# The Partition Function of Log-Gases with Multiple Odd Charges

Elisha D. Wolff  Jonathan M. Wells

###### Abstract

We use techniques in the shuffle algebra to present a formula for the
partition function of a one-dimensional log-gas comprised of particles of
(possibly) different integer charges at certain inverse temperature $\beta$ in
terms of the Berezin integral of an associated non-homogeneous alternating
tensor. This generalizes previously known results by removing the restriction
on the number of species of odd charge. Our methods provide a unified
framework extending the de Bruijn integral identities from classical
$\beta$-ensembles ($\beta=1,2,4$) to multicomponent ensembles, as well as to
iterated integrals of more general determinantal integrands.

Keywords: Partition function, Berezin integral, Pfaffian, Hyperpfaffian, Grand
canonical ensemble, Shuffle algebra

##  1 Introduction

Suppose a finite number of charged particles are placed on an infinite wire
represented by the real line. The charges of the particles are assumed to be
integers with the same sign, and the particles are assumed to repel each other
with logarithmic interactions. We assume any two particles of the same charge,
which we will call same _species_ , are indistinguishable. The wire is imbued
with a potential which discourages the particles from escaping to infinity in
either direction, and heat is applied to the system according to a parameter
we call the inverse temperature $\beta$.

We consider two ensembles:

  1. 1.

_The Canonical Ensemble_ , in which the number of particles of each species is
fixed; in this case, we say fixed population.

  2. 2.

_The Isocharge Grand Canonical Ensemble_ , in which the sum of the charges of
the particles is fixed, but the number of particles of each species is allowed
to vary; in this case, we say the total charge of the system is fixed.

In contrast, the _Grand Canonical Ensemble_ traditionally refers to the
ensemble in which the total number of particles is not fixed. For
computational purposes, it is beneficial to group configurations which share
the same total charge. The Grand Canonical Ensemble is then a disjoint union
(over all possible sums of charges) of our Isocharge ensembles.

In 2012, Sinclair [20] provided a closed form of the partition function for
both ensembles in terms of Berezin integrals (see subsection 2.2) of
alternating tensors, but only for certain $\beta$ and only for ensembles with
at most one species of odd charge. Here, we provide an alternative framework
which allows us to generalize the result to ensembles with arbitrary mix of
odd and even charges, albeit with the same limitations on $\beta$. These
Berezin integral expressions are analogous to the (Pfaffian) de Bruijn
integral identities [6] (for classical $\beta=1$ and $\beta=4$ ensembles). Our
methods are analogous to algebraic methods first used by Thibon and Luque [12]
in the classical single species cases. In subsection 3.1, we demonstrate how
these methods can also be applied to iterated integrals of more general
determinantal integrands.

By first conditioning on the number of particles of each species, the
partition function for the Isocharge Grand Canonical Ensemble is built up from
the partition functions of the Canonical type, revealing the former to be a
generating function of the latter as a function of the _fugacities_ of each
species (roughly, the probability of the occurrence of any one particle of a
given charge). In section 7, we produce analogous results for charged
particles placed on the unit circle in the complex plane.

###  1.1 Historical context

The $\beta$-ensembles are a well-studied collection of random matrices whose
eigenvalue densities take a common form, indexed by a non-negative, real
parameter $\beta$. Suppose $\mu$ is a continuous probability measure on
${\mathbb{R}}$ with Radon-Nikodym derivative $\frac{d\mu}{dx}=w(x)$. For each
$\beta\in{\mathbb{R}}_{>0}$, consider the $N$-point process specified by the
joint probability density

$\rho_{N}(x_{1},\ldots,x_{N})=\frac{1}{Z_{N}(\beta)N!}\prod_{i<j}\left|x_{j}-x_{i}\right|^{\beta}\prod_{i}w(x_{i})$

where $Z_{N}(\beta)$, which denotes the _partition function_ of $\beta$, is
the normalizing constant required for $\rho_{N}$ to be a probability density
function. This eigenvalue density function can be identified with the
Boltzmann factor of the previously discussed log-gas particles, as first
observed by Dyson [8], and further developed by Forrester in [11].

The _classical_ $\beta$-ensembles (with $\beta=1,2,4$ and
$w(x)=e^{-x^{2}/2}$), corresponding to Hermitian matrices with real, complex,
or quaternionic Gaussian entries (respectively), were first studied in the
1920s by Wishart in multivariate statistics [23] and the 1950s by Wigner in
nuclear physics [22]. In the subsequent decade, Dyson and Mehta [9] unified a
previously disparate collection of random matrix models by demonstrating that
the three classic $\beta$-ensembles are each variations of a single action on
random Hermitian matrices (representing the three associative division
algebras over ${\mathbb{R}}$). In [7], Dumitriu and Edelman provide
tridiagonal matrix models for $\beta$-ensembles of arbitrary positive $\beta$,
which are then used by Ramírez, Rider, and Virág in [16] to obtain the
asymptotic distribution of the largest eigenvalue.

For each $1\leq n\leq N$, define the $n^{\rm{th}}$ correlation function by

$R_{n}(x_{1},\ldots,x_{n})=\frac{1}{(N-n)!}\int_{{\mathbb{R}}^{N-n}}\rho_{N}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{N-n})\,dy_{1}\cdots
dy_{N-n}.$

It turns out that the correlation function for the classic $\beta$-ensembles
takes a particularly nice algebraic form. For example, when $\beta=2$, it can
be shown using only elementary matrix operations and Fubini’s Theorem that

$R_{n}(x_{1},\ldots,x_{n})=\frac{1}{Z_{N}(2)}\det(K(x_{i},x_{j})_{1\leq
i,j\leq n}),$

where the _kernel_ $K(x,y)$ is a certain square integrable function
${\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}$ that can most easily be
expressed in terms a family of polynomials which are orthogonal with respect
to the measure $\mu$. For this reason, we say the classical $\beta=2$ ensemble
is an example of a _determinantal_ point process. The details of this
derivation are given in [14]. Similarly, when $\beta=1$ or $4$,

$R_{n}(x_{1},\ldots,x_{n})=\frac{1}{Z_{N}(\beta)}{\rm{Pf}}(K(x_{i},x_{j})_{1\leq
i,j\leq n}),$

where ${\rm{Pf}}(A)=\sqrt{\det(A)}$ denotes the Pfaffian of an antisymmetric
matrix $A$, and where $K(x,y)$ is a certain $2\times 2$ matrix-valued function
whose entries are square-integrable, and which satisfies $K(x,y)^{T}=-K(y,x)$.
We then say the classical $\beta=1$ and $\beta=4$ ensembles are examples of
_Pfaffian_ point processes. This result was first shown for circular ensembles
by Dyson in [8], then for Gaussian ensembles by Mehta in [14] and then for
general weights ($\mu$) by Mehta and Mahoux in [13], except for the case
$\beta=1$ and $N$ odd. Finally, the last remaining case was given by Adler,
Forrester, and Nagao in [1]. Of fundamental concern in the theory of random
matrices is the behavior of eigenvalue statistics as $N\to\infty$. The
immediate advantage of these determinantal and Pfaffian expressions for the
correlation functions is that these matrix kernels do not essentially increase
in complexity as $N$ grows large, since the dimensions are the matrix kernel
are stable, and the entries are expressed as a sum whose asymptotics are well-
understood.

###  1.2 Hyperpfaffian partition functions

Derivations of the determinantal and Pfaffian expressions of the correlation
functions have been presented in numerous ways over the past several decades.
Of particular note is the method of Tracy and Widom [21], who first show that
the partition function is determinantal or Pfaffian, and then use matrix
identities and generating functions to obtain a corresponding form for the
correlation functions.

But recognizing the partition function $Z_{N}(\beta)$ as the determinant or
Pfaffian of a matrix of integrals of appropriately chosen orthogonal
polynomials is essential and nontrivial. One way to do this is to apply the
Andreif determinant identity [2] to the iterated integral which defines
$Z_{N}(\beta)$. This is immediate when $\beta=2$, and viewing the Pfaffian as
the square root of a determinant, this identity can also be applied (with some
additional finesse) when $\beta=1$ or $4$. However, viewing the Pfaffian in
the context of the exterior algebra allows us to extend the Andreif
determinant identity to analogous Pfaffian identities (referred to as the de
Bruijn integral identities [6]).

In 2002, Luque and Thibon [12] used techniques in the shuffle algebra to show
that when $\beta=L^{2}$ is an even square integer, the partition function
$Z_{N}(\beta)$ can be written as a Hyperpfaffian of an $L$-form whose
coefficients are integrals of Wronskians of suitable polynomials. Then in
2011, Sinclair [19] used other combinatorial methods to show that the result
also holds when $\beta=L^{2}$ is an odd square integer.

In this volume, we consider _multicomponent_ ensembles in which the joint
probability density functions generally have the form

$\rho_{N}(x_{1},\ldots,x_{N})=\frac{1}{Z_{f}(\beta)N!}\prod_{i<j}\left|x_{j}-x_{i}\right|^{\beta
f(i,j)}\prod_{i}w_{i}(x_{i}),$

where $f:{\mathbb{R}}^{2}\to{\mathbb{R}}$ specifies (possibly) different
exponents for each factor in the product. This setup was first investigated
for particles of charge $1$ and $2$ by Rider, Sinclair and Xu in [17], a model
closely associated to the eigenvalue densities for the real Ginibre ensembles
as discussed by Borodin and Sinclair in [4]. The limiting behavior for the two
species model in the circular case was later studied by Shum and Sinclair in
[18].

In 2012, Sinclair showed (using his same methods) the partition function
$Z_{f}(\beta)$ has a Berezin integral expression when each $\sqrt{\beta}L_{j}$
is an even integer. Herein, we extend to arbitrary positive integers
$\sqrt{\beta}L_{j}\in{\mathbb{Z}}_{>0}$ using shuffle algebra techniques,
analogous to the methods of Thibon and Luque. Note, single-component
$\beta$-ensembles are a subset of multicomponent ensembles in which
$f(i,j)=1$, and the Berezin integral is a generalization of the Hyperpfaffian.
Thus, one consequence of this work is a new, streamlined derivation of the
Hyperpfaffian partition functions for the single-component $\beta$-ensembles
in which $\beta=L^{2}$ is any square integer, even or odd.

Though multicomponent ensembles are the focus of this volume, our methods
further generalize to other ensembles. We explore many of these possibilities
in a forthcoming volume. One key observation is that $\rho_{N}$ can be written
as a determinant (typically easier than expressing $Z_{N}$ as a determinant).
More generally, we can replace the partition function $Z_{N}$ (or $Z_{f}$)
with an iterated integral of any determinant fitting certain criteria. In
subsection 3.1, we describe how to modify the statement of the main results of
this volume, producing a vast generalization of the de Bruijn integral
identities.

###  1.3 The multicomponent setup

Let $J\in{\mathbb{Z}}_{>0}$ be a positive integer, the maximum number of
distinct charges in the system. Let
$\vec{L}=(L_{1},L_{2},\ldots,L_{J})\in({\mathbb{Z}}_{>0})^{J}$ be a vector of
distinct positive integers which we will call the _charge vector_ of the
system. Let $\vec{M}\in({\mathbb{Z}}_{\geq 0})^{J}$ be a vector of non-
negative integers which we will call the _population vector_ of the system.
Each $M_{j}$ gives the number (possibly zero) of indistinguishable particles
of charge $L_{j}$. Let

${\textbf{x}}=(\textbf{x}^{1},\textbf{x}^{2},\ldots,\textbf{x}^{J})\in{\mathbb{R}}^{M_{1}}\times{\mathbb{R}}^{M_{2}}\times\cdots\times{\mathbb{R}}^{M_{J}}$

so that
$\textbf{x}^{j}=(x_{1}^{j},x_{2}^{j},\ldots,x_{M_{j}}^{j})\in{\mathbb{R}}^{M_{j}}$
for each $j$. We call x the _location vector_ of the system in which each
$x_{m}^{j}\in{\mathbb{R}}$ gives the location of a particle of charge $L_{j}$.
We call $\textbf{x}^{j}$ the _location vector_ for the species with charge
$L_{j}$. If some $M_{j}=0$, then we take $\textbf{x}^{j}$ to be the empty
vector.

