{(-1)^{M-N}Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\frac{1}{z_{\alpha}-x_{L}}\mid\omega\mid\frac{1}{w_{\beta}-x_{R}})\det_{1\leq
a,b\leq
M-N}\quantity(\sum_{\alpha,\beta=1}^{M}p_{a-1}(z_{\alpha})\widetilde{\omega}_{\alpha,\beta}q_{b-1}(w_{\beta}))$
(4.7b)

where $\widetilde{\omega}_{\alpha,\beta}$ is the inverse of
$\quantity(\frac{1}{z_{\alpha}-x_{L}}\mid\omega\mid\frac{1}{w_{\beta}-x_{R}})$.

###### Proof.

We first consider the case $M\leq N$. In this case, applying the Schur
polynomial expansion as before, we obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\sum_{\ell(\lambda),\ell(\mu)\leq
M}\det_{1\leq\alpha,\beta\leq
M}z_{\alpha}^{-N-\lambda_{\beta}+\beta-1}\det_{1\leq\alpha,\beta\leq
M}w_{\alpha}^{-N-\mu_{\beta}+\beta-1}$ $\displaystyle\hskip
80.00012pt\times\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ 1\leq
i,j\leq
N-M\end{subarray}}\begin{pmatrix}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})&(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{N-M-j})\\\ (x_{L}^{N-M-i}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})&(x_{L}^{N-M-i}\mid\omega\mid
x_{R}^{N-M-j})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\sum_{\ell(\lambda),\ell(\mu)\leq
M}\det_{1\leq\alpha,\beta\leq
M}z_{\alpha}^{-N-\lambda_{\beta}+\beta-1}\det_{1\leq\alpha,\beta\leq
M}w_{\alpha}^{-N-\mu_{\beta}+\beta-1}$ $\displaystyle\hskip
80.00012pt\times\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ 1\leq
i,j\leq
N-M\end{subarray}}\begin{pmatrix}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})&(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
q_{N-M-j})\\\ (p_{N-M-i}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})&(p_{N-M-i}\mid\omega\mid q_{N-M-j})\end{pmatrix}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\sum_{\ell(\lambda),\ell(\mu)\leq
M}\det_{1\leq\alpha,\beta\leq
M}z_{\alpha}^{-N-\lambda_{\beta}+\beta-1}\det_{1\leq\alpha,\beta\leq
M}w_{\alpha}^{-N-\mu_{\beta}+\beta-1}$
$\displaystyle\quad\times\det_{1\leq\alpha,\beta\leq
M}\quantity((x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})-\sum_{i,j=0}^{N-M-1}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
q_{i})(\mathsf{N}^{-1})_{i,j}(p_{j}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta}))\,.$ (4.8)

We remark that each element in the determinant is given by

$\displaystyle(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})-\sum_{i,j=0}^{N-M-1}(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
q_{i})(\mathsf{N}^{-1})_{i,j}(p_{j}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})$
$\displaystyle=(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})$
$\displaystyle\qquad-\int\prod_{k=L,R,L^{\prime},R^{\prime}}\differential{x}_{k}\mathrm{e}^{-V_{k}(x_{k})}x_{L}^{\lambda_{\alpha}+N-\alpha}\omega(x_{L},x_{R^{\prime}})\sum_{i,j=0}^{N-M-1}q_{i}(x_{R^{\prime}})(\mathsf{N}^{-1})_{i,j}p_{j}(x_{L^{\prime}})\omega(x_{L^{\prime}},x_{R})x_{L}^{\mu_{\beta}+N-\beta}$
$\displaystyle=(x_{L}^{\lambda_{\alpha}+N-\alpha}\mid\omega\mid
x_{R}^{\mu_{\beta}+N-\beta})$
$\displaystyle\qquad-\int\prod_{k=L,R}\differential{x}_{k}\differential{x}_{k^{\prime}}\mathrm{e}^{-V_{k}(x_{k})}x_{L}^{\lambda_{\alpha}+N-\alpha}\omega(x_{L},x_{R^{\prime}})K_{N-M}(x_{R^{\prime}},x_{L^{\prime}})\omega(x_{L^{\prime}},x_{R})x_{R}^{\mu_{\beta}+N-\beta}$
$\displaystyle=\int\prod_{k=L,R}\differential{x}_{k}\differential{x}_{k^{\prime}}\mathrm{e}^{-V_{k}(x_{k})}x_{L}^{\lambda_{\alpha}+N-\alpha}\omega(x_{L},x_{R^{\prime}})\quantity(\widetilde{\omega}(x_{R^{\prime}},x_{L^{\prime}})-K_{N-M}(x_{R^{\prime}},x_{L^{\prime}}))\omega(x_{L^{\prime}},x_{R})x_{R}^{\mu_{\beta}+N-\beta}$
$\displaystyle=\int\prod_{k=L,R}\differential{x}_{k}\differential{x}_{k^{\prime}}\mathrm{e}^{-V_{k}(x_{k})}x_{L}^{\lambda_{\alpha}+N-\alpha}\omega(x_{L},x_{R^{\prime}})\quantity(\sum_{k=N-M}^{\infty}\psi_{k}(x_{R^{\prime}})\phi_{k}(x_{L^{\prime}}))\omega(x_{L^{\prime}},x_{R})x_{R}^{\mu_{\beta}+N-\beta}\,.$
(4.9)

