# The Schur Expansion of Characteristic Polynomials
and Random Matrices

Taro<EMAIL_ADDRESS>& Edward A.
<EMAIL_ADDRESS>
###### Abstract

We develop a new framework to compute the exact correlators of characteristic
polynomials, and their inverses, in random matrix theory. Our results hold for
general potentials and incorporate the effects of an external source. In
matrix model realizations of string theory, these correspond to correlation
functions of exponentiated “(anti-)branes” in a given background of “momentum
branes”. Our method relies on expanding the (inverse) determinants in terms of
Schur polynomials, then re-summing their expectation values over the allowed
representations of the symmetric group. Beyond unifying previous, seemingly
disparate calculations, this powerful technique immediately delivers two new
results: 1) the full finite $N$ answer for the correlator of inverse
determinant insertions in the presence of a matrix source, and 2) access to an
interesting, novel regime $M>N$, where the number of inverse determinant
insertions $M$ exceeds the size of the matrix $N$.

* Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, France
$\dagger$ Kadanoff Center for Theoretical Physics, University of Chicago,
Chicago, IL 60637

###### Contents

  1. 1 Introduction and Summary of Results
  2. 2 Characteristic Polynomials without Source
    1. 2.1 Schur Polynomial Average
    2. 2.2 Characteristic Polynomial Average
    3. 2.3 Inverse Characteristic Polynomial Average
  3. 3 Generalizing to Non-Zero Source
    1. 3.1 Schur Polynomial Average with Source
    2. 3.2 Characteristic Polynomial Average with Source
    3. 3.3 Inverse Characteristic Polynomial Average with Source
      1. 3.3.1 Zero Source Limit
      2. 3.3.2 The Case with $M>N$
      3. 3.3.3 Characteristic Polynomial/Source Duality
  4. 4 Discussion

##  1 Introduction and Summary of Results

Characteristic polynomials, and their inverses, serve as detailed probes of
systems described by random matrix ensembles. Indeed, while expectation values
of simple traces neatly encapsulate collective properties of the eigenvalues,
determinant insertions can capture single eigenvalue effects. For example,
their one point functions encode the effective potential for one-eigenvalue
instantons [1, 2].

The finite $N$ correlation functions of characteristics polynomials have been
known for quite some time. They can be written in terms of a determinant built
out of the relevant orthogonal polynomials $P_{k}(x)$ [3, 4],

$\frac{1}{\mathcal{Z}_{N}}\int\differential{X}e^{-\Tr
V(X)}\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)=\frac{\displaystyle\det_{1\leq\alpha,\beta\leq
M}\left[P_{\alpha+N-1}(z_{\beta})\right]}{\Delta_{M}(Z)}$ (1.1)

where $X$ is a size $N$ Hermitian matrix, $P_{k}(x)$ is a monic orthogonal
polynomial of degree $k$ (2.10), and

$\displaystyle\Delta_{M}(z)=\prod_{\alpha<\beta}^{M}(z_{\alpha}-z_{\beta})=\det_{1\leq\alpha,\beta\leq
M}\quantity[p_{M-\alpha}(z_{\beta})]$ (1.2)

is the Van-der-Monde determinant for the matrix
$Z=\text{diag}(z_{1},...,z_{M})$,333Not to be confused with the partition
function $\displaystyle\mathcal{Z}_{N}=\int\differential{X}e^{-\Tr V(X)}$. See
Eq. (2.12) for the normalization of the integral applied in this paper. with
$p_{k}(z)=z^{k}+\cdots$ an arbitrary monic polynomial of degree $k$.

From the string theory perspective, the addition of determinants inside the
matrix integral corresponds to adding coherent states of branes, via the
immediate relation to the loop operator
$\det(z-X)=\exp\left(\Tr\log(z-X)\right)$. Taking the inverse determinant
corresponds simply to changing the sign in the exponent; in other words to
negative tension branes, sometimes dubbed “anti-branes”.444Another reason for
this name comes from the fact that  $\lim_{y\rightarrow
z}\frac{\det(z-X)}{\det(y-X)}=1$ (1.3) The two insertions thus cancel each
other out as we take a coincident limit, much the same way that a particle and
its anti-particle annihilate each other. A similar determinantal formula
composed out of the Hilbert transform of said orthogonal polynomials computes
the correlator of inverse characteristic polynomials [5, 6, 7, 8]:

$\frac{1}{\mathcal{Z}_{N}}\int\differential{X}e^{-\Tr
V(X)}\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}=\frac{\tilde{\mathcal{Z}}_{N-M}}{\tilde{\mathcal{Z}}_{N}}\frac{\displaystyle\det_{1\leq\alpha,\beta\leq
M}\left[\tilde{P}_{N-\alpha}(z_{\beta})\right]}{\Delta_{M}(Z)}$ (1.4)

with

$\tilde{P}_{k}(z)=\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{P_{k}(x)}{z-x}.$
(1.5)

See Eq.(2.12) for the definition of the prefactor appearing in Eq. (1.4).

Since the orthogonal polynomials cannot carry negative powers, the $\alpha$
dependence on the RHS of (1.4) immediately begs the question as to what
happens when $M>N$. We will see this known formula indeed fails there. As far
as we are aware, the case of $M>N$ has been overlooked. Our method allows us
to probe this novel regime, giving instead one of the main technical results
of this paper

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}=\frac{\tilde{\mathcal{Z}}_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}\qquad(M\geq N).$ (1.6)

with the Hilbert transform

$\tilde{p}_{k}(z)=\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{p_{k}(x)}{z-x}.$
(1.7)

Adding an external source to a matrix ensemble provides a powerful new tool in
studying random matrix theory (RMT) [9]. First, it serves as a generating
function for correlation functions, as familiar from quantum field theory.
Secondly, taking a Wignerian perspective and treating the random matrix as
some quantum mechanical Hamiltonian, it allows us to study the effects of
random perturbations around some fixed known Hamiltonian, the external source
in the matrix integral. The applications do not stop there. For example, by
adding a Gaussian potential centered about the source, Ref. [10] has also
recently studied the transition between RMT and a fixed Hamiltonian in the
context of Jackiw-Teitelboim (JT) gravity. Sending the width of the Gaussian
to zero effectively collapses the matrix integral of [11], picking out the
source as the Hamiltonian. In yet another guise, Refs. [12, 1] pioneered the
use of an external source in the topological string context. They viewed it as
a way of inserting branes, who boundary parameter was related to that of the
usual determinantal brane via a Fourier transform. These are sometimes
referred to as “momentum branes”

$\int\differential{X}e^{-\Tr V(X)+i\Tr
AX}=\frac{C_{N}}{\Delta_{N}(A)}\det_{1\leq i,j\leq
N}\left[Q_{N-i}(\mathsf{a}_{j})\right]$ (1.8)

with the constant $C_{N}$ defined in (3.5) and the transformed orthogonal
polynomial $Q_{k}(\mathsf{a})$ defined as

$Q_{k}(\mathsf{a})=\int\frac{\differential{x}}{2\pi}e^{-V(x)+i\mathsf{a}x}p_{k}(x).$
(1.9)

See, for example, [3] for more technical details. This Fourier transform in
Eq. (1.9) shows in what sense Eq. (1.8) represents the insertion of “momentum
branes” if we interpret Eq. (1.1) as “position brane” correlators.