The particles are assumed to interact logarithmically on an infinite wire so
that the contribution of energy to the system by two particles of charge
$L_{j}$ and $L_{k}$ at locations $x_{m}^{j}$ and $x_{n}^{k}$ respectively is
given by $-L_{j}L_{k}\log|x_{n}^{k}-x_{m}^{j}|$. If $U$ is the potential on
the system, then at inverse temperature $\beta$, the total potential energy of
the system is given by

$E_{\vec{M}}({\textbf{x}})=\beta\sum_{j=1}^{J}L_{j}\sum_{m=1}^{M_{j}}U(x_{m}^{j})-\beta\sum_{j=1}^{J}L_{j}^{2}\sum_{m<n}\log|x_{n}^{j}-x_{m}^{j}|-\beta\sum_{j<k}L_{j}L_{k}\sum_{m=1}^{M_{j}}\sum_{n=1}^{M_{k}}\log|x_{n}^{k}-x_{m}^{j}|.$

The first type of iterated sum accounts for the potential $U$, the second type
of iterated sum accounts for interactions between particles of the same charge
$L_{j}$, and the third type of iterated sum accounts for the interactions
between particles of distinct charges $L_{j}$ and $L_{k}$.

With this setup, the relative density of states (corresponding to varying
location vectors x) is given by the Boltzmann factor

$\displaystyle\Omega_{\vec{M}}({\textbf{x}})$
$\displaystyle=\exp(-E_{\vec{M}}({\textbf{x}}))$
$\displaystyle=\prod_{j=1}^{J}\prod_{m=1}^{M_{j}}\exp\left(-\beta
L_{j}U(x_{m}^{j})\right)\times\prod_{j=1}^{J}\prod_{m<n}\left|x_{n}^{j}-x_{m}^{j}\right|^{\beta
L_{j}^{2}}\times\prod_{j<k}\prod_{m=1}^{M_{j}}\prod_{n=1}^{M_{k}}\left|x_{n}^{k}-x_{m}^{j}\right|^{\beta
L_{j}L_{k}}.$

Later, it will be convenient to write $W_{\vec{M}}({\textbf{x}})$ in place of
the first of the three iterated products above. In the case when
$\sqrt{\beta}L_{j}\in{\mathbb{Z}}$ for all $j$, we will also write $\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|$ in place of the product of the
remaining two iterated products above. In subsection 4.4, we construct the
matrix $V^{\vec{L},\vec{M}}({\textbf{x}})$ of which this is the determinant.
Then the probability of finding the system in a state corresponding to a
location vector x is given by the joint probability density function

$\rho_{\vec{M}}({\textbf{x}})=\frac{\Omega_{\vec{M}}({\textbf{x}})}{Z_{\vec{M}}M_{1}!M_{2}!\cdots
M_{J}!}=\frac{W_{\vec{M}}({\textbf{x}})\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|}{Z_{\vec{M}}M_{1}!M_{2}!\cdots
M_{J}!},$

where the _partition function_ (of the Canonical Ensemble) $Z_{\vec{M}}$ is
the normalization constant given by

$\displaystyle Z_{\vec{M}}$ $\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\int_{{\mathbb{R}}^{M_{1}}}\cdots\int_{{\mathbb{R}}^{M_{J}}}\Omega_{\vec{M}}({\textbf{x}})\,d\nu^{M_{1}}(\textbf{x}^{1})\,d\nu^{M_{2}}(\textbf{x}^{2})\cdots
d\nu^{M_{J}}(\textbf{x}^{J})$ $\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\int_{{\mathbb{R}}^{M_{1}}}\cdots\int_{{\mathbb{R}}^{M_{J}}}\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|\,d\mu_{1}^{M_{1}}(\textbf{x}^{1})\,d\mu_{2}^{M_{2}}(\textbf{x}^{2})\cdots
d\mu_{J}^{M_{J}}(\textbf{x}^{J}),$

with Lebesgue measure $\nu^{M_{j}}$ on ${\mathbb{R}}^{M_{j}}$ and
$d\mu_{j}(x)=w_{j}(x)\,dx=\exp\left(-\beta L_{j}U(x)\right)dx$.

Note, the factorial denominators appear since particles of the same charge are
indistinguishable, giving many different representatives for each state. In
particular, the integrand is invariant under permutation of
$\\{x_{1}^{j},x_{2}^{j},\ldots,x_{M_{j}}^{j}\\}$ for any $j$ fixed. At this
point, it is necessary to assume the potential $U$ is one for which
$Z_{\vec{M}}$ is finite. Also, replacing $\beta$ with
$\beta^{\prime}=\beta/b^{2}$ and replacing each $L_{j}$ with
$L_{j}^{\prime}=bL_{j}$ leaves $\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|$ unchanged. Then replacing $U$ with
$U^{\prime}=bU$ leaves $W_{\vec{M}}({\textbf{x}})$ unchanged. Thus, for
computational purposes, we can always assume $\beta=1$.

Next, allowing the number of particles of each species to vary, let
$P(\vec{M})$ be the probability of finding the system with population vector
$\vec{M}$. Let $\vec{z}=(z_{1},\dots,z_{J})\in({\mathbb{R}}_{>0})^{J}$ be a
vector of positive real numbers called the _fugacity vector_. Classically, the
probability $P(\vec{M})$ is given by

$P(\vec{M})=z_{1}^{M_{1}}z_{2}^{M_{2}}\cdots
z_{J}^{M_{J}}\frac{Z_{\vec{M}}}{Z_{N}},$

where $Z_{N}$ is the partition function of the Isocharge Grand Canonical
Ensemble (corresponding to fixed total charge $N$) given by

$Z_{N}=\sum_{\vec{L}\cdot\vec{M}=N}z_{1}^{M_{1}}z_{2}^{M_{2}}\cdots
z_{J}^{M_{J}}Z_{\vec{M}}.$

In the above expression, the vector $\vec{L}$ of allowed charges is fixed, so
we are summing over allowed population vectors $\vec{M}$. A population vector
is valid only when the sum of the charges $\sum_{j=1}^{J}L_{j}M_{j}$ is equal
to the prescribed total charge $N$.

This $Z_{N}$ is the primary object of interest to us. In addition to the
obvious dependence on charge vector $\vec{L}$ and total charge $N$, this
expression also varies with potential $U$ and inverse temperature $\beta$. The
potential $U$ dictates the external forces experienced by each particle
individually, affecting the measures against which we are integrating. The
inverse temperature $\beta$ influences the strength of the interactions
between the particles, affecting the exponents on the interaction terms in the
Boltzmann factor. Taking the fugacity vector $\vec{z}$ to be a vector of
indeterminants, $Z_{N}$ is a polynomial in these indeterminants which
generates the partition functions of the Canonical Ensembles.

Recall, each $Z_{\vec{M}}$ is an iterated integral in many variables. Our goal
here is not to compute these integrals for any particular choice of several
parameters. Instead, we will show, in general, how to reduce the difficulty of
the computation to (sums of products of) only single or double integrals.
Sinclair (2012) showed for certain values of $\beta$ (for which
$\sqrt{\beta}L_{j}\in{\mathbb{Z}}$ for all $j$), $Z_{N}$ can be expressed as
the Berezin integral (with respect to the volume form on ${\mathbb{R}}^{N}$)
of the exponential of an explicit element (alternating tensor, also _form_) in
the exterior algebra $\bigwedge({\mathbb{R}}^{N})$ in the case when at most
one of the $L_{j}$ is odd. In the case when exactly one of the $L_{j}$ is odd,
say $L_{1}$, this required the additional restriction that $N$ be even, which
would force $M_{1}$ to be even. We will show his expression can be extended to
arbitrary $\vec{L}$ (for which any number of the $L_{j}$ may be odd), and in
the case when $N$ is odd, we give an analogous Berezin integral expression
with respect to the volume form on ${\mathbb{R}}^{N+1}$.

##  2 Preliminary definitions

In this section, we introduce a mix of conventions and definitions which
simplify the statement of our main results. First, for any positive integer
$N$, let $\underline{N}$ denote the set $\\{1,\ldots,N\\}$. Assuming positive
integers $K\leq N$, let $\mathfrak{t}:\underline{K}\nearrow\underline{N}$
denote a strictly increasing function from $\underline{K}$ to $\underline{N}$,
meaning

$1\leq\mathfrak{t}(1)<\mathfrak{t}(2)<\cdots<\mathfrak{t}(K)\leq N.$

It will be convenient to use these increasing functions to track indices used
in denoting minors of matrices and elements of exterior algebras, among other
things (often in place of, but sometimes in conjunction with, permutations).
For example, given an $N\times N$ matrix $V$, $V_{\mathfrak{t}}$ might denote
the $K\times K$ minor composed of the rows
$\mathfrak{t}(1),\ldots,\mathfrak{t}(K)$, taken from the first $K$ columns of
$V$. More conventions related to indexing and permutations (which are relevant
to the proofs) are introduced in section 4 but are not necessary for the
statement of our main results.

###  2.1 Wronskians

For any non-negative integer $l$, define the $l^{\rm{th}}$ modified
differential operator $D^{l}$ by

$D^{l}f(x)=\frac{1}{l!}\frac{d^{l}f}{dx^{l}},$

with $D^{0}f(x)=f(x)$. Define the modified Wronskian, Wr$(\vec{f},x)$, of a
family, $\vec{f}=\\{f_{n}\\}_{n=1}^{L}$, of $L$ many sufficiently
differentiable functions by

${\rm{Wr}}(\vec{f},x)=\det\left[D^{l-1}f_{n}(x)\right]_{n,l=1}^{L}.$

We call this the _modified_ Wronskian because it differs from the typical
Wronskian (used in the study of elementary differential equations to test for
linear dependence of solutions) by a combinatorial factor of
$\prod_{l=1}^{L}l!$.

A _complete_ $N$-family of monic polynomials is a collection
$\vec{p}=\\{p_{n}\\}_{n=1}^{N}$ such that each $p_{n}$ is monic of degree
$n-1$. Given $\mathfrak{t}:\underline{L}\nearrow\underline{N}$, denote
$\vec{p}_{\mathfrak{t}}=\\{p_{\mathfrak{t}(k)}\\}_{k=1}^{L}$. Then the
(modified) Wronskian of $\vec{p}_{\mathfrak{t}}$ is given by

${\rm{Wr}}(\vec{p}_{\mathfrak{t}},x)=\det\left[D^{l-1}p_{\mathfrak{t}(k)}(x)\right]_{k,l=1}^{L}.$

###  2.2 The Berezin integral

Let $\varepsilon_{1},\ldots,\varepsilon_{N}$ be a basis for
${\mathbb{R}}^{N}$. For any injection
$\mathfrak{t}:\underline{K}\to\underline{N}$, let
$\varepsilon_{\mathfrak{t}}\in\bigwedge^{K}({\mathbb{R}}^{N})$ denote

$\varepsilon_{\mathfrak{t}}=\varepsilon_{\mathfrak{t}(1)}\wedge\varepsilon_{\mathfrak{t}(2)}\wedge\cdots\wedge\varepsilon_{\mathfrak{t}(K)}.$

Then
$\\{\varepsilon_{\mathfrak{t}}\,|\,\mathfrak{t}:\underline{K}\nearrow\underline{N}\\}$
is a basis for $\bigwedge^{K}({\mathbb{R}}^{N})$. In particular,
$\bigwedge^{N}({\mathbb{R}}^{N})$ is a one-dimensional subspace we call the
_determinantal line_ , spanned by

$\varepsilon_{{\rm{vol}}}=\varepsilon_{\rm{id}}=\varepsilon_{1}\wedge\varepsilon_{2}\wedge\cdots\wedge\varepsilon_{N},$

which we call the _volume form_ (in ${\mathbb{R}}^{N}$). For each $0<n\leq N$,
define
$\frac{\partial}{\partial\varepsilon_{n}}:\bigwedge^{K}({\mathbb{R}}^{N})\to\bigwedge^{K-1}({\mathbb{R}}^{N})$
on basis elements by