Therefore, we obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\sum_{i=N-M}^{\infty}\int\prod_{k=L,R}\differential{x}_{k}\differential{x}_{k^{\prime}}\mathrm{e}^{-V_{k}(x_{k})}\frac{\omega(x_{L},x_{R^{\prime}})\psi_{i}(x_{R^{\prime}})\phi_{i}(x_{L^{\prime}})\omega(x_{L^{\prime}},x_{R})}{(z_{\alpha}-x_{L})(w_{\beta}-x_{R})})$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\mathrm{e}^{-V_{L}(z_{\alpha})}\mathrm{e}^{-V_{R}(w_{\beta})}\sum_{i=N-M}^{\infty}\widetilde{\phi}_{i}(z_{\alpha})\widetilde{\psi}_{i}(w_{\beta}))$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\mathrm{e}^{-\operatorname{tr}V_{L}(Z)}\mathrm{e}^{-\operatorname{tr}V_{R}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\widetilde{K}_{N-M}(w_{\beta},z_{\alpha}))\,.$ (4.10)

This completes the derivation of the formula (4.7a).
We then consider the case $M\geq N$. In this case, we similarly obtain the
formula (4.7b) as follows,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}$
$\displaystyle\times\sum_{\begin{subarray}{c}0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty\\\
0\leq\mu_{N}\leq\cdots\leq\mu_{1}\leq\infty\end{subarray}}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\det_{1\leq i,j\leq
N}(x_{L}^{\lambda_{i}+N-i}\mid\omega\mid
x_{R}^{\mu_{j}+N-j})\det_{\begin{subarray}{c}j=1,\ldots,N\\\
\beta=1,\ldots,M\\\
b=1,\ldots,M-N\end{subarray}}\begin{pmatrix}w_{\beta}^{-\mu_{j}+j-(N+1)}\\\
q_{b-1}(w_{\beta})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\frac{1}{N!^{2}}\sum_{\begin{subarray}{c}0\leq
r_{1},\cdots,r_{N}\leq\infty\\\ 0\leq s_{1},\cdots,s_{N}\leq\infty\\\
r_{i}\neq r_{j},s_{i}\neq
s_{j}\end{subarray}}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{pmatrix}z_{\alpha}^{-r_{i}-1}\\\
p_{a-1}(z_{\alpha})\end{pmatrix}\det_{1\leq i,j\leq
N}(x_{L}^{r_{i}}\mid\omega\mid
x_{R}^{s_{j}})\det_{\begin{subarray}{c}j=1,\ldots,N\\\ \beta=1,\ldots,M\\\
b=1,\ldots,M-N\end{subarray}}\begin{pmatrix}w_{\beta}^{-s_{j}-1}\\\
q_{b-1}(w_{\beta})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq a,b\leq
M-N\end{subarray}}\begin{pmatrix}\displaystyle\sum_{r,s=0}^{\infty}z_{\alpha}^{-r-1}w_{\beta}^{-s-1}(x_{L}^{r}\mid\omega\mid
x_{R}^{s})&q_{b-1}(w_{\beta})\\\ p_{a-1}(z_{\alpha})&0\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq a,b\leq
M-N\end{subarray}}\begin{pmatrix}\quantity(\frac{1}{z_{\alpha}-x_{L}}\mid\omega\mid\frac{1}{w_{\beta}-x_{R}})&q_{b-1}(w_{\beta})\\\
p_{a-1}(z_{\alpha})&0\end{pmatrix}$
$\displaystyle=\frac{(-1)^{M-N}Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{1\leq\alpha,\beta\leq
M}\quantity(\frac{1}{z_{\alpha}-x_{L}}\mid\omega\mid\frac{1}{w_{\beta}-x_{R}})\det_{1\leq
a,b\leq
M-N}\quantity(\sum_{\alpha,\beta=1}^{M}p_{a-1}(z_{\alpha})\widetilde{\omega}_{\alpha,\beta}q_{b-1}(w_{\beta}))$
(4.11)