It is thus natural to consider an ensemble with both an external source and
the addition of (inverse) determinants. In Ref. [13], the characteristic
polynomial average in the presence of an external matrix source was computed,
generalizing the formulas above to

$\displaystyle\int\differential{X}e^{-\Tr V(X)+i\Tr
AX}\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)=$
$\displaystyle\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{bmatrix}p_{N+M-\beta}(z_{\alpha})&p_{N-j}(z_{\alpha})\\\
Q_{N+M-\beta}(\mathsf{a}_{i})&Q_{N-j}(\mathsf{a}_{i})\end{bmatrix}.$ (1.10)

See Section 3.2 for the details.

In this paper, we will reproduce all the above formula using the Schur
expansion of the (inverse) characteristic polynomial. Our method therefore
unifies the otherwise seemingly disparate techniques used to obtain the
results above. Furthermore, it will allow us to compute the inverse
determinant correlators in the presence of a source, first for the case $M\leq
N$:

$\displaystyle\int\differential{X}e^{-\Tr V(X)+i\Tr
AX}\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}=$
$\displaystyle\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{\alpha},z_{\beta})&Q_{N-b}(\mathsf{a}_{\alpha})\\\
\tilde{Q}(\mathsf{a}_{a},z_{\beta})&Q_{N-b}(\mathsf{a}_{a})\end{bmatrix}$
(1.11)

with the new function $\tilde{Q}(\mathsf{a},z)$ defined as

$\tilde{Q}(\mathsf{a},z)=\int\frac{\differential{x}}{2\pi}e^{-V(x)+i\mathsf{a}x}\frac{1}{z-x}.$
(1.12)

while for $M\geq N$, we obtain a generalization of Eq. (1.6),

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{i},z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}\qquad(M\geq N).$ (1.13)

Together, these two formula constitute the other main new result of this work,
and a useful application of our Schur polynomial method.

The goal of this paper is to provide a usable introduction to the application
of Schur polynomials towards computing exact determinant and inverse
determinant correlators in RMT. We have chosen a rather pedagogical tone,
spelling out several of the intermediate steps, as they may not be familiar to
many readers.

We begin with the study of characteristic polynomials, and their inverses,
without external matrix source. Section 2.1 introduces the Schur polynomial
expansion and computes its expectation value. In sections 2.2 and 2.3, we
perform the sum over the relevant representations of the symmetric group,
reproducing the familiar formulas discussed above. Section 3 generalizes this
approach to non-zero source. In particular, section 3.3 finds the exact
correlator of inverse determinant insertions in this ensemble, a computation
which had resisted previous methods. We conclude with a discussion focused
around the interpretation of the $M>N$ result and a string theory perspective
on the Schur expansion.

##  2 Characteristic Polynomials without Source

###  2.1 Schur Polynomial Average

We first explore the Schur polynomial method in the more familiar context of
RMT without an external field. The Schur polynomial $s_{\lambda}(X)$,
parametrized by a partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N})\in\mathbb{Z}_{\geq
0}^{N}$, has a determinantal expression

$s_{\lambda}(X)=s_{\lambda}(x_{1},\ldots,x_{N})=\frac{\displaystyle\det_{1\leq
i,j\leq N}\left[x_{i}^{\lambda_{j}+N-j}\right]}{\Delta_{N}(X)}.$ (2.1)

We will need to evaluate the Schur polynomial average,

$\displaystyle\expectationvalue{s_{\lambda}(X)}$
$\displaystyle=\frac{1}{\mathcal{Z}_{N}}\int\differential{X}e^{-\Tr
V(X)}s_{\lambda}(X).$ (2.2)

Diagonalizing the Hermitian matrix $X$, and applying expression (1.2), we
obtain555 Diagonalization of the rank $N$ Hermitian matrix measure gives rise
to
$\displaystyle\int\differential{X}=\frac{\operatorname{Vol}U(N)}{|\mathfrak{S}_{N}|\times\operatorname{Vol}U(1)^{N}}\int\prod_{i=1}^{N}\differential{x}_{i}\Delta_{N}(X)^{2},$
(2.3) where $\mathfrak{S}_{N}$ is the degree $N$ symmetric group to be
identified with the Weyl group of $U(N)$, and $U(1)^{N}$ is the maximal Cartan
torus of $U(N)$ with $|\mathfrak{S}_{N}|=N!$ and
$\operatorname{Vol}U(1)=2\pi$. See [14] for details.

$\displaystyle\expectationvalue{s_{\lambda}(X)}$
$\displaystyle=\frac{\operatorname{Vol}U(N)}{N!\mathcal{Z}_{N}}\int\prod_{i=1}^{N}\frac{\differential{x}_{i}}{2\pi}e^{-V(x_{i})}\det_{1\leq
i,j\leq N}\left[p_{N-j}(x_{i})\right]\det_{1\leq i,j\leq
N}\left[x_{i}^{\lambda_{j}+N-j}\right],$ (2.4)

where the volume of $U(N)$ group is given by the double gamma function,

$\displaystyle\operatorname{Vol}U(N)=\frac{(2\pi)^{N(N+1)/2}}{\Gamma_{2}(N+1)}\,,\qquad\Gamma_{2}(N+1)=\prod_{k=1}^{N-1}k!\,.$
(2.5)

Now we may apply the Andréif–Heine identity

$\displaystyle\frac{1}{N!}\int\prod_{i=1}^{N}\differential{x}_{i}\det_{1\leq
i,j\leq N}\left[\psi_{j}(x_{i})\right]\det_{1\leq i,j\leq
N}\left[\phi_{j}(x_{i})\right]=\det_{1\leq i,j\leq
N}\quantity[\int\differential{x}\psi_{i}(x)\phi_{j}(x)],$ (2.6)

which gives rise to the determinantal formula for the Schur polynomial
average,

$\displaystyle\expectationvalue{s_{\lambda}(X)}=\frac{\operatorname{Vol}U(N)}{\mathcal{Z}_{N}}\det_{1\leq
i,j\leq N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}]$ (2.7)

where we define the inner product together with the weight function
$e^{-V(x)}$,

$\displaystyle\innerproduct{\psi}{\phi}=\int\frac{\differential{x}}{2\pi}\,e^{-V(x)}\psi(x)\phi(x)\,.$
(2.8)

#### Normalization

For the trivial partition $\lambda=\emptyset$, we obtain

$\displaystyle\expectationvalue{s_{\emptyset}(X)}=\frac{\operatorname{Vol}U(N)}{\mathcal{Z}_{N}}\det_{1\leq
i,j\leq
N}\quantity[\innerproduct{p_{N-i}}{x^{N-j}}]=\frac{\operatorname{Vol}U(N)}{\mathcal{Z}_{N}}\det_{1\leq
i,j\leq N}\quantity[\innerproduct{p_{N-i}}{p_{N-j}}]\,.$ (2.9)