$\frac{\partial}{\partial\varepsilon_{n}}\varepsilon_{\mathfrak{t}}=\begin{cases}(-1)^{k}\varepsilon_{\mathfrak{t}(1)}\wedge\cdots\wedge\varepsilon_{\mathfrak{t}(k-1)}\wedge\varepsilon_{\mathfrak{t}(k+1)}\wedge\cdots\wedge\varepsilon_{\mathfrak{t}(K)}&\text{
if }k=\mathfrak{t}^{-1}(n)\\\ 0&\text{ otherwise}\end{cases},$

and then extend linearly. If $n\in\mathfrak{t}(\underline{K})$, meaning
$\varepsilon_{n}$ appears as a factor in $\varepsilon_{\mathfrak{t}}$, then
$\frac{\partial\varepsilon_{\mathfrak{t}}}{\partial\varepsilon_{n}}$ is the
result of permuting $\varepsilon_{n}$ to the front and then removing it,
picking up a sign associated with changing the order in which the basis
elements occur. If $\varepsilon_{\mathfrak{t}}$ does not have
$\varepsilon_{n}$ as a factor, then
$\frac{\partial\varepsilon_{\mathfrak{t}}}{\partial\varepsilon_{n}}=0$. Given
an injection $\mathfrak{s}:\underline{L}\to\underline{N}$, we define the
Berezin integral [3] (with respect to $\varepsilon_{\mathfrak{s}}$) as a
linear operator $\bigwedge({\mathbb{R}}^{N})\to\bigwedge({\mathbb{R}}^{N})$
given by

$\int\varepsilon_{\mathfrak{t}}\,d\varepsilon_{\mathfrak{s}}=\int\varepsilon_{\mathfrak{t}}\,d\varepsilon_{\mathfrak{s}(1)}\,d\varepsilon_{\mathfrak{s}(2)}\cdots
d\varepsilon_{\mathfrak{s}(L)}=\frac{\partial}{\partial\varepsilon_{\mathfrak{s}(L)}}\cdots\frac{\partial}{\partial\varepsilon_{\mathfrak{s}(2)}}\frac{\partial}{\partial\varepsilon_{\mathfrak{s}(1)}}\varepsilon_{\mathfrak{t}}.$

Our main results are stated in terms of Berezin integrals with respect to the
volume form $\varepsilon_{{\rm{vol}}}\in\bigwedge^{N}({\mathbb{R}}^{N})$.
Note, if $\varepsilon_{\mathfrak{t}}\in\bigwedge^{K}({\mathbb{R}}^{N})$ for
any $K<N$, then

$\int\varepsilon_{\mathfrak{t}}\,d\varepsilon_{{\rm{vol}}}=0$

because $\varepsilon_{\mathfrak{t}}$ is missing some $\varepsilon_{k}$ as a
factor. Thus, the Berezin integral with respect to $\varepsilon_{{\rm{vol}}}$
is a projection operator
$\bigwedge({\mathbb{R}}^{N})\to\bigwedge^{N}({\mathbb{R}}^{N})\cong{\mathbb{R}}$.
In particular, if $\sigma\in S_{N}$, then

$\int\varepsilon_{\sigma}\,d\varepsilon_{{\rm{vol}}}={\mathrm{sgn}}(\sigma).$

###  2.3 Exponentials of forms

For $\omega\in\bigwedge({\mathbb{R}}^{N})$ and positive integer $m$, we write

$\omega^{\wedge m}=\omega\wedge\cdots\wedge\omega,$

with $\omega$ appearing as a factor $m$ times. By convention, $\omega^{\wedge
0}=1$. We then define the exponential

$\exp(\omega)=\sum_{m=0}^{\infty}\frac{\omega^{\wedge m}}{m!}.$

Moreover, suppose $\omega=\omega_{1}+\omega_{2}+\cdots+\omega_{J}$ where each
$\omega_{j}\in\bigwedge^{L_{j}}({\mathbb{R}}^{N})$ and each $L_{j}$ even, then
(we say each $\omega_{j}$ is a homogeneous even form of length $L_{j}$ and) it
is easily verified

$\exp({\omega})=\exp({\omega_{1}+\cdots+\omega_{J}})=\exp({\omega_{1}})\wedge\cdots\wedge\exp({\omega_{J}}).$

In our main results, the forms we are exponentiating are non-homogeneous.
However, we get a homogeneous form in the case when we only have one species
of particle. In that case, exactly one summand in the exponential will live at
the determinantal line. Assuming $\omega\in\bigwedge^{L}({\mathbb{R}}^{N})$
with $LM=N$, we get

$\int\exp(\omega)\,d\varepsilon_{{\rm{vol}}}=\int\sum_{m=0}^{\infty}\frac{\omega^{\wedge
m}}{m!}\,d\varepsilon_{{\rm{vol}}}=\int\frac{\omega^{\wedge
M}}{M!}\,d\varepsilon_{{\rm{vol}}}={\rm{PF}}(\omega),$

where PF$(\omega)$ is the _Hyperpfaffian_ of $\omega$, the real number
coefficient on $\varepsilon_{{\rm{vol}}}$ in $\frac{\omega^{\wedge M}}{M!}$.
Thus, this Berezin integral is the appropriate generalization of the
Hyperpfaffian. To avoid confusing this Berezin integral with other integrals
which appear in our computations, we’ll write

${\rm{BE}}_{{\rm{vol}}}(\omega)=\int\exp(\omega)\,d\varepsilon_{{\rm{vol}}},$

where the subscript on the left hand side indicates which form we are
integrating with respect to.

The partition function of an ensemble with a single species has been shown to
have a Hyperpfaffian expression (for certain $\beta$) [19]. As we generalize
to ensembles with multiple species (and therefore non-homogeneous forms), we
replace the Hyperpfaffian with the more general Berezin integral (of an
exponential).

##  3 Statement of results

Recall the setup from subsection 1.3. Let $\vec{p}$ be a complete $N$-family
of monic polynomials. Define

$\gamma_{j}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\int_{{\mathbb{R}}}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},x)\,d\mu_{j}(x)\,\,\varepsilon_{\mathfrak{t}},$

and define

$\eta_{j,k}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{L_{k}}\nearrow\underline{N}}\int\int_{x<y}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},x){\rm{Wr}}(\vec{p}_{\mathfrak{s}},y)\,d\mu_{j}(x)d\mu_{k}(y)\,\,\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}.$

######  Theorem 3.1.

If all $L_{j}$ are even, then

$Z_{N}={\rm{BE}}_{{\rm{vol}}}\left(\sum_{j=1}^{J}z_{j}\gamma_{j}\right).$

The above theorem is Sinclair’s (2012) for which we give a different proof and
then the following generalization:

######  Theorem 3.2.

If the first $r$ many $L_{j}$ are even and $N$ is even, then

$Z_{N}={\rm{BE}}_{{\rm{vol}}}\left(\sum_{j=1}^{r}z_{j}\gamma_{j}+\sum_{j=r+1}^{J}\sum_{k=r+1}^{J}z_{j}z_{k}\eta_{j,k}\right).$

######  Theorem 3.3.

If the first $r$ many $L_{j}$ are even and $N$ is odd, then

$Z_{N}={\rm{BE}}_{{\rm{vol}}_{1}}\left(\sum_{j=1}^{r}z_{j}\gamma_{j}+\sum_{j=r+1}^{J}\sum_{k=r+1}^{J}z_{j}z_{k}\eta_{j,k}+\sum_{j=r+1}^{J}z_{j}\gamma_{j}\wedge\varepsilon_{N+1}\right),$

where ${\rm{BE}}_{{\rm{vol}}_{1}}$ includes the Berezin integral with respect
to
$\varepsilon_{{\rm{vol}}_{1}}=\varepsilon_{{\rm{vol}}}\wedge\varepsilon_{N+1}\in\bigwedge^{N+1}({\mathbb{R}}^{N+1})$.

For our methods, it is necessary to extend the basis by $\varepsilon_{N+1}$ so
that the new volume form $\varepsilon_{{\rm{vol}}_{1}}$ has even length $N+1$.
More generally, we can write

$\varepsilon_{{\rm{vol}}_{k}}=\varepsilon_{{\rm{vol}}}\wedge\xi_{k}=\varepsilon_{{\rm{vol}}}\wedge\varepsilon_{N+1}\wedge\varepsilon_{N+2}\wedge\cdots\wedge\varepsilon_{N+k}.$

Then for any
$\omega\in\bigwedge({\mathbb{R}}^{N})\leq\bigwedge({\mathbb{R}}^{N+k})$, we
have

$\int\omega\,d\varepsilon_{{\rm{vol}}}=\int\omega\wedge\varepsilon_{N+1}\wedge\cdots\wedge\varepsilon_{N+k}\,\,d\varepsilon_{{\rm{vol}}}\,d\varepsilon_{N+1}\cdots
d\varepsilon_{N+k}=\int\omega\wedge\xi_{k}\,\,d\varepsilon_{{\rm{vol}}_{k}}.$

Thus, we can embed any Berezin integral computation in a higher dimension as
desired.

######  Corollary.

In the single species case ($N$ indistinguishable particles of charge $L$), we
get the known Hyperpfaffian expression (Sinclair 2011):

$Z_{N}={\rm{BE}}_{{\rm{vol}}_{k}}(\omega)={\rm{PF}}(\omega),$

where $\omega$ and $k$ depends on $N$ and $L$.

  1. 1.

If $L$ is even, then $\omega=\gamma_{1}$ and
${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}}$.

  2. 2.

If $L$ is odd and $N$ is even, then $\omega=\eta_{1,1}$ and
${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}}$.

  3. 3.

If $L$ is odd and $N$ is odd, then $\omega=\eta_{1,1}+\gamma_{1}\wedge\xi_{L}$
and ${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}_{L}}$.

Note, we extend by $\xi_{L}$ instead of just $\xi_{1}=\varepsilon_{N+1}$ in
case 3 only so that $\gamma_{1}\wedge\xi_{L}$ is a $2L$-form and therefore
$\omega$ is homogeneous. Every choice of $k$ produces a different but equally
valid Berezin integral expression. We obtain the (Pfaffian) de Bruijn integral
identities for classical $\beta=1$ and $\beta=4$ when $L=1$ and $L=2$,
respectively.

To prove these theorems, we start by giving general methods (in the shuffle
algebra) for manipulating iterated integrals of determinantal integrands in
section 4. The conclusion of these methods is a generalization of the de
Bruijn integral identities, given in subsection 3.1. In section 5, we apply
these identities first to $Z_{\vec{M}}$, the partition function of the
Canonical Ensemble with arbitrary but fixed population $\vec{M}$. In section
6, we sum over all possible population vectors $\vec{M}$ to obtain $Z_{N}$,
the partition function of the Isocharge Grand Canonical Ensemble.

###  3.1 Generalized de Bruijn identities

Let $N=L_{1}+\cdots+L_{J}$. Define $K_{j}=\sum_{k=1}^{j}L_{k}$. Let
$B(\vec{x})$ be an $N\times N$ matrix whose entries are single variable
integrable functions of variables $\vec{x}=(x_{1},\ldots,x_{J})$. Explicitly,
the first $L_{1}$ many columns are functions of $x_{1}$, the second $L_{2}$
many columns are functions of $x_{2}$, and so on up through $x_{J}$. For
$\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}$, let
$B_{\mathfrak{t}}(x_{j})$ denote the $L_{j}\times L_{j}$ minor of $B(\vec{x})$
given by

$B_{\mathfrak{t}}(x_{j})=\left[B(\vec{x})_{\mathfrak{t}(l),n+K_{j-1}}\right]_{l,n=1}^{L_{j}},$

equivalently obtained from $B(\vec{x})$ by taking the rows
$\mathfrak{t}(1),\ldots,\mathfrak{t}(L_{j})$ from the $L_{j}$ many columns in
the same variable $x_{j}$. Define

$\gamma_{j}^{B}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\int_{{\mathbb{R}}}\det
B_{\mathfrak{t}}(x_{j})\,dx_{j}\,\varepsilon_{\mathfrak{t}},$

and define

$\eta_{j,k}^{B}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{L_{k}}\nearrow\underline{N}}\int\int_{x_{j}<x_{k}}\det
B_{\mathfrak{t}}(x_{j})\cdot\det
B_{\mathfrak{s}}(x_{k})\,dx_{j}\,dx_{k}\,\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}.$

######  Theorem 3.4.

Suppose the first $r$ many $L_{j}$ are even, then

$\int_{-\infty<x_{1}<\ldots<x_{J}<\infty}\det B(\vec{x})\,dx_{1}\cdots
dx_{J}=\int\omega\,d\varepsilon_{{\rm{vol}}},$

where $\omega$ is defined as follows:

  1. 1.