This completes the proof. ∎

###  4.3 Mixed pair correlation

We consider the mixed-type pair correlation function of the characteristic
polynomials.

######  Proposition 4.5.

Let $Z=\operatorname{diag}(z_{1},\ldots,z_{M})$ and
$W=\operatorname{diag}(w_{1},\ldots,w_{M})$. The following determinantal
formulas hold for the mixed-pair correlation for $M\leq N$.

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}\alpha=1,\ldots,M\\\
\beta=1,\ldots,2M\end{subarray}}\begin{pmatrix}P_{N+M-\beta}(z_{\alpha})\\\
\widetilde{Q}_{N+M-\beta}(w_{\alpha})\end{pmatrix}\,,$ (4.12a)
$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})^{-1}\det(w_{\alpha}-X_{R})}$
$\displaystyle=\frac{Z_{N-M}/Z_{N}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}\alpha=1,\ldots,M\\\
\beta=1,\ldots,2M\end{subarray}}\begin{pmatrix}\widetilde{P}_{N+M-\beta}(z_{\alpha})\\\
Q_{N+M-\beta}(w_{\alpha})\end{pmatrix}\,.$ (4.12b)

###### Proof.

Applying the Schur polynomial expansion and the co-factor expansion as before,
we obtain the following,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X_{L})\det(w_{\alpha}-X_{R})^{-1}}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\sum_{\ell(\lambda)\leq
M}\det_{1\leq\alpha,\beta\leq
M}\quantity(w_{\alpha}^{-\lambda_{\beta}+\beta-N-1})\det_{\begin{subarray}{c}i=1,\ldots,N+M\\\
\alpha,\beta=1,\ldots,M\\\
k=M+1,\ldots,N\end{subarray}}\begin{pmatrix}z_{\alpha}^{N+M-i}\\\
(x_{L}^{N+M-i}\mid\omega\mid x_{R}^{\lambda_{\beta}+N-\beta})\\\
(x_{L}^{N+M-i}\mid\omega\mid x_{R}^{N-k})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\sum_{\ell(\lambda)\leq
M}\det_{1\leq\alpha,\beta\leq
M}\quantity(w_{\alpha}^{-\lambda_{\beta}+\beta-N-1})\det_{\begin{subarray}{c}i=1,\ldots,N+M\\\
\alpha,\beta=1,\ldots,M\\\
k=M+1,\ldots,N\end{subarray}}\begin{pmatrix}p_{N+M-i}(z_{\alpha})\\\
(p_{N+M-i}\mid\omega\mid x_{R}^{\lambda_{\beta}+N-\beta})\\\
(p_{N+M-i}\mid\omega\mid x_{R}^{N-k})\end{pmatrix}$
$\displaystyle=\frac{Z_{N}^{-1}}{\Delta_{M}(Z)\Delta_{M}(W)}\det_{\begin{subarray}{c}i=1,\ldots,N+M\\\
\alpha,\beta=1,\ldots,M\\\
k=M+1,\ldots,N\end{subarray}}\begin{pmatrix}p_{N+M-i}(z_{\alpha})\\\
\widetilde{q}_{N+M-i}(w_{\beta})\\\ (p_{N+M-i}\mid\omega\mid
x_{R}^{N-k})\end{pmatrix}\,.$ (4.13)