Taking the orthogonal polynomial as a monic polynomial $p_{k}(x)\to P_{k}(x)$,
such that

$\displaystyle\innerproduct{P_{i}}{P_{j}}=h_{i}\,\delta_{i,j}\,,$ (2.10)

the Schur polynomial average reads

$\displaystyle\expectationvalue{s_{\emptyset}(X)}=\frac{\operatorname{Vol}U(N)}{\mathcal{Z}_{N}}\prod_{i=1}^{N}h_{i-1}.$
(2.11)

Recalling $s_{\emptyset}(X)=1$, we obtain the partition function

$\displaystyle\mathcal{Z}_{N}=\operatorname{Vol}U(N)\tilde{\mathcal{Z}}_{N}\,,\qquad\tilde{\mathcal{Z}}_{N}=\prod_{i=1}^{N}h_{i-1},$
(2.12)

so that

$\displaystyle\expectationvalue{s_{\lambda}(X)}=\frac{1}{\tilde{\mathcal{Z}}_{N}}\det_{1\leq
i,j\leq N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}].$ (2.13)

###  2.2 Characteristic Polynomial Average

Based on the Schur polynomial average discussed above, we wish to compute the
following correlation functions exactly in $N$ and $M$,

$\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)}=\frac{1}{\mathcal{Z}_{N}}\int\differential{X}e^{-\Tr
V(X)}\prod_{\alpha=1}^{M}\det(z_{\alpha}-X).$ (2.14)

For this purpose, we apply the Schur polynomial expansion of the
characteristic polynomial product

$\displaystyle\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)=\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}s_{\lambda^{\vee}}(Z)s_{\lambda}(X)$
(2.15)

where $Z=(z_{1},\ldots,z_{M})$ and $X=(x_{1},\ldots,x_{N})$. This expansion is
due to the Cauchy sum formula of the Schur polynomial [15]. We denote the
transposition of the partition $\lambda$ by $\lambda^{\text{T}}$. Then, the
dual partition $\lambda^{\vee}$ is defined as follows:

$\displaystyle\lambda^{\vee}=(\lambda_{1}^{\vee},\lambda_{2}^{\vee},\ldots,\lambda_{M}^{\vee})=(N-\lambda_{M}^{\text{T}},N-\lambda_{M-1}^{\text{T}},\ldots,N-\lambda_{1}^{\text{T}})\,,$
(2.16)

which is graphically expressed in Fig. 1.

$N$$M$$\lambda\ =$$\lambda^{\vee}\ =$ Figure 1: An example of a partition
$\lambda$ and its dual $\lambda^{\vee}$ for the case of $N=4$ and $M=3$:
$\lambda=(3,2,1,1)$ and $\lambda^{\vee}=(3,2,0)$, while
$\lambda^{\text{T}}=(4,2,1)$.

Using the Schur polynomial average (2.13), the characteristic polynomial
correlator is given by

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{N}\det(z_{\alpha}-X)}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{\lambda^{\vee}_{\beta}+M-\beta}]\expectationvalue{s_{\lambda}(X)}$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N}^{-1}}{\Delta_{M}(Z)}\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{\lambda^{\vee}_{\beta}+M-\beta}]\det_{1\leq i,j\leq
N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}].$ (2.17)

In order to carry out the summation over partitions $\lambda$, we first
briefly recall the generalized co-factor expansion of an $(N+M)\times(N+M)$
matrix $B$ along the first $M$ rows. Let $S$ by the set of all $M$-element
subsets of $\\{1,2,...,N+M\\}$, $H$ the set $\\{1,..,M\\}$ (this is our choice
of expanding along the first $M$ rows) and $H^{\prime}$ the complement of $H$
(the set $\\{M+1,..,N+M\\}$). The expansion then reads

$\det(B)=\sum_{L\in S}(-1)^{(\sum_{h\in H}h+\sum_{l\in
L}l)}\det(B_{H,L})\det(B_{H^{\prime},L^{\prime}})$ (2.18)

where $B_{H,L}$ is the square minor obtained by deleting rows and columns in
$H$ and $L$ respectively. $B_{H^{\prime},L^{\prime}}$ is similarly defined,
with $L^{\prime}$ the complement of $L$, i.e. $L\cup L^{\prime}=S$.

We now wish to identify the summation over the partition $\lambda$ as a rank
$M$ co-factor expansion of the following rank $(N+M)$ determinant,

$\displaystyle\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{\lambda^{\vee}_{\beta}+M-\beta}]\det_{1\leq i,j\leq
N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}]$
$\displaystyle=\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ 1\leq
i,j\leq
N\end{subarray}}\begin{bmatrix}z_{\alpha}^{N+M-\beta}&z_{\alpha}^{N-j}\\\
\innerproduct{p_{N-i}}{x^{N+M-\beta}}&\innerproduct{p_{N-i}}{x^{N-j}}\end{bmatrix}=\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{bmatrix}p_{N+M-\beta}(z_{\alpha})&p_{N-j}(z_{\alpha})\\\
\innerproduct{p_{N-i}}{p_{N+M-\beta}}&\innerproduct{p_{N-i}}{p_{N-j}}\end{bmatrix}\,.$
(2.19)

where in going to the last equality, we used the linearity of the inner-
product and the invariance of the determinant under linear transformations on
its rows to trade $x^{N+M-\beta}\rightarrow p_{N+M-\beta}(x)$, and similarly
for $z$.

Remark that $\lambda\subseteq(M^{N})$, so that $0\leq\lambda_{j}+N-j\leq
N+M-1$ for $j=1,\ldots,N$. In other words, there are always $\lambda_{j}$ such
that we may obtain any $N$-column subset of the matrix
$\innerproduct{p_{N-i}}{x^{N+M-a}}$. The sum over $\lambda$ is thus indeed a
sum over the different subsets of $N$ columns of the matrix
$\innerproduct{p_{N-i}}{x^{N+M-a}}$. We may rewrite the rank-$N$ matrix as

$\displaystyle\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}]_{i,j=1,\ldots,N}=\begin{bmatrix}\bra{p_{N-1}}\\\
\vdots\\\
\bra{p_{0}}\end{bmatrix}\quantity[\ket{x^{N+M-1}}\cdots|\widehat{x^{\lambda_{1}^{\vee}+M-1}}\rangle\cdots|\widehat{x^{\lambda_{M}^{\vee}}}\rangle\cdots\ket{x^{0}}]$
(2.20)

where the hat is a removal symbol. This reflects the fact that
$\lambda^{\vee}$ plays the same role as $L$ in (2.18) (the elements $l\in L$
are related via $l\leftrightarrow N+\beta-\lambda^{\vee}_{\beta}$). Finally,
we used that $NM-|\lambda^{\vee}|=|\lambda|$ to match the grading appearing in
Eq. 2.18. We may express all this graphically as in Fig. 2.