If $N$ is even, then

$\omega=\frac{1}{\left(r+\frac{J-r}{2}\right)!}\bigwedge_{j=1}^{r}\gamma_{j}^{B}\wedge\bigwedge_{m=1}^{(J-r)/2}\eta_{r+2m-1,r+2m}^{B}.$

  2. 2.

If $N$ is odd, then

$\omega=\frac{1}{\left(r+1+\frac{J-r-1}{2}\right)!}\bigwedge_{j=1}^{r}\gamma_{j}^{B}\wedge\bigwedge_{m=1}^{(J-r-1)/2}\eta_{r+2m-1,r+2m}^{B}\wedge\gamma_{J}^{B}.$

For $1\leq j\leq r$, $L_{j}$ is even, and $\gamma_{j}^{B}$ is an even
$L_{j}$-form. For the $L_{j}$ which are odd, $\eta_{j,k}^{B}$ combines minors
of odd $L_{j}\times L_{j}$ dimensions with minors of odd $L_{k}\times L_{k}$
dimensions to produce an even $(L_{j}+L_{k})$-form. In case 1, the requirement
that $N$ be even means there are an even number of odd $L_{j}$ to be paired
down into $(J-r)/2$ pairs. In case 2, there are an odd number of odd $L_{j}$,
so $\gamma_{J}^{B}$ remains as an odd $L_{J}$-form.

Compared to Theorems 3.2 and 3.3, $\det B(\vec{x})$ takes the place of the
Boltzmann factor $\Omega_{\vec{M}}(\textbf{x})$, so that $\det
B_{\mathfrak{t}}(x_{j})$ and $\det B_{\mathfrak{s}}(x_{k})$ take the places of
${\rm{Wr}}(\vec{p}_{\mathfrak{t}},x)$ and
${\rm{Wr}}(\vec{p}_{\mathfrak{s}},y)$, respectively. Recall (from subsection
1.3), the partition function $Z_{N}$ of the Isocharge Grand Canonical Ensemble
is a linear combination of the partition functions $Z_{\vec{M}}$ of the
Canonical Ensembles, with the sum taken over all possible populations
$\vec{M}$. For each $\vec{M}$, we get a different Boltzmann factor integrand
$\Omega_{\vec{M}}(\textbf{x})$. Moreover, the defining integral is taken over
all of ${\mathbb{R}}^{N}$. The wedge products which appear in Theorem 3.4
should be thought of as just one of the summands which appears in the
expansion of the exponentials which appear in Theorems 3.2 and 3.3.

Note, we assume the functions which make up $B(\vec{x})$ are suitably
integrable so that all integrals which appear in $\gamma_{j}^{B}$ and
$\eta_{j,k}^{B}$ are finite. However, we do not assume any resemblance between
the $L_{j}$ many columns in $x_{j}$ and the $L_{k}$ many columns in $x_{k}$.
Assuming some additional consistency, we obtain a Hyperpfaffian analogue of
the de Bruijn integral identities.

######  Corollary.

Suppose $L_{1}=\cdots=L_{J}=L$. Under the additional assumption that
$\gamma_{j}^{B}=\gamma$ for all $j$, and $\eta_{j,k}^{B}=\eta$ for all $j,k$
(typically because the entries of $B(\vec{x})$ in one variable $x_{j}$ are the
same as the entries in any other variable $x_{k}$),

$\int_{-\infty<x_{1}<\ldots<x_{J}<\infty}\det B(\vec{x})\,dx_{1}\cdots
dx_{J}={\rm{BE}}_{{\rm{vol}}_{k}}(\omega)={\rm{PF}}(\omega),$

where $\omega$ and $k$ depend on $M$ and $L$.

  1. 1.

If $L$ is even, then $\omega=\gamma$ and
${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}}$.

  2. 2.

If $L$ is odd and $M$ is even, then $\omega=\eta$ and
${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}}$.

  3. 3.

If $L$ is odd and $M$ is odd, then $\omega=\eta+\gamma\wedge\xi_{L}$ and
${\rm{BE}}_{{\rm{vol}}_{k}}={\rm{BE}}_{{\rm{vol}}_{L}}$.

Further generalization in Theorem 3.4 is still possible if desired. Suppose
instead the first $L_{1}$ columns of $B(\vec{x})$ are made up of functions,
not necessarily single variable, of variables $x_{1},\ldots,x_{a}$, and the
next $L_{2}$ columns of $B(\vec{x})$ are made up of functions of variables
$x_{a+1},\ldots,x_{b}$. If $L_{1}$ is even, then we replace $\gamma_{1}^{B}$
with

$\gamma_{1}^{B}=\sum_{\mathfrak{t}:\underline{L_{1}}\nearrow\underline{N}}\int_{-\infty<x_{1}<\cdots<x_{a}<\infty}\det
B_{\mathfrak{t}}(x_{1},\ldots,x_{a})\,dx_{1}\cdots
dx_{a}\,\varepsilon_{\mathfrak{t}},$

which now features iterated integrals of the multivariate minors. If $L_{1}$
and $L_{2}$ are odd, then we replace $\eta_{1,2}^{B}$ with

$\eta_{1,2}^{B}=\sum_{\mathfrak{t}:\underline{L_{1}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{L_{2}}\nearrow\underline{N}}\int_{-\infty<x_{1}<\cdots<x_{b}<\infty}\det
B_{\mathfrak{t}}(x_{1},\ldots,x_{a})\cdot\det
B_{\mathfrak{s}}(x_{a+1},\ldots,x_{b})\,dx_{1}\cdots
dx_{b}\,\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}.$

In general, we integrate the minors with respect to whichever variables appear
with respect to the same total order on the domain induced by the original
integral of $\det B(\vec{x})$.

##  4 Algebraic lemmas

For any injection $\mathfrak{t}:\underline{K}\to\underline{N}$, let
$Q_{\mathfrak{t}}$ denote $Q_{\mathfrak{t}(1),\ldots,\mathfrak{t}(K)}$
whenever it is clear from context $Q$ admits $K$ many indices, and let
$Q_{\mathfrak{t}}$ denote $\\{Q_{\mathfrak{t}(k)}\\}_{k=1}^{K}$ whenever it is
clear from context $Q$ admits only one index. For any permutation $\sigma\in
S_{K}$, we can view $\sigma:\underline{K}\to\underline{K}$ as a bijection and
then write $Q_{\mathfrak{t}\circ\sigma}$ to denote
$Q_{\mathfrak{t}\circ\sigma(1),\ldots,\mathfrak{t}\circ\sigma(K)}$ or
$\\{Q_{\mathfrak{t}\circ\sigma(k)}\\}_{k=1}^{K}$ as appropriate in context.

For example, when $V$ is a matrix, $V_{\mathfrak{t}}$ is a minor. We should
think of $V_{\mathfrak{t}}$ as a single object with $K$ many indices (which
indicate a choice of $K$ many rows $\mathfrak{t}(1),\ldots,\mathfrak{t}(K)$
from which our minor is constructed). Similarly, if
$\omega\in\bigwedge^{K}({\mathbb{R}}^{N})$, then $A_{\mathfrak{t}}$ might
denote the coefficient of $\varepsilon_{\mathfrak{t}}$ (equivalently, an entry
in a $K$-dimensional hyper array). In contrast, if
$\vec{f}=\\{f_{k}\\}_{k=1}^{N}$ is a family of functions, then
$\vec{f}_{\mathfrak{t}}=\\{f_{\mathfrak{t}(k)}\\}_{k=1}^{K}$ is a subfamily of
$K$ functions, each indexed by a single integer. In our statement of our main
results, we use these increasing function subscripts in both ways, but it is
clear from context how these subscripts should be applied differently to
different objects.

Let $\mathfrak{t}:\underline{K}\nearrow\underline{N}$ denote a strictly
increasing function from $\underline{K}$ to $\underline{N}$. Note, every
injection $\mathfrak{s}:\underline{K}\to\underline{N}$ can be written uniquely
as $\mathfrak{s}=\mathfrak{t}\circ\sigma$ for some
$\mathfrak{t}:\underline{K}\nearrow\underline{N}$ and $\sigma\in S_{K}$. For
any $\mathfrak{t}:\underline{K}\nearrow\underline{N}$, there exists a unique
complementary $\mathfrak{t}^{\prime}:\underline{N-K}\nearrow\underline{N}$
with
$\mathfrak{t}(\underline{K})\cup\mathfrak{t}^{\prime}(\underline{N-K})=\underline{N}$.
Define ${\mathrm{sgn}}(\mathfrak{t})$ to be the signature of the permutation
$\sigma\in S_{N}$ given by

$\sigma(k)=\begin{cases}\mathfrak{t}(k)&\text{ if }k\in\underline{K}\\\
\mathfrak{t}^{\prime}(k-K)&\text{ if
}k\in\underline{N}\setminus\underline{K}\end{cases}.$

Equivalently,

${\mathrm{sgn}}(\mathfrak{t})=\int\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{t}^{\prime}}\,d\varepsilon_{{\rm{vol}}}.$

For any $\Lambda=(\lambda_{1},\ldots,\lambda_{K})$ which partitions $N$ and
$\mathfrak{t}:\underline{N}\to\underline{M}$, write
$\mathfrak{t}=(\mathfrak{t}_{1}|\ldots|\mathfrak{t}_{K})$ to indicate a
decomposition of $\mathfrak{t}$ in which $\mathfrak{t}_{1}$ is the restriction
of $\mathfrak{t}$ to the first $\lambda_{1}$ positive integers, and each
$\mathfrak{t}_{k}$ is the restriction of $\mathfrak{t}$ to the next
$\lambda_{k}$ positive integers. For convenience, we will treat each
$\mathfrak{t}_{k}$ as having domain $\underline{\lambda_{k}}$ instead of the
appropriate subset of $\underline{N}$ of size $\lambda_{k}$.

Conversely, for any
$\mathfrak{t}_{1}:\underline{\lambda_{1}}\to\underline{M},\ldots,\mathfrak{t}_{K}:\underline{\lambda_{K}}\to\underline{M}$,
we can construct
$(\mathfrak{t}_{1}|\ldots|\mathfrak{t}_{K})=\mathfrak{t}:\underline{N}\to\underline{M}$
by defining for each $n\in\underline{N}$,
$\mathfrak{t}(n)=\mathfrak{t}_{k}\left(n-\sum_{j=1}^{k-1}\lambda_{j}\right)$
where this $k$, which depends on $n$, is the largest $k$ for which the
difference inside the parentheses is positive. As before, it will be
convenient to identify the restrictions of this new $\mathfrak{t}$ with the
original $\mathfrak{t}_{1},\ldots,\mathfrak{t}_{K}$ even though the domains
aren’t exactly the same.

In general, this is just a bookkeeping device which gives us a convenient
notation for a choice of indices from the codomain $\underline{M}$. For
example, when $\sigma:\underline{N}\to\underline{N}$ is a permutation, we can
think of $\sigma$ as sorting $N$ many possible indices into $K$ blocks of
different sizes $\lambda_{1},\ldots,\lambda_{K}$ as specified by the images of
the $\sigma_{k}$.

###  4.1 Decomposition of the symmetric group

For any $\Lambda=(\lambda_{1},\ldots,\lambda_{K})$ which partitions $N$, let
${\rm{H}}(\Lambda)\subseteq S_{N}$ denote the _Young subgroup_ , meaning
${\rm{H}}(\Lambda)\cong S_{\lambda_{1}}\times\cdots\times S_{\lambda_{K}}$.
Viewing $\sigma$ as a function from $\underline{N}$ to $\underline{N}$, we can
write the decomposition (with respect to $\Lambda$) as
$\sigma=(\sigma_{1}|\ldots|\sigma_{K})$. Then we should think of these
$\sigma_{k}$ belonging to the appropriate $S_{\lambda_{k}}$.

Let Sh$(\Lambda)\subseteq S_{N}$ denote the subset of _shuffle permutations_.
These are permutations which satisfy $\sigma(i)<\sigma(j)$ whenever

$\lambda_{1}+\cdots+\lambda_{k}<i<j\leq\lambda_{1}+\cdots+\lambda_{k+1}.$

Each shuffle permutation represents a way to iteratively riffle shuffle $K$
stacks of $\lambda_{1},\ldots,\lambda_{K}$ many cards into a single pile of
$N$ cards while preserving the original ordering within each of the $K$
stacks. Equivalently, the shuffle permutations are the $\sigma\in S_{N}$ for
which each $\sigma_{k}$ is a strictly increasing function from
$\underline{\lambda_{k}}$ to $\underline{N}$.