Then, the determinant part is given by

$\displaystyle\det_{\begin{subarray}{c}i=1,\ldots,N+M\\\
\alpha,\beta=1,\ldots,M\\\
k=M+1,\ldots,N\end{subarray}}\begin{pmatrix}p_{N+M-i}(z_{\alpha})\\\
\widetilde{q}_{N+M-i}(w_{\beta})\\\ (p_{N+M-i}\mid\omega\mid
x_{R}^{N-k})\end{pmatrix}$
$\displaystyle=\det_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta=1,\ldots,M\\\
k,l=1,\ldots,N-M\end{subarray}}\begin{pmatrix}p_{N+M-\gamma}(z_{\alpha})&p_{N-\delta}(z_{\alpha})&p_{N-M-l}(z_{\alpha})\\\
\widetilde{q}_{N+M-\gamma}(w_{\beta})&\widetilde{q}_{N-\delta}(w_{\beta})&\widetilde{q}_{N-M-l}(w_{\beta})\\\
(p_{N+M-\gamma}\mid\omega\mid q_{N-M-k})&(p_{N-\delta}\mid\omega\mid
q_{N-M-k})&(p_{N-M-l}\mid\omega\mid q_{N-M-k})\end{pmatrix}$
$\displaystyle=\det_{\begin{subarray}{c}\alpha,\beta,\gamma,\delta=1,\ldots,M\\\
k,l=1,\ldots,N-M\end{subarray}}\begin{pmatrix}P_{N+M-\gamma}(z_{\alpha})&P_{N-\delta}(z_{\alpha})&P_{N-M-l}(z_{\alpha})\\\
\widetilde{Q}_{N+M-\gamma}(w_{\beta})&\widetilde{Q}_{N-\delta}(w_{\beta})&\widetilde{Q}_{N-M-l}(w_{\beta})\\\
0&0&h_{N-M-l}\,\delta_{N-M-l,N-M-k}\end{pmatrix}$
$\displaystyle=Z_{N-M}\det_{\alpha,\beta,\gamma,\delta=1,\ldots,M}\begin{pmatrix}P_{N+M-\gamma}(z_{\alpha})&P_{N-\delta}(z_{\alpha})\\\
\widetilde{Q}_{N+M-\gamma}(w_{\beta})&\widetilde{Q}_{N-\delta}(w_{\beta})\end{pmatrix}\,.$
(4.14)

This completes the derivation of (4.12a). The other formula (4.12b) can be
also derived in the same way. ∎

######  Remark 4.6.

For $M=1$, the mixed-pair correlation functions are given by

$\displaystyle\expectationvalue{\frac{\det(z-X_{L})}{\det(w-X_{R})}}$
$\displaystyle=\frac{Z_{N-1}}{Z_{N}}\det\begin{pmatrix}P_{N}(z)&P_{N-1}(z)\\\
\widetilde{Q}_{N}(w)&\widetilde{Q}_{N-1}(w)\end{pmatrix}$
$\displaystyle=\frac{1}{h_{N-1}}\left(P_{N}(z)\widetilde{Q}_{N-1}(w)-P_{N-1}(z)\widetilde{Q}_{N}(w)\right)\,,$
(4.15a) $\displaystyle\expectationvalue{\frac{\det(w-X_{R})}{\det(z-X_{L})}}$
$\displaystyle=\frac{Z_{N-1}}{Z_{N}}\det\begin{pmatrix}\widetilde{P}_{N}(z)&\widetilde{P}_{N-1}(z)\\\
{Q}_{N}(w)&{Q}_{N-1}(w)\end{pmatrix}$
$\displaystyle=\frac{1}{h_{N-1}}\left(\widetilde{P}_{N}(z){Q}_{N-1}(w)-\widetilde{P}_{N-1}(z)\widetilde{Q}_{N}(w)\right)\,.$
(4.15b)

These expressions suggest that the mixed-pair correlation could be also
written in terms of the associated CD kernel. See [SF03, BDS03, BS06, EKR15]
for details. We leave this issue for the future study.

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