$N-j$$N$$M$$M-\beta$ Figure 2: We can append to the Young Tableau of Fig. 1 an
$N\times N$ square block to left and an $M\times M$ block below. The number of
total white squares in each row are labeled by $\lambda_{j}+N-j$. The number
of total shaded squares in each column are labeled by
$\lambda^{\vee}_{\beta}+M-\beta$. For this particular example of partition
$\lambda$, we would thus have
$B_{H^{\prime},L^{\prime}}=\innerproduct{p_{N-i}}{x^{a}}$ for
$a\in\\{1,2,4,6\\}$ and $B_{H,L}=z_{\beta}^{c}$ for $c\in\\{0,3,5\\}$.

Taking a monic polynomial as the orthogonal polynomial (2.10) as before, the
lower left block in Eq. (2.2) vanishes and the determinant factorizes into the
product of determinants for each block. We thus obtain the determinantal
formula of the characteristic polynomial average,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{N}\det(z_{\alpha}-X)}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\quantity[P_{N+\alpha-1}(z_{\beta})]\,,$ (2.21)

which reproduces formula (1.1).

###  2.3 Inverse Characteristic Polynomial Average

We now wish to compute the following correlation functions, exactly in $N$ and
$M$,

$\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}=\frac{1}{\mathcal{Z}_{N}}\int\differential{X}e^{-\Tr
V(X)}\prod_{\alpha=1}^{M}\frac{1}{\det(z_{\alpha}-X)}.$ (2.22)

The Cauchy sum formula also allows us to expand the inverse characteristic
polynomial in terms of a sum of products of Schur polynomials

$\prod_{\alpha=1}^{M}\frac{1}{\det(z_{\alpha}-X)}=\frac{1}{\det(Z)^{N}}\sum_{\ell(\lambda)\leq\min(M,N)}s_{\lambda}(Z^{-1})s_{\lambda}(X)$
(2.23)

where $X=(x_{1},..,x_{N})$ are the eigenvalues of the matrix $X$ and
$\ell(\lambda)=\lambda^{\text{T}}_{1}$. This is a sum over a subset of
representations of the symmetric group. We will first assume $M<N$, so that
$\lambda_{i}=0$ for all $i>M$, i.e. the relevant Young tableaux all have at
most $M$ rows, $\ell(\lambda)\leq M$.

Note we can rewrite part of Eq. (2.23) as

$\frac{s_{\lambda}(Z^{-1})}{\det(Z)^{N}}=\frac{\displaystyle\det_{1\leq\alpha,\beta\leq
M}\left[z_{\alpha}^{-\lambda_{\beta}+\beta-(N+1)}\right]}{\Delta_{M}(Z)}.$
(2.24)

While we have already computed the expectation value of $s_{\lambda}(x)$, we
now want to exploit the fact that $\lambda_{i}=0$ for $i>M$. To make this
split manifest, let Greek letters $\alpha,\beta...\in(1,\ldots,M)$ and reserve
early Latin alphabet letters $a,b\in(M+1,\ldots,N)$. Then we may rewrite Eq.
(2.13) in block form

$\displaystyle\expectationvalue{s_{\lambda}(X)}$
$\displaystyle=\frac{1}{\tilde{\mathcal{Z}}_{N}}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\innerproduct{p_{N-\alpha}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{p_{N-\alpha}}{p_{N-b}}\\\
\innerproduct{p_{N-a}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{p_{N-a}}{p_{N-b}}\end{bmatrix}$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N-M}}{\tilde{\mathcal{Z}}_{N}}\det_{1\leq\alpha,\beta\leq
M}\quantity[\innerproduct{P_{N-\alpha}}{x^{\lambda_{\beta}+N-\beta}}]$ (2.25)

In going to the second line, we again took $p_{k}\rightarrow P_{k}$ via linear
transformations on the rows. The upper right block then vanishes due to their
orthogonality and the determinant factorizes. The prefactor
$\tilde{\mathcal{Z}}_{N-M}=\prod_{a=M+1}^{N}h_{N-a}$ arises as the determinant
of the lower right block.

Having computed the expectation value of the Schur polynomial, we can now
write Eq. (2.22) as

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)}\sum_{0\leq\lambda_{M}\leq...\leq\lambda_{1}\leq\infty}\det_{1\leq\alpha,\beta\leq
M}\left[z_{\alpha}^{-\lambda_{\beta}+\beta-(N+1)}\right]\det_{1\leq\alpha,\beta\leq
M}\quantity[\innerproduct{P_{N-\alpha}}{x^{\lambda_{\beta}+N-\beta}}].$ (2.26)

To perform this sum, first define the shifted variable
$r_{\alpha}=\lambda_{\alpha}+M-\alpha$, and note that even if
$\lambda_{\alpha+1}=\lambda_{\alpha}$, $r_{\alpha}>r_{\alpha+1}$, namely
$(r_{1}>r_{2}>\cdots>r_{M})\in\mathbb{Z}^{M}_{\geq 0}$. We can then rewrite
Eq. (2.3) as 666The factor of $1/M!$ below comes from the lack of ordering in
the sum over $r_{\alpha}$.

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}=$
$\displaystyle\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)M!}\sum_{\begin{subarray}{c}0\leq
r_{1},...,r_{M}\leq\infty\\\ r_{\alpha}\neq
r_{\beta}\end{subarray}}\det_{1\leq\alpha,\beta\leq M}\left[z_{\alpha}^{M-N-
r_{\beta}-1}\right]\det_{1\leq\alpha,\beta\leq
M}\quantity[\innerproduct{P_{N-\alpha}}{x^{-M+r_{\beta}+N}}]$ $\displaystyle=$
$\displaystyle\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\left[\sum_{r=0}^{\infty}z_{\alpha}^{M-N-r-1}\innerproduct{P_{N-\beta}}{x^{N-M+r}}\right]$
(2.27)

where we used a discrete equivalent of the Andréif–Heine identity to trade the
sum of products of determinants for a determinant of the sum of products.