Let ${\rm{Sh}}^{\circ}(\Lambda)\subseteq{\rm{Sh}}(\Lambda)$ denote the subset
of _ordered_ shuffle permutations. These are shuffle permutations which also
satisfy

$\sigma(1)<\sigma(\lambda_{1}+1)<\sigma(\lambda_{1}+\lambda_{2}+1)<\cdots<\sigma(\lambda_{1}+\cdots+\lambda_{K-1}+1).$

Using the decomposition $\sigma=(\sigma_{1}|\ldots|\sigma_{K})$, we can
conveniently rewrite the above condition as

$\sigma_{1}(1)<\sigma_{2}(1)<\cdots<\sigma_{K}(1).$

Let Bl$(\Lambda)\subseteq S_{N}$ denote the subset of _block permutations_. A
block permutation represents shuffling a deck of cards by first separating the
deck into a pile of the first $\lambda_{1}$ cards, a pile of the second
$\lambda_{2}$ cards, and so on, then reassembling the deck without interlacing
the piles or shuffling within any of the piles. Using the decomposition
$\sigma=(\sigma_{1}|\ldots|\sigma_{K})$, block permutations are permutations
for which each $\sigma_{k}:\underline{\lambda_{k}}\to\underline{N}$ acts by
$\sigma_{k}(j)=(j-1)+\sigma_{k}(1)$.

Clearly, any block permutation is determined entirely by the action on the
first element of each block (of $\lambda_{k}$ elements), of which there are
$K$ many. Given a block permutation $\sigma$, define $\theta_{\sigma}\in
S_{K}$ to be the unique permutation for which
$\theta_{\sigma}(i)<\theta_{\sigma}(j)$ if and only if
$\sigma_{i}(1)<\sigma_{j}(1)$. Heuristically, $\sigma$ moves blocks of
$\lambda_{k}$ consecutive elements together. $\theta_{\sigma}$ is the unique
action on the $K$ many blocks determined by $\sigma$. The map
$\sigma\mapsto\theta_{\sigma}$ is bijective, so ${\rm{Bl}}(\Lambda)\cong
S_{K}$.

Note, because the blocks have (possibly) different sizes $\lambda_{j}$, block
permutations do not (in general) preserve partitions. Define

$\Lambda^{\sigma}=(\lambda_{\theta_{\sigma}^{-1}(1)},\ldots,\lambda_{\theta_{\sigma}^{-1}(K)}),$

obtained from $\Lambda$ by reordering its entries according to
$\theta_{\sigma}$.

In the sequel, we make use of the following Lemma which decomposes
permutations in the symmetric group as products of ordered shuffles, block
permutations, and young permutations (composed in the opposite order).

######  Lemma 4.1.

Let $\Lambda=(\lambda_{1},\ldots,\lambda_{K})$ be a partition of $N$. Given
any $\varphi\in S_{N}$, there exists unique permutations
$\tau\in{\rm{H}}(\Lambda)$, $\pi\in{\rm{Bl}}(\Lambda)$, and
$\sigma\in{\rm{Sh}}^{\circ}(\Lambda^{\pi})$ so that
$\varphi=\sigma\circ\pi\circ\tau$.

For completeness, we include a proof of this lemma in Appendix A.

###  4.2 Chen’s lemma

For a set $X$ and a ring $R$, let $R\langle X\rangle$ denote the free unital
algebra on $X$ over $R$. Given $u=u_{1}\cdots u_{k}$ and
$u^{\prime}=u_{k+1}\cdots u_{n}$, define an operation $\shuffle$ on $R\langle
X\rangle$ as follows:

$u\shuffle u^{\prime}=\sum_{\sigma\in{\rm{Sh}}(k,n-k)}u_{\sigma^{-1}(1)}\cdots
u_{\sigma^{-1}(n)}$

and by $e\shuffle e=e$ for the empty word $e\in R\langle X\rangle$. Denote by
$R\langle X\rangle_{\shuffle}$ the algebra $R\langle X\rangle$ with (shuffle)
multiplication $\shuffle$, which is called the Shuffle Algebra on $X$. The
shuffle product was first introduced by Eilenberg and Mac Lane in [10].

Let $\mathcal{H}$ be the Hilbert space $L^{2}({\mathbb{R}})$ of square
integrable functions with respect to Lebesgue measure on ${\mathbb{R}}$, and
suppose $H$ is a finite-dimensional subspace of $\mathcal{H}$ with basis $X$.
We assume $H$ is large enough to contain all functions of interest to us.

For any $\sigma\in S_{k}$, let $\Delta_{k}(\sigma)\subset(a,b)^{k}$ denote the
region where $a<x_{\sigma^{-1}(1)}<\cdots<x_{\sigma^{-1}(k)}<b$. For each non-
negative integer $k$, define a linear functional $\langle\cdot\rangle_{k}$ on
the $k^{\rm{th}}$ graded component of ${\mathbb{R}}\langle X\rangle\cong T(V)$
by

$\langle f_{1}\otimes\cdots\otimes
f_{k}\rangle_{k}=\int_{\Delta_{k}({\mathrm{id}})}f_{1}(x_{1})\cdots
f_{K}(x_{k})\,dx_{1}\cdots dx_{k}.$

Note, $T(V)$ denotes the tensor algebra to which ${\mathbb{R}}\langle
X\rangle$ is isomorphic. Though not strictly necessary, we write
$f_{1}\otimes\cdots\otimes f_{k}\in T(V)$ instead of $f_{1}\cdots
f_{k}\in{\mathbb{R}}\langle X\rangle$ to avoid confusing concatenation with
function multiplication.

This collection $\\{\langle\cdot\rangle_{k}\\}_{k=0}^{\infty}$ defines a
functional $\langle\cdot\rangle$ on ${\mathbb{R}}\langle X\rangle$ whereby
$\langle\cdot\rangle$ acts as $\langle\cdot\rangle_{k}$ on the $k^{\rm{th}}$
graded component of a non-homogeneous tensor. The following lemma, due to Chen
[5], asserts that this operator $\langle\cdot\rangle$ is actually an algebra
homomorphism from $T(V)_{\shuffle}$ to ${\mathbb{R}}$:

######  Lemma 4.2 (Chen).

If $f,g\in{\mathbb{R}}\langle X\rangle$, then $\langle f\shuffle
g\rangle=\langle f\rangle\langle g\rangle$.

A major hurdle in computing the partition function $Z_{\vec{M}}$ is the
absolute value inside the integral. We will remove the absolute value by
decomposing the domain of integration into these totally ordered subsets.
However, when we change the domain of integration, we lose the ability to use
Fubini’s Theorem. Chen’s Lemma serves the role of Fubini’s Theorem, provided
we can demonstrate the integrand to have the appropriate form.

###### Proof.

We can assume $f=f_{1}\otimes\cdots\otimes f_{k}$ is a pure tensor of length
$k$ and $g=f_{k+1}\otimes\cdots\otimes f_{n}$ is a pure tensor of length $n-k$
(because $\langle\cdot\rangle$ is linear and $\shuffle$ distributes over
addition). Then $\langle f\shuffle g\rangle=\langle f\shuffle g\rangle_{n}$ is
an $n$-fold iterated integral over $\Delta_{n}({\mathrm{id}})$. Likewise,
$\langle f\rangle\langle g\rangle=\langle f\rangle_{k}\langle g\rangle_{n-k}$
is the product of a $k$-fold iterated integral over
$\Delta_{k}({\mathrm{id}})$ and an $(n-k)$-fold iterated integral over
$\Delta_{n-k}({\mathrm{id}})$. Under the integrability criteria on $X$,
Fubini’s Theorem tells us this is the same as an $n$-fold iterated integral
over the product space
$\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}})$.

Key to the proof of Chen’s Lemma is the observation that

$\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}})=Z\cup\bigcup_{\sigma\in{\rm{Sh}}(k,n-k)}\Delta_{n}(\sigma),$

where $Z\subset(a,b)^{n}$ has measure $0$. To this end, let $Y$ be the set of
points in $(a,b)^{n}$ whose coordinates are all distinct, and then let
$Z=((a,b)^{n}\setminus
Y)\cap(\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}}))$. If
$\vec{x}\in\left(\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}})\right)\setminus
Z\subset Y$, then its coordinates are all distinct and can be ordered
according to some permutation, meaning $\vec{x}\in\Delta_{n}(\sigma)$ for a
unique $\sigma\in S_{n}$. Since
$\vec{x}\in\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}})$, this
permutation is one which separately preserves the relative order of the first
$k$ coordinates and the relative order of the remaining $n-k$ coordinates,
meaning $\sigma\in{\rm{Sh}}(k,n-k)$. Thus,

$\displaystyle\langle f\rangle\langle g\rangle$
$\displaystyle=\int_{\Delta_{k}({\mathrm{id}})\times\Delta_{n-k}({\mathrm{id}})}f_{1}(x_{1})\cdots
f_{n}(x_{n})\,dx_{1}\cdots dx_{n}$
$\displaystyle=\sum_{\sigma\in{\rm{Sh}}(k,n-k)}\int_{\Delta_{n}(\sigma)}f_{1}(x_{1})\cdots
f_{n}(x_{n})\,dx_{1}\cdots dx_{n}.$

By relabeling the variables as $x_{j}=y_{\sigma(j)}$, we can rewrite this sum
as

$\displaystyle=\sum_{\sigma\in{\rm{Sh}}(k,n-k)}\int_{\Delta_{n}({\mathrm{id}})}f_{1}(y_{\sigma(1)})\cdots
f_{n}(y_{\sigma(n)})\,dy_{1}\cdots dy_{n}$
$\displaystyle=\sum_{\sigma\in{\rm{Sh}}(k,n-k)}\int_{\Delta_{n}({\mathrm{id}})}f_{\sigma^{-1}(1)}(y_{1})\cdots
f_{\sigma^{-1}(n)}(y_{n})\,dy_{1}\cdots dy_{n}$ $\displaystyle=\langle
f\shuffle g\rangle.$

∎

Until section 7, we will only need $(a,b)=(-\infty,\infty)={\mathbb{R}}$, but
we have shown Chen’s Lemma to hold for more general intervals. Also, we do not
need our functions to be real-valued. We just need the codomain to be
associative and commutative, provided our functions are appropriately
integrable. In particular, we will use complex-valued functions on real domain
$[0,2\pi)$ for the circular ensembles in section 7.

###  4.3 Exterior shuffle algebra

Let
$\\{Q_{\mathfrak{t}}^{j}\,|\,j\in\underline{J},\mathfrak{t}:\underline{L_{j}}\to\underline{N}\\}$
be a subset of an alphabet $I$, and let $F$ be a field of characteristic 0. We
use single integer superscripts to emphasize there are different $Q$’s which
admit different numbers of integer subscripts, chosen by $\mathfrak{t}$’s of
different sizes. For each $\mathfrak{t}:\underline{L_{j}}\to\underline{N}$,
define $A_{\mathfrak{t}}^{j}\in F\langle I\rangle$ by

$A_{\mathfrak{t}}^{j}=\sum_{\tau\in
S_{L_{j}}}{\mathrm{sgn}}(\tau)Q_{\mathfrak{t}\circ\tau}^{j}.$

We call this the _antisymmetrization_ of $Q_{\mathfrak{t}}^{j}$. We should
think of this as analogous to taking a pure tensor in the tensor algebra and
then adding to it all possible orderings of the basis vectors with signs. Let
$V$ be a rank $N$ free module over $R=F\langle I\rangle_{\shuffle}$. Define
$\alpha_{j}\in\bigwedge_{R}V$ by

$\alpha_{j}=\sum_{\mathfrak{t}:\underline{L_{j}}\to\underline{N}}Q_{\mathfrak{t}}^{j}\,\varepsilon_{\mathfrak{t}}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}A_{\mathfrak{t}}^{j}\,\varepsilon_{\mathfrak{t}}.$

We call this an _antisymmetrized_ $L_{j}$-form.

Let $\vec{M}=(M_{1},\ldots,M_{J})$, and let $\vec{L}=(L_{1},\ldots,L_{J})$
such that each $L_{j}$ is even. Let $N=\vec{L}\cdot\vec{M}$, and let
$\Lambda=(L_{1},\ldots,L_{1},L_{2},\ldots,L_{2},\ldots,L_{J},\ldots,L_{J})=(\lambda_{1},\lambda_{2},\ldots,\lambda_{K})$
with each $L_{j}$ appearing $M_{j}$ times. Let $K=\sum_{j=1}^{J}M_{j}$. For
$\sigma\in S_{N}$, write $\sigma=(\sigma_{1}|\cdots|\sigma_{K})$ so that each
$\sigma_{k}$ is the restriction of $\sigma:\underline{N}\to\underline{N}$ to a
subset of size $\lambda_{k}=L_{j}$ of which there are $M_{j}$ many. In
particular, when $\sigma\in{\rm{Sh}}(\Lambda)$, we have
$\sigma_{k}:\underline{L_{j}}\nearrow\underline{N}$.

######  Lemma 4.3.

Under the above assumptions (particularly requiring each $L_{j}$ to be even)
and definitions (such as how to obtain $\alpha_{j}$ from the
$Q^{j}_{\mathfrak{t}}$), we have

$\sum_{\sigma\in S_{N}}{\mathrm{sgn}}(\sigma)Q_{\sigma_{1}}^{1}\cdots
Q_{\sigma_{K}}^{J}=\int\frac{\alpha_{1}^{\wedge_{\shuffle}M_{1}}\wedge_{\shuffle}\cdots\wedge_{\shuffle}\alpha_{J}^{\wedge_{\shuffle}M_{J}}}{K!}\,d\varepsilon_{{\rm{vol}}}.$

In particular, when $J=1$, the right hand side is
${\rm{PF}}_{\shuffle}(\alpha_{1})$, where the subscript $\shuffle$ is added to
emphasize the coefficients $A_{\mathfrak{t}}^{j}$ are in $F\langle
I\rangle_{\shuffle}$ in which multiplication of coefficients is done by
$\shuffle$.

Though this lemma holds for a more general collection of
$Q_{\mathfrak{t}}^{j}$, we should think of the left hand side as being an
$N\times N$ determinant. A single $Q_{\sigma_{k}}^{j}$ is a product of $L_{j}$
many entries from the matrix. $A_{\sigma_{k}}^{j}$, the antisymmetrization of
$Q_{\sigma_{k}}^{j}$, is the determinant of an $L_{j}\times L_{j}$ minor
selected by $\sigma_{k}$. In summary, this Lemma 4.3 transforms any
determinantal integrand into one for which Chen’s Lemma will apply. This is
functionally similar to the Laplace expansion of the determinant (over
complimentary $L_{j}\times L_{j}$ minors) which Sinclair uses in his proofs.

###### Proof.

Starting with the right hand side, note each factor $\alpha_{j}$ is a sum of
$A_{\mathfrak{t}}^{j}\varepsilon_{\mathfrak{t}}$’s. If we expand the product
of the sums, each summand will be a product of some
$A_{\mathfrak{t}}^{j}\varepsilon_{\mathfrak{t}}$’s. Any time
$\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}$ and
$\mathfrak{s}:\underline{L_{k}}\nearrow\underline{N}$ have overlapping ranges,
$\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}=0$. Thus, each
nonzero summand corresponds to a permutation
$(\mathfrak{t}_{1}|\ldots|\mathfrak{t}_{K})=\sigma\in S_{N}$. Since each
$\mathfrak{t}_{k}$ is an increasing function, we have
$\sigma\in{\rm{Sh}}(\Lambda)$. We use the Berezin integral $\int
d\varepsilon_{{\rm{vol}}}$ because it sends
$\varepsilon_{\sigma}=\varepsilon_{\mathfrak{t}_{1}}\wedge\cdots\wedge\varepsilon_{\mathfrak{t}_{K}}$
to ${\mathrm{sgn}}(\sigma)$ and picks up the coefficients
$A_{\sigma_{1}}^{1}\shuffle\cdots\shuffle A_{\sigma_{K}}^{J}$. We can rewrite
the right hand side as

${\rm{RHS}}=\frac{1}{K!}\sum_{\sigma\in{\rm{Sh}}(\Lambda)}{\mathrm{sgn}}(\sigma)A_{\sigma_{1}}^{1}\shuffle\cdots\shuffle
A_{\sigma_{K}}^{J}.$

We eliminate the factorial denominator by requiring
$\sigma_{1}(1)<\sigma_{2}(1)<\ldots<\sigma_{K}(1)$.

${\rm{RHS}}=\sum_{\sigma\in{\rm{Sh}}^{\circ}(\Lambda)}{\mathrm{sgn}}(\sigma)A_{\sigma_{1}}^{1}\shuffle\cdots\shuffle
A_{\sigma_{K}}^{J}.$

Expanding each $A_{\sigma_{k}}^{j}$ according to the definition, we get

${\rm{RHS}}=\sum_{\sigma\in{\rm{Sh}}^{\circ}(\Lambda)}{\mathrm{sgn}}(\sigma)\left(\sum_{\tau_{1}\in
S_{L_{1}}}{\mathrm{sgn}}(\tau_{1})Q_{\sigma_{1}\circ\tau_{1}}^{1}\right)\shuffle\cdots\shuffle\left(\sum_{\tau_{K}\in
S_{L_{J}}}{\mathrm{sgn}}(\tau_{K})Q_{\sigma_{K}\circ\tau_{K}}^{J}\right).$

Collect the $\tau_{k}$’s as a single element of
${\rm{H}}(\Lambda)\cong(S_{L_{1}})^{M_{1}}\times\cdots\times(S_{L_{J}})^{M_{J}}$
with
${\mathrm{sgn}}(\tau)={\mathrm{sgn}}(\tau_{1})\cdots{\mathrm{sgn}}(\tau_{K})$,
so that

${\rm{RHS}}=\sum_{\sigma\in{\rm{Sh}}^{\circ}(\Lambda)}{\mathrm{sgn}}(\sigma)\sum_{\tau\in{\rm{H}}(\Lambda)}{\mathrm{sgn}}(\tau)Q_{\sigma_{1}\circ\tau}^{1}\shuffle\cdots\shuffle
Q_{\sigma_{K}\circ\tau}^{J}.$

Next, we apply an identity of the $\shuffle$ operation. Note, we are shuffling
individual letters together, not strings of letters, so the sum is over
$\pi\in S_{K}={\rm{Sh}}(1,\ldots,1)$. The action on the subscripts can also be
viewed as a permutation of the $K$ many “blocks” of $N$ as prescribed by the
partition $\Lambda$. Thus,

$\displaystyle{\rm{RHS}}$
$\displaystyle=\sum_{\sigma\in{\rm{Sh}}^{\circ}(\Lambda)}{\mathrm{sgn}}(\sigma)\sum_{\tau\in{\rm{H}}(\Lambda)}{\mathrm{sgn}}(\tau)\sum_{\pi\in
S_{K}}Q_{\sigma_{\pi(1)}\circ\tau}^{1}\cdots Q_{\sigma_{\pi(K)}\circ\tau}^{J}$
$\displaystyle=\sum_{\varphi\in
S_{N}}{\mathrm{sgn}}(\varphi)Q_{\varphi_{1}}^{1}\cdots Q_{\varphi_{K}}^{J}.$

Note, the equality in the last line is not an obvious one but follows from
Lemma 4.1. In the context of Lemma 4.3, all of the block sizes $L_{j}$ are
even, so ${\mathrm{sgn}}(\pi)=1$ for all $\pi\in{\rm{Bl}}(\Lambda)$. Thus,
${\mathrm{sgn}}(\varphi)={\mathrm{sgn}}(\sigma\circ\pi\circ\tau)={\mathrm{sgn}}(\sigma){\mathrm{sgn}}(\tau)$.

∎

###  4.4 Confluent Vandermonde determinant

In this subsection, we demonstrate that the joint probability density function
can be written as a determinant (to which we can apply Lemma 4.3). Fix charge
vector $\vec{L}$, population vector $\vec{M}$, and location vector x as in
subsection 1.3. Recall $N=\sum_{j=1}^{J}M_{j}L_{j}$. Let
$\vec{f}=\left\\{f_{n}\right\\}_{n=1}^{N}$ be a family of
$\max(L_{1},\ldots,L_{J})-1$ times differentiable functions. For each $j$,
define the $N\times L_{j}$ matrix

$V^{L_{j}}(x)=\left[D^{l-1}f_{n}(x)\right]_{n,l=1}^{N,L_{j}}.$

For each $\textbf{x}^{j}\in{\mathbb{R}}^{M_{j}}$, define the $N\times
M_{j}L_{j}$ matrix

$V^{L_{j},M_{j}}(\textbf{x}^{j})=\begin{bmatrix}V^{L_{j}}(x_{1}^{j})&V^{L_{j}}(x_{2}^{j})&\cdots&V^{L_{j}}(x_{M_{j}}^{j})\end{bmatrix}.$

Finally, define the $N\times N$ matrix

$V^{\vec{L},\vec{M}}({\textbf{x}})=\begin{bmatrix}V^{L_{1},M_{1}}(\textbf{x}^{1})&V^{L_{2},M_{2}}(\textbf{x}^{2})&\cdots&V^{L_{J},M_{J}}(\textbf{x}^{J})\end{bmatrix},$

in which each variable $x_{m}^{j}$ appears in $L_{j}$ many consecutive
columns, generated from $\vec{f}$ by taking derivatives. The Wronskians which
appear in section 3 are merely the determinants of the univariate $L_{j}\times
L_{j}$ minors of this matrix.

With the additional restriction that $\vec{f}$ be a complete $N$-family of
monic polynomials, we call $V^{\vec{L},\vec{M}}({\textbf{x}})$ the confluent
Vandermonde matrix (with respect to shape $\vec{L},\vec{M}$) in variables x.
Under these conditions (on $\vec{f}$), it is well known [15] that

$\det
V^{\vec{L},\vec{M}}({\textbf{x}})=\prod_{j=1}^{J}\prod_{m<n}\left(x_{n}^{j}-x_{m}^{j}\right)^{L_{j}^{2}}\times\prod_{j<k}\prod_{m=1}^{M_{j}}\prod_{n=1}^{M_{k}}\left(x_{n}^{k}-x_{m}^{j}\right)^{L_{j}L_{k}}.$

Note, we can only construct whole numbers of columns for each variable. This
is where our restrictions on $\vec{L}$ and $\beta$ come from. On the physical
side, we only consider whole number charges for our particles. Using the above
confluent Vandermonde determinant identity, we get

$\Omega_{\vec{M}}({\textbf{x}})=W_{\vec{M}}(\textbf{x})\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|.$

Moreover, if we define the $N\times L_{j}$ matrices

$H^{L_{j}}(x)=\exp(-U(x))V^{L_{j}}(x)=\left[\exp(-U(x))\cdot
D^{l-1}f_{n}(x)\right]_{n,l=1}^{N,L_{j}}$

and the combined $N\times N$ matrix $H^{\vec{L},\vec{M}}({\textbf{x}})$, then

$\left|\det
H^{\vec{L},\vec{M}}({\textbf{x}})\right|=\prod_{j=1}^{J}\prod_{m=1}^{M_{j}}\exp\left(-L_{j}U(x_{m}^{j})\right)\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|=W_{\vec{M}}(\textbf{x})\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|.$

Thus, integration of $V^{\vec{L},\vec{M}}({\textbf{x}})$ with respect to the
$d\mu_{j}$’s is equivalent to integration of
$H^{\vec{L},\vec{M}}({\textbf{x}})$ with respect to Lebesgue measure. For us
to use Lemma 4.3, it is important the entire integrand
$\Omega_{\vec{M}}(\textbf{x})$ be determinantal, with the extra weight
functions $W_{\vec{M}}(\textbf{x})$ incorporated into
$H^{\vec{L},\vec{M}}({\textbf{x}})$.