We can now explicitly perform the sum over $r$, and recognize it as the
geometric series expansion
$(z_{\alpha}-x)^{-1}=\sum_{r=0}^{\infty}z^{-r-1}x^{-r}$. This leaves us with

$\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}=\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)}\det\left[z_{\alpha}^{M-N}\int\frac{\differential{x}}{2\pi}e^{-V(x)}P_{N-\beta}(x)\frac{x^{N-M}}{z_{\alpha}-x}\right]$
(2.28)

To recover the more familiar expression, first rewrite

$\frac{x^{N-M}}{z-x}=\frac{z^{N-M}}{z-x}-\frac{z^{N-m}-x^{N-m}}{z-x}=\frac{z^{N-M}}{z-x}+\mathcal{O}(x^{N-M-1})$
(2.29)

and note that, by the orthogonality,

$\int\frac{\differential{x}}{2\pi}e^{-V(x)}P_{N-\beta}(x)\mathcal{O}(x^{N-(M+1)})=0.$
(2.30)

Recalling the definition of the Hilbert transform (1.5), this gives us our
final expression

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\left[\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{P_{N-\beta}(x)}{z_{\alpha}-x}\right]$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N-M}/\tilde{\mathcal{Z}}_{N}}{\Delta_{M}(Z)}\det_{1\leq\alpha,\beta\leq
M}\left[\tilde{P}_{N-\beta}(z_{\alpha})\right]$ (2.31)

which reproduces the classic formula for inverse characteristic polynomials in
terms of a determinant over Hilbert-transforms of the orthogonal polynomials,
as advertised in Eq.(1.4). This shows a different (and also simpler)
derivation of the inverse characteristic polynomial formula, originally found
in [5].

#### The Case with $M>N$

Let us now consider the opposite situation $M>N$. In this case, a different
manipulation is necessary in the Cauchy sum formula (2.23). We now have the
condition $\ell(\lambda)\leq\operatorname{min}(N,M)=N$, so that
$\lambda_{\alpha}=0$ for all $\alpha>N$.

While there are no immediate simplification to be made to our expression for
$\expectationvalue{s_{\lambda}(X)}$, we may instead exploit the following
block matrix structure for the $M$-variable Schur polynomial (2.24),

$\displaystyle\frac{s_{\lambda}(Z^{-1})}{\det(Z)^{N}}=\frac{1}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{bmatrix}.$ (2.32)

Thus, the inverse characteristic polynomial average admits the following
expansion over partitions,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}$
$\displaystyle=\frac{\tilde{\mathcal{Z}}_{N}^{-1}}{\Delta_{M}(Z)}\sum_{0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{bmatrix}\det_{1\leq i,j\leq
N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}].$ (2.33)

To perform the summation over the partition, we reintroduce
$r_{i}=\lambda_{i}+N-i$ obeying the strictly decreasing condition
$r_{1}>r_{2}>\cdots>r_{N}\geq 0$. We can use a slight generalization of the
discrete Andréif–Heine identity accommodating the determinant of different
size matrices 777This is easiest to see by expanding the determinants using
the Levi-Civita symbol. to obtain

$\displaystyle\sum_{0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{bmatrix}\det_{1\leq i,j\leq
N}\quantity[\innerproduct{p_{N-i}}{x^{\lambda_{j}+N-j}}]$
$\displaystyle=\det_{\begin{subarray}{c}i=1,\ldots,N\\\ \alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\displaystyle\sum_{r=0}^{\infty}z_{\alpha}^{-r-1}\innerproduct{p_{N-i}}{x^{r}}\\\
z_{\alpha}^{a-1}\end{bmatrix}=\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix},$ (2.34)

where we define the Hilbert transform of the monic polynomial,

$\displaystyle\tilde{p}_{k}(z)=\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{p_{k}(x)}{z-x}.$
(2.35)

We have also used the invariance of the determinant under linear
transformation to convert the monomial $z_{\alpha}^{a-1}$ to an arbitrary
monic polynomial $p_{a-1}(z_{\alpha})$. This gives us the following
determinantal formula for the inverse characteristic polynomial in the case of
$M>N$,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}=\frac{\tilde{\mathcal{Z}}_{N}^{-1}}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}.$ (2.36)

To the best of our knowledge, the inverse characteristic polynomial formula
for $M>N$ has not appeared before in the literature. It represents one of the
main results of this paper. We have thus seen how our analysis based on the
Schur polynomial expansion provides a systematic method to calculate various
averages of the characteristic polynomials, and their inverses.

##  3 Generalizing to Non-Zero Source

###  3.1 Schur Polynomial Average with Source

In order to calculate the characteristic polynomials with the source term, we
first consider the (non-normalized) Schur polynomial average in the presence
of an external matrix source,

$\displaystyle\expectationvalue{s_{\lambda}(X)}_{A}=\int\differential{X}e^{-\Tr
V(X)+i\Tr AX}s_{\lambda}(X)\,.$ (3.1)

To evaluate this average, we use the Harish-Chandra–Itzykson–Zuber (HCIZ)
formula [16]

888 We remark
$\displaystyle\frac{1}{\operatorname{Vol}U(N)}\int_{U(N)}\differential{U}e^{i\Tr
XUYU^{-1}}\ \xrightarrow{X\to\mathbbm{1}_{N}}\
\frac{1}{\operatorname{Vol}U(N)}\int_{U(N)}\differential{U}e^{i\Tr Y}=e^{i\Tr
Y}.$ (3.2) This fixes the normalization of the integral.

$\displaystyle\frac{1}{\operatorname{Vol}U(N)}\int_{U(N)}\differential{U}e^{i\Tr
XUYU^{-1}}=\frac{\Gamma_{2}(N+1)(-i)^{N(N-1)/2}}{\Delta_{N}(X)\Delta_{N}(Y)}\det_{1\leq
i,j\leq N}\quantity[e^{ix_{i}y_{i}}],$ (3.3)

which yields

$\displaystyle\expectationvalue{s_{\lambda}(X)}_{A}$
$\displaystyle=\frac{C_{N}}{\Delta_{N}(A)}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{\differential{x}_{i}}{2\pi}e^{-V(x_{i})}\det_{1\leq
i,j\leq N}\quantity[e^{i\mathsf{a}_{i}x_{j}}]\det_{1\leq i,j\leq
N}\quantity[x_{i}^{\lambda_{j}+N-j}]$
$\displaystyle=\frac{C_{N}}{\Delta_{N}(A)}\det_{1\leq i,j\leq
N}\quantity[\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{\lambda_{j}+N-j}}].$ (3.4)

with $(\mathsf{a}_{i})_{i=1,\ldots,N}$ the eigenvalues of $A$ and the
normalization constant

$\displaystyle
C_{N}=\operatorname{Vol}U(N)\times\Gamma_{2}(N+1)\times(-i)^{N(N-1)/2}=(2\pi)^{N(N+1)/2}(-i)^{N(N-1)/2}.$
(3.5)

We use this result in the following to compute the characteristic polynomial,
and its inverse, with non-zero source.

###  3.2 Characteristic Polynomial Average with Source

We first compute the expectation value of multiple determinant insertions, in
the presence of source. In this Section, we reproduce the results of [13], in
particular its Eq. (3.10).