As an example, consider one charge 2 particle, one charge 3 particle, and
three charge 1 particles with potential $U(x)=x^{2}$. This gives us
$\vec{L}=(2,3,1)$, $\vec{M}=(1,1,3)$, and $N=8$. For simplicity, we will use
the variables $\textbf{x}=(a,b,c_{1},c_{2},c_{3})$. Let
$\vec{f}=\\{x^{n-1}\\}_{n=1}^{N}$. Then the three columns corresponding to the
charge 3 particle are

$V^{3}(b)=\begin{bmatrix}1&0&0\\\ b&1&0\\\ b^{2}&2b&1\\\ b^{3}&3b^{2}&3b\\\
\vdots&&\vdots\\\ b^{7}&7b^{6}&21b^{5}\end{bmatrix}.$

In the third column, we have not just the second derivative but also a
denominator of $2!$. One consequence of these $l!$ denominators in $D^{l-1}$
is that we get 1’s on this diagonal. We also get three columns corresponding
to the three charge 1 particles

$V^{1,3}(c_{1},c_{2},c_{3})=\begin{bmatrix}1&1&1\\\ c_{1}&c_{2}&c_{3}\\\
c_{1}^{2}&c_{2}^{2}&c_{3}^{2}\\\ \vdots&&\vdots\\\
c_{1}^{7}&c_{2}^{7}&c_{3}^{7}\end{bmatrix}.$

Together, the full $8\times 8$ confluent Vandermonde matrix is

$V^{\vec{L},\vec{M}}(\textbf{x})=\begin{bmatrix}1&0&1&0&0&1&1&1\\\
a&1&b&1&0&c_{1}&c_{2}&c_{3}\\\
a^{2}&2a&b^{2}&2b&1&c_{1}^{2}&c_{2}^{2}&c_{3}^{2}\\\
a^{3}&3a^{2}&b^{3}&3b^{2}&3b&c_{1}^{3}&c_{2}^{3}&c_{3}^{3}\\\
\vdots&&&&&&&\vdots\\\
a^{7}&7a^{6}&b^{7}&7b^{6}&21b^{5}&c_{1}^{7}&c_{2}^{7}&c_{3}^{7}\end{bmatrix}.$

We obtain $H^{\vec{L},\vec{M}}(\textbf{x})$ by multiplying the first two
columns by $\exp(-a^{2})$, the next three columns by $\exp(-b^{2})$, and the
last three columns by the appropriate $\exp(-c_{j}^{2})$. This changes the
determinant by

$W_{\vec{M}}(\textbf{x})=\exp(-2a^{2})\exp(-3b^{2})\exp(-c_{1}^{2})\exp(-c_{2}^{2})\exp(-c_{3}^{2}).$

###  4.5 Absolute value of determinants

Though our Boltzmann factor integrand $\Omega_{\vec{M}}(\textbf{x})$ is
recognizably determinantal, we still need to remove the absolute value before
we can apply Lemma 4.3 (and later Lemma 4.2). This can be done by decomposing
the domain of integration into subsets over which the sign of the determinant
is constant. Namely, we use totally ordered subsets $\Delta_{N}(\sigma)$ over
which the differences in the confluent Vandermonde determinant never change
signs. These smaller domains of integration are exactly the ones which allow
us to apply (Chen’s) Lemma 4.2.

As in section 3, suppose $L_{j}$ is even for $1\leq j\leq r$. Let
$K_{e}=\sum_{j=1}^{r}M_{j}$ be the total number of particles with even charge,
and let $K_{o}=\sum_{j=r+1}^{J}M_{j}$ be the total number of particles with
odd charge. Relabel

$y_{1}=x_{1}^{1},\hskip 28.45274pty_{2}=x_{2}^{1},\hskip
28.45274pt\cdots\hskip 28.45274pty_{M_{1}}=x_{M_{1}}^{1},\hskip
28.45274pty_{M_{1}+1}=x_{1}^{2}\hskip 28.45274pt\cdots\hskip
28.45274pty_{K_{e}}=x_{M_{r}}^{r}$

so that $\vec{y}=(\textbf{x}^{1},\textbf{x}^{2},\ldots,\textbf{x}^{r})$ gives
the locations of the particles of even charge. Similarly, relabel

$w_{1}=x_{1}^{r+1},\hskip 28.45274pt\cdots\hskip
28.45274ptw_{M_{r+1}}=x_{M_{r+1}}^{r+1},\hskip
28.45274ptw_{M_{r+1}+1}=x_{1}^{r+2}\hskip 28.45274pt\cdots\hskip
28.45274ptw_{K_{e}}=x_{M_{J}}^{J}$

so that $\vec{w}=(\textbf{x}^{r+1},\textbf{x}^{r+2},\ldots,\textbf{x}^{J})$
gives the locations of the particles of odd charge. Define $\lambda_{j}^{e}$
to be the $L_{k}$ which corresponds to $y_{j}$ so that

$\Lambda^{e}=(\lambda_{1}^{e},\ldots,\lambda_{K_{e}}^{e})=(L_{1},\ldots,L_{1},L_{2},\ldots,L_{2},\ldots,L_{r},\ldots,L_{r})$

gives the list of even charges, with each $L_{j}$ appearing $M_{j}$ times.
Similarly define $\lambda_{j}^{o}$ for corresponding $w_{j}$ so that

$\Lambda^{o}=(\lambda_{1}^{o},\ldots,\lambda_{K_{o}}^{o})=(L_{r+1},\ldots,L_{r+1},L_{r+2},\ldots,L_{r+2},\ldots,L_{J},\ldots,L_{J})$

gives the list of the odd charges. We will treat all charges $\lambda_{j}^{e}$
and $\lambda_{j}^{o}$ as distinct until it is relevant to recall which charges
are repeated (and how many times each).

For each $(\sigma,\tau)\in S_{K_{e}}\times S_{K_{o}}$, define
$V_{\sigma\tau}^{\vec{L},\vec{M}}({\textbf{x}})$ to be the matrix

$V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})=\begin{bmatrix}V^{\lambda^{e}_{\sigma^{-1}(1)}}(y_{\sigma^{-1}(1)})&\cdots&V^{\lambda^{e}_{\sigma^{-1}(K_{e})}}(y_{\sigma^{-1}(K_{e})})&V^{\lambda^{o}_{\tau^{-1}(1)}}(w_{\tau^{-1}(1)})&\cdots&V^{\lambda^{o}_{\tau^{-1}(K_{o})}}(w_{\tau^{-1}(K_{o})})\end{bmatrix},$

obtained from $V^{\vec{L},\vec{M}}({\textbf{x}})$ by permuting the columns so
that the columns with $y_{\sigma^{-1}(1)}$ come first (of which there are
$\lambda^{e}_{\sigma^{-1}(1)}$ many), then all of the columns with
$y_{\sigma^{-1}(2)}$ come next (of which there are
$\lambda^{e}_{\sigma^{-1}(2)}$ many), and so on until the $y_{j}$ are
exhausted, doing the same for the $w_{j}$.

Using the same confluent Vandermonde determinant identity from subsection 4.4,
we get

$\displaystyle\det V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})$
$\displaystyle=\prod_{j<k}(y_{\sigma^{-1}(k)}-y_{\sigma^{-1}(j)})^{\lambda^{e}_{\sigma^{-1}(k)}\lambda^{e}_{\sigma^{-1}(j)}}\times\prod_{j=1}^{K_{e}}\prod_{k=1}^{K_{o}}(w_{\sigma^{-1}(k)}-y_{\tau^{-1}(j)})^{\lambda^{o}_{\sigma^{-1}(k)}\lambda^{e}_{\tau^{-1}(j)}}$
$\displaystyle\hskip
28.45274pt\times\prod_{j<k}(w_{\tau^{-1}(k)}-w_{\tau^{-1}(j)})^{\lambda^{o}_{\tau^{-1}(k)}\lambda^{o}_{\tau^{-1}(j)}}.$

Next, consider
${\textbf{x}}=(\vec{y},\vec{w})\in\Delta_{K_{e}}(\sigma)\times\Delta_{K_{o}}(\tau)$
in which the even charged particles (located by $\vec{y}$) are ordered
according to $\sigma$ and the odd charged particles (located by $\vec{w}$) are
ordered according to $\tau$. In particular,
$w_{\tau^{-1}(k)}>w_{\tau^{-1}(j)}$ whenever $j<k$. Thus, all differences in
the third product are positive. Additionally, each difference in the first and
second products have even exponents $\lambda^{e}_{j}$. Thus,

$\left|\det V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})\right|=\det
V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})$

on the domain $\Delta_{K_{e}}(\sigma)\times\Delta_{K_{o}}(\tau)$.

Note, permuting the variables $\vec{y}$ involves permuting blocks of even
numbers of columns at a time, leaving the determinant of
$V^{\vec{L},\vec{M}}({\textbf{x}})$ unchanged. In contrast, permuting
variables $\vec{w}$ involves permuting blocks of odd numbers of columns at a
time, changing the determinant by ${\mathrm{sgn}}(\tau)$. Thus,

$\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|=\left|{\mathrm{sgn}}(\tau)\det
V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})\right|=\det
V^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})$

on the domain $\Delta_{K_{e}}(\sigma)\times\Delta_{K_{o}}(\tau)$. Analogous
results hold if we replace $V^{\vec{L},\vec{M}}({\textbf{x}})$ with
$H^{\vec{L},\vec{M}}({\textbf{x}})$ (as defined in subsection 4.4).

Recall the example from subsection 4.4 in which we have a single particle of
even charge 2. Then $K_{e}=1$, $\vec{y}=(a)$, and $\Lambda^{e}=(2)$. There is
one particle of odd charge 3, and there are three particles of odd charge 1.
Then $K_{o}=4$, $\vec{w}=(b,c_{1},c_{2},c_{3})$, and $\Lambda^{o}=(3,1,1,1)$.
Let $\tau$ be the permutation which swaps $b$ with $c_{1}$. The new matrix
(with permuted columns) is given by

$V^{\vec{L},\vec{M}}_{{\mathrm{id}},\tau}(\textbf{x})=\begin{bmatrix}1&0&1&1&0&0&1&1\\\
a&1&c_{1}&b&1&0&c_{2}&c_{3}\\\
a^{2}&2a&c_{1}^{2}&b^{2}&2b&1&c_{2}^{2}&c_{3}^{2}\\\
a^{3}&3a^{2}&c_{1}^{3}&b^{3}&3b^{2}&3b&c_{2}^{3}&c_{3}^{3}\\\
\vdots&&&&&&&\vdots\\\
a^{7}&7a^{6}&c_{1}^{7}&b^{7}&7b^{6}&21b^{5}&c_{2}^{7}&c_{3}^{7}\end{bmatrix}$

Note, swapping two variables requires more than just swapping two columns. We
swap the entire three-column block $V^{3}(b)$ with the one-column block
$V^{1}(c_{1})$.

##  5 Canonical ensemble

With the modification to the integrand outlined in subsection 4.5, we can now
decompose $Z_{\vec{M}}$ into integrals without absolute value, provided we
divide the domain of integration appropriately. Explicitly,

$\displaystyle Z_{\vec{M}}$ $\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\int_{{\mathbb{R}}^{K_{e}}}\int_{{\mathbb{R}}^{K_{o}}}\Omega_{\vec{M}}({\textbf{x}})\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}}$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{e}}(\sigma)}\int_{\Delta_{K_{o}}(\tau)}\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|W_{\vec{M}}({\textbf{x}})\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}}$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{e}}(\sigma)}\int_{\Delta_{K_{o}}(\tau)}\det
H^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}},$

summing over all totally ordered subsets
$\Delta_{K_{e}}(\sigma)\subset{\mathbb{R}}^{K_{e}}$ and
$\Delta_{K_{o}}(\tau)\subset{\mathbb{R}}^{K_{o}}$.