Applying the Schur polynomial expansion together with the formula (3.4), we
obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{N}\det(z_{\alpha}-X)}_{A}$
$\displaystyle=\frac{1}{\Delta_{M}(Z)}\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{N-\lambda_{\beta}^{\text{T}}+\beta-1}]\expectationvalue{s_{\lambda}(X)}_{A}$
$\displaystyle=\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{N-\lambda_{\beta}^{\text{T}}+\beta-1}]\det_{1\leq
i,j\leq
N}\left[\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{N+\lambda_{j}-j}}\right].$
(3.6)

We can again recognize this as the minor expansion on the first $M$ rows of
the following $(N+M)\times(N+M)$ matrix,

$\displaystyle\sum_{\lambda\subseteq(M^{N})}(-1)^{|\lambda|}\det_{1\leq\alpha,\beta\leq
M}\quantity[z_{\alpha}^{N-\lambda_{\beta}^{\text{T}}+\beta-1}]\det_{1\leq
i,j\leq N}\quantity[\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{\lambda_{j}+N-j}}]$
$\displaystyle=\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ 1\leq
i,j\leq
N\end{subarray}}\begin{bmatrix}z_{\alpha}^{N+M-\beta}&z_{\alpha}^{N-j}\\\
\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{N+M-\beta}}&\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{N-j}}\end{bmatrix}=\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{bmatrix}p_{N+M-\beta}(z_{\alpha})&p_{N-j}(z_{\alpha})\\\
\innerproduct{e^{i\mathsf{a}_{i}x}}{p_{N+M-\beta}}&\innerproduct{e^{i\mathsf{a}_{i}x}}{p_{N-j}}\end{bmatrix}\,.$
(3.7)

where in going to the last equality, we again used the linearity of the inner-
product and the invariance of the determinant under linear transformations on
its rows to trade $x^{N+M-\beta}\rightarrow p_{N+M-\beta}(x)$. We can thus
write the final result as

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{N}\det(z_{\alpha}-X)}_{A}$
$\displaystyle=\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ 1\leq i,j\leq
N\end{subarray}}\begin{bmatrix}p_{N+M-\beta}(z_{\alpha})&p_{N-j}(z_{\alpha})\\\
Q_{N+M-\beta}(\mathsf{a}_{i})&Q_{N-j}(\mathsf{a}_{i})\end{bmatrix}\,$ (3.8)

where we borrowed the notation of [13]

$Q_{k}(\mathsf{a})=\innerproduct{e^{i\mathsf{a}x}}{p_{k}}=\int\frac{\differential{x}}{2\pi}e^{-V(x)+i\mathsf{a}x}p_{k}(x).$
(3.9)

This shows the formula presented in (1.10).

###  3.3 Inverse Characteristic Polynomial Average with Source

We now wish to generalize our calculation to the inverse characteristic
polynomial with non-zero source

$\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=\int\differential{X}e^{-\Tr
V(x)+i\Tr AX}\prod_{\alpha=1}^{M}\frac{1}{\det(z_{\alpha}-X)}.$ (3.10)

This computation is the main technical result of this paper.

With the expectation value of $s_{\lambda}(x)$ in the presence of the source
$A$ as in (3.4), we use the Cauchy sum formula to expand the inverse
determinants in terms of Schur polynomials. We begin with the case $M<N$. This
means $\lambda_{i}=0$ for $i>M$. Then, we obtain a block matrix structure
similar in spirit to the case without the source term (2.25),

$\displaystyle\expectationvalue{s_{\lambda}(X)}_{A}$
$\displaystyle=\frac{C_{N}}{\Delta_{N}(A)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\innerproduct{e^{i\mathsf{a}_{\alpha}x}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{e^{i\mathsf{a}_{\alpha}x}}{p_{N-b}}\\\
\innerproduct{e^{i\mathsf{a}_{a}x}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{e^{i\mathsf{a}_{a}x}}{p_{N-b}}\end{bmatrix}.$
(3.11)

None of the blocks in the above matrix vanish. Hence, the determinant no
longer factorizes into a determinant of the $M\times M$ and $(N-M)\times(N-M)$
diagonal blocks. This is the main complication relative to the case without
source.

Using the same notation as in Section 2.3, we now want to perform the
following sum over representations

$\displaystyle\frac{1}{M!}\sum_{\begin{subarray}{c}0\leq
r_{1},...,r_{M}\leq\infty\\\ r_{\alpha}\neq
r_{\beta}\end{subarray}}\det_{1\leq\alpha,\beta\leq M}\left[z_{\alpha}^{M-N-
r_{\beta}-1}\right]\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ M+1\leq
a,b\leq
N\end{subarray}}\begin{bmatrix}\innerproduct{e^{i\mathsf{a}_{\alpha}x}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{e^{i\mathsf{a}_{\alpha}x}}{p_{N-b}}\\\
\innerproduct{e^{i\mathsf{a}_{a}x}}{x^{\lambda_{\beta}+N-\beta}}&\innerproduct{e^{i\mathsf{a}_{a}x}}{p_{N-b}}\end{bmatrix}.$
(3.12)

After yet another application of the generalized discrete Andréif–Heine
identity and performing the geometric sum, we arrive at

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=$
$\displaystyle\frac{C_{N}}{\Delta_{N}(A)\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\innerproduct{e^{i\mathsf{a}_{\alpha}}}{\frac{(x/z_{\beta})^{N-M}}{z_{\beta}-x}}&\innerproduct{e^{i\mathsf{a}_{\alpha}}}{p_{N-b}}\\\\[5.0pt]
\innerproduct{e^{i\mathsf{a}_{a}}}{\frac{(x/z_{\beta})^{N-M}}{z_{\beta}-x}}&\innerproduct{e^{i\mathsf{a}_{a}}}{p_{N-b}}\end{bmatrix}.$
(3.13)

The final simplification arises by noting that we can rewrite

$z^{M-N}\left(\frac{x^{N-M}}{z-x}\right)=\frac{1}{z-x}+\sum_{s=0}^{N-M-1}\frac{x^{s}}{z^{s+1}},$
(3.14)

so that the first $M$ columns can be decomposed as

$\displaystyle
z_{\beta}^{M-N}\innerproduct{e^{i\mathsf{a}_{i}x}}{\frac{x^{N-M}}{z_{\beta}-x}}$
$\displaystyle=\innerproduct{e^{i\mathsf{a}_{i}x}}{\frac{1}{z_{\beta}-x}}+\sum_{b=M+1}^{N}z_{\beta}^{-N+b-1}\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{N-b}}.$
(3.15)

The determinant is insensitive to this linear superposing of the other $N-M$
columns, so we can drop these other terms. Our final result therefore reads,
as advertised:

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=$
$\displaystyle\frac{C_{N}}{\Delta_{N}(A)\Delta_{M}(Z)}\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{\alpha},z_{\beta})&Q_{N-b}(\mathsf{a}_{\alpha})\\\
\tilde{Q}(\mathsf{a}_{a},z_{\beta})&Q_{N-b}(\mathsf{a}_{a})\end{bmatrix}$
(3.16)

with the new function $\tilde{Q}(\mathsf{a},z)$ defined as in Eq. (1.12). The
structure of this result agrees with a particular limit of a supermatrix model
studied by one of the authors in [17] (see the $M=p=0$ limit of Eq. (3.3) in
that paper). The normalization could not be computed there due to the use of a
supermatrix generalization of the HCIZ angular integral.

####  3.3.1 Zero Source Limit

We can further check our formula by considering the limit of vanishing source,
$A\rightarrow 0$. This limit is not completely straightforward because of Van-
der-Monde determinant $\Delta_{N}(A)$ appearing in the denominator. We
therefore need to expand the numerator to $\mathcal{O}(\mathsf{a}_{i}^{N-1})$.
Begin by writing (the unusual labeling of $k$ in the Taylor expansion will be
useful below)

$\displaystyle\tilde{Q}(\mathsf{a}_{i},z_{\beta})=$
$\displaystyle\sum_{k=1}^{N}\frac{1}{(N-k)!}\tilde{Q}^{(N-k)}(0,z_{\beta})\mathsf{a}_{i}^{N-k}+\mathcal{O}(\mathsf{a}_{i}^{N}),$
(3.17a) $\displaystyle Q_{N-b}(\mathsf{a}_{i})=$
$\displaystyle\sum_{k=1}^{N}\frac{1}{(N-k)!}Q^{(N-k)}_{N-b}(0)\mathsf{a}_{i}^{N-k}+\mathcal{O}(\mathsf{a}_{i}^{N}),$
(3.17b)

where

$\displaystyle\tilde{Q}^{(k)}(0,z_{\beta})=$
$\displaystyle\left.\frac{\partial^{k}\tilde{Q}(\mathsf{a}_{i},z_{\beta})}{\partial\mathsf{a}_{i}^{k}}\right|_{\mathsf{a}_{i}=0}=i^{k}\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{x^{k}}{z_{\beta}-x},$
(3.18a) $\displaystyle Q^{(k)}_{N-b}(0)=$
$\displaystyle\left.\frac{\partial^{k}Q_{N-b}(\mathsf{a}_{i})}{\partial\mathsf{a}_{i}^{k}}\right|_{\mathsf{a}_{i}=0}=i^{k}\int\frac{\differential{x}}{2\pi}e^{-V(x)}x^{k}p_{N-b}(x).$
(3.18b)

Interpreting the sum in the Taylor expansions of Eq. (3.17) as matrix
multiplication, and using $\det(AB)=\det(A)\det(B)$, we arrive at the
following expansion for the determinant

$\displaystyle\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq M\\\ M+1\leq
a,b\leq
N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{\alpha},z_{\beta})&Q_{N-b}(\mathsf{a}_{\alpha})\\\
\tilde{Q}(\mathsf{a}_{a},z_{\beta})&Q_{N-b}(\mathsf{a}_{a})\end{bmatrix}$
$\displaystyle=\det_{1\leq i,j\leq
N}\left[\frac{1}{(N-j)!}\mathsf{a}_{i}^{N-j}\right]\det_{\begin{subarray}{c}1\leq\alpha,\beta\leq
M\\\ M+1\leq a,b\leq
N\end{subarray}}\begin{bmatrix}\tilde{Q}^{(N-\alpha)}(0,z_{\beta})&Q^{(N-\alpha)}_{N-b}(0)\\\
\tilde{Q}^{(N-a)}(0,z_{\beta})&Q^{(N-a)}_{N-b}(0)\end{bmatrix}+\mathcal{O}(\mathsf{a}_{i}^{N}).$
(3.19)

The upper right block $Q_{N-b}^{(N-\alpha)}(0)$ vanishes by orthogonality if
we specialize the monic polynomial $p_{k}(x)\to P_{k}(x)$, so that the second
determinant factorizes into a determinant of each diagonal block. With the
Van-der-Monde determinant written as $\Delta_{N}(A)=\det_{1\leq i,j\leq
N}(\mathsf{a}_{i}^{N-j})$, and using the invariance of the determinant under
linear transformation of the rows, we can simplify the above to

$\displaystyle\frac{\tilde{\mathcal{Z}}_{N-M}}{\Gamma_{2}(N+1)(-i)^{N(N-1)/2}}\Delta_{N}(A)\det_{1\leq\alpha,\beta\leq
M}\left[\int\frac{\differential{x}}{2\pi}e^{-V(x)}\frac{P_{N-\alpha}(x)}{z_{\beta}-x}\right]+\mathcal{O}(\mathsf{a}_{i}^{N})$
(3.20)

where $\tilde{\mathcal{Z}}_{N-M}=\prod_{a=M+1}^{N}h_{N-a}$ comes from
evaluating the determinant of lower right block. It is trivial now to take the
limit $\mathsf{a}_{i}\rightarrow 0$ and recuperate the result of Section 2.3
(recalling the relative factor of $\mathcal{Z}_{N}$ used in the normalization
there).

####  3.3.2 The Case with $M>N$

So far, we have assumed the condition $M\leq N$ in the computation. We now
consider the opposite situation $M>N$, and calculate the inverse
characteristic polynomial average.

In this case, we similarly apply the determinantal formula (2.32), and put to
use the Schur polynomial average with source (3.4). We then obtain

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}$
$\displaystyle=\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\sum_{0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{bmatrix}\det_{1\leq i,j\leq
N}\quantity[\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{\lambda_{j}+N-j}}].$ (3.21)

Introducing $r_{i}=\lambda_{i}+N-i$ obeying $r_{1}>r_{2}>\cdots>r_{N}\geq 0$
as before, we can again use the generalized discrete Andréif–Heine identity to
write

$\displaystyle\sum_{0\leq\lambda_{N}\leq\cdots\leq\lambda_{1}\leq\infty}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}z_{\alpha}^{-\lambda_{i}+i-(N+1)}\\\
z_{\alpha}^{a-1}\end{bmatrix}\det_{1\leq i,j\leq
N}\quantity[\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{\lambda_{j}+N-j}}]$
$\displaystyle=\det_{\begin{subarray}{c}i=1,\ldots,N\\\ \alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\displaystyle\sum_{r=0}^{\infty}z_{\alpha}^{-r-1}\innerproduct{e^{i\mathsf{a}_{i}x}}{x^{r}}\\\
z_{\alpha}^{a-1}\end{bmatrix}=\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{i},z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}.$ (3.22)

To convert $z_{\alpha}^{a-1}$ to $p_{a-1}(z_{\alpha})$ in the last equality,
we have used the invariance of the determinant under linear transformation.
Hence, we arrive at the inverse characteristic polynomial formula for $M>N$,

$\displaystyle\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=\frac{C_{N}}{\Delta_{M}(Z)\Delta_{N}(A)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{Q}(\mathsf{a}_{i},z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}.$ (3.23)

#### Zero Source Limit

The zero source limit $A\to 0$ of the formula (3.23) is similarly considered
as in Section 3.3.1. Applying the Taylor expansion for the function
$\tilde{Q}(\mathsf{a}_{i},z_{\alpha})$ with respect to the source (3.17), we
straightforwardly obtain

$\displaystyle\lim_{A\to
0}\expectationvalue{\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)^{-1}}_{A}=\frac{\operatorname{Vol}U(N)}{\Delta_{M}(Z)}\det_{\begin{subarray}{c}i=1,\ldots,N\\\
\alpha=1,\ldots,M\\\
a=1,\ldots,M-N\end{subarray}}\begin{bmatrix}\tilde{p}_{N-i}(z_{\alpha})\\\
p_{a-1}(z_{\alpha})\end{bmatrix}.$ (3.24)

This reproduces the previous result (2.36), again remembering the relative
normalization constant $\mathcal{Z}_{N}$ (2.12).

####  3.3.3 Characteristic Polynomial/Source Duality

The characteristic polynomial with the source shows an interesting duality
under $Z\leftrightarrow A$ [4, 18, 13]. We demonstrate this duality using the
formula obtained in this paper.

Let us focus on the Gaussian case for simplicity, $V(x)=x^{2}/2$. Then,
$P_{k}$ and $Q_{k}$ are essentially equivalent to each other in this case.
Furthermore, we have

$\displaystyle e^{\frac{1}{2}\mathsf{a}^{2}}\tilde{Q}(\mathsf{a},z)$
$\displaystyle=\int\frac{\differential{x}}{2\pi}\frac{e^{-\frac{1}{2}(x-i\mathsf{a})^{2}}}{z-x}=\int\frac{\differential{x}}{2\pi}\frac{e^{-\frac{1}{2}x^{2}}}{z-i\mathsf{a}-x}=\int\frac{\differential{x}}{2\pi}\frac{e^{-\frac{1}{2}(x+z)^{2}}}{-i\mathsf{a}-x}=e^{-\frac{1}{2}z^{2}}\tilde{Q}(iz,-i\mathsf{a}).$
(3.25)

Comparing the two expressions for the inverse characteristic polynomial (3.16)
and (3.23), we explicitly see the duality $Z\leftrightarrow A$ in the Gaussian
case. This duality had been derived for the Gaussian case via an integrating
in-out-in-out procedure, rewriting the inverse determinants in terms of a
bosonic integral (see section 4.2 of [13]).

##  4 Discussion

#### Eigenvalue “Fermi Sea” Picture and the $M>N$ Regime for Inverse
Determinant Correlators

The $M>N$ regime seems not to have been previously considered in the
literature. One reason is perhaps that the condition $M\leq N$ or $M\geq N$
matters little for the better studied determinant insertions. There are
multiple ways to see this. First, in the case of non-zero source, Eq. (1.10)
is the determinant of an $(N+M)\times(N+M)$ matrix, with sub-blocks of size
$M$ and $N$. Contrast this with the inverse characteristic polynomial result
Eq. (1.11), which is the determinant of a $\operatorname{max}(M,N)$ sized
matrix with a $\operatorname{min}(M,N)$ sized sub-block structure. Even in the
case without source, $N$ is a lower bound on the index of the orthogonal
polynomials appearing in Eq. (1.1), while it appears as an upper bound in Eq.
(1.4).

In the “Fermi Sea” picture, the full probability distribution for the $n$
eigenvalues is nothing but the square of a Slater determinant ground state
wavefunction for $n$ fermions in the potential $V(x)$. Eq. (1.1) may also be
viewed as a Slater determinant wavefunction for fermions with position labels
$(z_{\alpha})_{\alpha=1,\ldots,M}$. The role of the Van-der-Monde in the
denominator is to cancel the $\Delta_{M}(Z)^{2}$ contained in the matrix
measure $\differential{Z}$, against which we would normalize the square of the
wavefunction (see the discussion around Eq. (2.10) of [19]). One way to
interpret the shift by $N$ in Eq. (1.1) is to say that determinant insertions
correspond to particle excitations above the Fermi level. We can always
populate higher energy levels, there is no cutoff.

Eq. (1.4) may then similarly be viewed as a Slater determinant wavefunction.
From this perspective, inverse characteristic polynomials more closely
resemble hole-excitations.999That said, this interpretation offers little as
to the appearance of the Hilbert transform. There is a natural cutoff, namely
the depth of the Fermi sea. How then do we make sense of $M>N$? One
speculative take on Eq. (3.23) is to say that we first make $N$ hole
excitations. We then “re-populate” the Fermi sea with the remaining $M-N$
fermions, starting from the ground state. We identify these filled levels with
the $(M-N)$-sized sub-block
$\left[p_{a-1}(z_{\beta})\right]_{a=1,\ldots,M-N}$.

#### String Theory Perspective on the Schur Expansion of Characteristic
Polynomials

Matrix models provide one of the most concrete ways to study open/closed
string duality. In particular, we can explicitly implement the idea of
“closing up holes” on the open string worldsheet, replacing them with a
superposition of closed string vertex operators. On the matrix side, this
corresponds simply to rewriting the determinant operator as
$\det(z-X)=\exp\Tr\log(z-X)$ and Taylor expanding for large $z$:

$\prod_{\alpha=1}^{M}\det(z_{\alpha}-X)=\det_{M\times
M}(Z)\exp\quantity(-\sum_{k=1}^{\infty}t_{k}\Tr X^{k})$ (4.1)

with

$t_{k}=\frac{1}{k}\Tr_{M}(Z^{-k}).$ (4.2)

The $t_{k}$’s are the so-called Miwa variables, and play the role of couplings
for the closed string worldsheet “vertex operators” $\Tr X^{k}$. They play a
crucial role in the Kontsevich duality [20]. Eq. (4.2) shows exactly how the
insertion of D-branes can equivalently be re-expressed in terms of a new
closed string background.

One interesting take on the Schur expansion of the characteristic polynomial
is as a decomposition of the brane insertions in terms of an orthogonal set of
multi-closed-string “operators”.101010We thank Rajesh Gopakumar for discussion
on this point. Orthogonality is guaranteed by the character nature of the
Schur polynomials under the $U(N)$ integral. We use the term “multi-closed-
string” because the Schur polynomials, when written in terms of traces,
contain products of traces. The role of the Miwa variable is played here by
$s_{\lambda^{\vee}}(Z)$.111111This Schur polynomial may be also interpreted as
the Wilson loop contribution with respect to the background $U(M)$ gauge field
[21]. This interpretation is inspired by [22], which discusses the “Schur
Polynomial Basis” for the matrix quantum harmonic oscillator in terms of a
closed string Fock basis.121212Of course, we are dealing with a simple matrix
integral here, and not a matrix quantum mechanics, so the notion of operator
and state on the matrix side does not quite carry over (but they are sensible
notions on the worldsheet).

## Acknowledgments

EAM wishes to thank Bruno Balthazar, Rajesh Gopakumar, Jorrit Kruthoff and
Phil Saad for useful discussions on the meaning of (inverse) characteristic
polynomials, and their Schur expansion, in matrix models of string theory and
JT gravity. EAM would also like to thank Pavel Wiegmann for a detailed
discussion of orthogonal polynomials in RMT. The work of TK 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. EAM is supported by a Kadanoff fellowship at
the University of Chicago, and would like to acknowledge the support of the
Jones Endowment for Physics Research.

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