###  5.1 When all $L_{j}$ are even

For $\pi=(\pi_{1}|\ldots|\pi_{K_{e}})\in S_{N}$ and complete $N$-family of
monic polynomials $\vec{p}=\\{p_{n}\\}_{n=1}^{N}$, define

$Q_{\pi_{j}}^{k}(x)=\prod_{l=1}^{\lambda^{e}_{k}}\exp(-U(x))\cdot
D^{l-1}p_{\pi_{j}(l)}(x),$

so that we can write

$\det H^{\vec{L},\vec{M}}_{\sigma}(\vec{y})=\sum_{\pi\in
S_{N}}{\mathrm{sgn}}(\pi)Q_{\pi_{1}}^{\sigma^{-1}(1)}(y_{\sigma^{-1}(1)})\cdots
Q_{\pi_{K_{e}}}^{\sigma^{-1}(K_{e})}(y_{\sigma^{-1}(K_{e})}).$

Note, each $Q^{k}_{\pi_{j}}$ is a product of $\lambda_{k}^{e}$ many matrix
entries which we have grouped together. These entries are taken from the
$\lambda_{k}^{e}$ many derivative columns for the one variable $y_{k}$.
Writing the determinant this way allows to invoke Lemma 4.3 which gives us

$\det
H^{\vec{L},\vec{M}}_{\sigma}(\vec{y})=\int\frac{\alpha_{\sigma^{-1}(1)}\wedge_{\shuffle}\cdots\wedge_{\shuffle}\alpha_{\sigma^{-1}(K_{e})}}{K_{e}!}\,d\varepsilon_{{\rm{vol}}},$

where each $\alpha_{k}$ is defined by

$\alpha_{k}(x)=\sum_{\mathfrak{t}:\underline{\lambda^{e}_{k}}\nearrow\underline{N}}A_{\mathfrak{t}}^{k}(x)\,\,\varepsilon_{\mathfrak{t}}=\sum_{\mathfrak{t}:\underline{\lambda^{e}_{k}}\nearrow\underline{N}}\sum_{\tau\in
S_{\lambda^{e}_{k}}}{\mathrm{sgn}}(\tau)Q_{\mathfrak{t}\circ\tau}^{k}(x)\,\,\varepsilon_{\mathfrak{t}}=\sum_{\mathfrak{t}:\underline{\lambda^{e}_{k}}\nearrow\underline{N}}\exp(-\lambda^{e}_{k}U(x)){\rm{Wr}}(\vec{p}_{\mathfrak{t}},x)\,\,\varepsilon_{\mathfrak{t}}.$

As mentioned in subsection 4.4, these Wronskians are the determinants of the
univariate $L_{j}\times L_{j}$ minors of the confluent Vandermonde matrix.
Recall from section 3,

$\gamma_{j}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\int_{\mathbb{R}}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},x)\,d\mu_{j}(x)\,\varepsilon_{\mathfrak{t}},$

with $d\mu_{j}(x)=\exp(-L_{j}U(x))\,dx$. Starting from

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\int_{\Delta_{K_{e}}(\sigma)}\det
H^{\vec{L},\vec{M}}_{\sigma}(\vec{y})\,dy_{1}\cdots dy_{K_{e}},$

relabeling the variables $x_{j}=y_{\sigma^{-1}(j)}$ produces

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\int_{\Delta_{K_{e}}({\mathrm{id}})}\det
H^{\vec{L},\vec{M}}_{\sigma}(\vec{x})\,dx_{1}\cdots dx_{K_{e}}.$

By Lemma 4.3, the determinant can be rewritten in the exterior shuffle
algebra. Thus,

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\left\langle\int\frac{\alpha_{\sigma^{-1}(1)}\wedge_{\shuffle}\cdots\wedge_{\shuffle}\alpha_{\sigma^{-1}(K_{e})}}{K_{e}!}\,d\varepsilon_{{\rm{vol}}}\right\rangle.$

Applying (Chen’s) Lemma 4.2 sends integration of shuffle products to ordinary
products of integrals. Thus,

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\int\frac{\gamma_{1}^{\wedge
M_{1}}\wedge\cdots\wedge\gamma_{J}^{\wedge
M_{J}}}{K_{e}!}\,d\varepsilon_{{\rm{vol}}}.$

Note, there exist $M_{j}$ many $k$ for which $\lambda_{k}^{e}=L_{j}$, so each
factor $\gamma_{j}$ appears $M_{j}$ times. This happens independent of
$\sigma\in S_{K_{e}}$, of which there are $|S_{K_{e}}|=K_{e}!$ many. Thus,

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\int\gamma_{1}^{\wedge
M_{1}}\wedge\cdots\wedge\gamma_{J}^{\wedge M_{J}}\,d\varepsilon_{{\rm{vol}}}.$

We have now proven the following lemma:

######  Lemma 5.1.

If all $L_{j}$ are even, then

$Z_{\vec{M}}=\int\frac{\gamma_{1}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{\gamma_{J}^{\wedge
M_{J}}}{M_{J}!}\,d\varepsilon_{{\rm{vol}}}.$

###  5.2 Even number of odd charges

In this subsection, we assume all $L_{j}$ are odd, but the total number of
particles $K_{o}=\sum_{j=1}^{J}M_{j}=2K$ is even. This happens, for example,
when total charge $N$ is even. Recall from section 3,

$\eta_{j,k}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{L_{k}}\nearrow\underline{N}}\int\int_{x<y}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},x){\rm{Wr}}(\vec{p}_{\mathfrak{s}},y)\,d\mu_{j}(x)d\mu_{k}(y)\,\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}.$

Following the same argument as in subsection 5.1, we can apply our Lemma 4.3
provided there exists antisymmetrized _even_ forms $\alpha_{j,k}$ such that
each $\eta_{j,k}$ is obtained from $\alpha_{j,k}$ by applying our functional
$\langle\cdot\rangle$ to the coefficients of $\alpha_{j,k}$. To this end,
define $\alpha_{j,k}$ by

$\alpha_{j,k}=\sum_{\mathfrak{t}:\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}\nearrow\underline{N}}A_{\mathfrak{t}}^{j,k}\varepsilon_{\mathfrak{t}}=\sum_{\mathfrak{t}:\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}\nearrow\underline{N}}\,\sum_{\pi\in
S_{\lambda_{j}^{o}+\lambda^{o}_{k}}}{\mathrm{sgn}}(\pi)Q_{\mathfrak{t}\circ\pi}^{j,k}\,\varepsilon_{\mathfrak{t}},$

where

$Q_{\mathfrak{t}\circ\pi}^{j,k}(x,y)=\prod_{l=1}^{\lambda_{j}^{o}}\exp(-U(x))\cdot
D^{l-1}p_{\mathfrak{t}\circ\pi(l)}(x)\prod_{l=1}^{\lambda^{o}_{k}}\exp(-U(y))\cdot
D^{l-1}p_{\mathfrak{t}\circ\pi(\lambda_{j}^{o}+l)}(y),$

then $\alpha_{j,k}$ is an antisymmetrized
$(\lambda_{j}^{o}+\lambda^{o}_{k})$-form by construction. As before, we should
think of $Q_{\mathfrak{t}}^{j,k}$ as taking one entry from each of the
$\lambda^{o}_{j}$ many derivative columns for one variable $x$ and then one
entry from each of the $\lambda^{o}_{k}$ many derivative columns for the next
variable $y$. Pairing an odd number of entries with another odd number of
entries produces an even form. It remains to be shown that the
antisymmetrizations $A_{\mathfrak{t}}^{j,k}$ are composed of complementary
Wronskian minors.

For
$\mathfrak{t}:\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}\nearrow\underline{N}$,
let $H_{\tau,\mathfrak{t}}^{\vec{L},\vec{M}}(w_{j},w_{k})$ denote the
$(\lambda_{j}^{o}+\lambda^{o}_{k})\times(\lambda_{j}^{o}+\lambda^{o}_{k})$
minor of $H_{\tau}^{\vec{L},\vec{M}}(\vec{w})$ comprised of rows
$\mathfrak{t}(1),\ldots,\mathfrak{t}(\lambda_{j}^{o}+\lambda^{o}_{k})$ taken
from the $\lambda_{j}^{o}$ columns in $w_{j}$ and the $\lambda^{o}_{k}$
columns in $w_{k}$. When viewed as a two variable function,
$A_{\mathfrak{t}}^{j,k}$ is the determinant of this minor. Explicitly,

$A_{\mathfrak{t}}^{j,k}(w_{j},w_{k})=\det
H_{\tau,\mathfrak{t}}^{\vec{L},\vec{M}}(w_{j},w_{k}).$

For
$\mathfrak{t}_{1}:\underline{\lambda_{j}^{o}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}$,
let $H_{\tau,\mathfrak{t}_{1}}^{\vec{L},\vec{M}}(w_{j})$ denote the
$\lambda_{j}^{o}\times\lambda_{j}^{o}$ minor of
$H_{\tau,\mathfrak{t}}^{\vec{L},\vec{M}}(w_{j},w_{k})$ comprised of rows
$\mathfrak{t}_{1}(1),\ldots,\mathfrak{t}_{1}(\lambda_{j}^{o})$ taken from the
$\lambda_{j}^{o}$ columns in $w_{j}$. Similarly, for
$\mathfrak{t}_{2}:\underline{\lambda^{o}_{k}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}$,
let $H_{\tau,\mathfrak{t}_{2}}^{\vec{L},\vec{M}}(w_{k})$ denote a
$\lambda^{o}_{k}\times\lambda^{o}_{k}$ minor in $w_{k}$. By the Laplace
expansion of the determinant,

$\displaystyle\det
H_{\tau,\mathfrak{t}}^{\vec{L},\vec{M}}(w_{j},w_{k})\,\varepsilon_{\mathfrak{t}}$
$\displaystyle=\sum_{\mathfrak{t}_{1}:\underline{\lambda_{j}^{o}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\sum_{\mathfrak{t}_{2}:\underline{\lambda^{o}_{k}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\det
H_{\tau,\mathfrak{t}_{1}}^{\vec{L},\vec{M}}(w_{j})\det
H_{\tau,\mathfrak{t}_{2}}^{\vec{L},\vec{M}}(w_{k})\cdot{\mathrm{sgn}}(\mathfrak{t}_{1},\mathfrak{t}_{2})\,\varepsilon_{\mathfrak{t}}$
$\displaystyle=\sum_{\mathfrak{t}_{1}:\underline{\lambda_{j}^{o}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\sum_{\mathfrak{t}_{2}:\underline{\lambda^{o}_{k}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\det
H_{\tau,\mathfrak{t}_{1}}^{\vec{L},\vec{M}}(w_{j})\det
H_{\tau,\mathfrak{t}_{2}}^{\vec{L},\vec{M}}(w_{k})\cdot\varepsilon_{\mathfrak{t}_{1}}\wedge\varepsilon_{\mathfrak{t}_{2}}$
$\displaystyle=\sum_{\mathfrak{t}_{1}:\underline{\lambda_{j}^{o}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\sum_{\mathfrak{t}_{2}:\underline{\lambda^{o}_{k}}\nearrow\underline{\lambda_{j}^{o}+\lambda^{o}_{k}}}\exp(-\lambda^{o}_{j}U(w_{j})){\rm{Wr}}(\vec{p}_{\mathfrak{t}_{1}},w_{j})\exp(-\lambda^{o}_{k}U(w_{k})){\rm{Wr}}(\vec{p}_{\mathfrak{t}_{2}},w_{k})\cdot\varepsilon_{\mathfrak{t}_{1}}\wedge\varepsilon_{\mathfrak{t}_{2}}.$

Thus, $\eta_{j,k}$ is the result of applying $\langle\cdot\rangle_{2}$ to the
two-variable coefficients of $\alpha_{j,k}$ as desired.

Proceeding as we did in subsection 5.1 (applying Lemma 4.3 in the third line
below and Lemma 4.2 in the last line),

$\displaystyle Z_{\vec{M}}$ $\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\sum_{\tau\in S_{K_{o}}}\int_{\Delta_{K_{o}}(\tau)}\det
H^{\vec{L},\vec{M}}_{\tau}(\vec{w})\,dw_{1}\cdots dw_{K_{o}}$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{o}}({\mathrm{id}})}\det
H^{\vec{L},\vec{M}}_{\tau}(\vec{x})\,dx_{1}\cdots dx_{K_{e}}$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\tau\in
S_{K_{o}}}\left\langle\int\frac{\alpha_{\tau^{-1}(1),\tau^{-1}(2)}\wedge_{\shuffle}\cdots\wedge_{\shuffle}\alpha_{\tau^{-1}(K_{o}-1),\tau^{-1}(K_{o})}}{K!}\,d\varepsilon_{{\rm{vol}}}\right\rangle$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\tau\in
S_{K_{o}}}\int\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\,d\varepsilon_{{\rm{vol}}},$

where $M_{\tau}^{j,k}$ is the number of times
$\lambda^{o}_{\tau^{-1}(2n-1)}=L_{j}$ while
$\lambda^{o}_{\tau^{-1}(2n)}=L_{k}$, and $K=K_{o}/2=\sum_{j,k}M_{\tau}^{j,k}$
is the total number of factors in the wedge product. Note, these
$M_{\tau}^{j,k}$ exponents depend on $\tau$, so we’re not able to drop the
sum. However, the $M_{j}$ many particles with the same charge $L_{j}$ are
indistinguishable. Restricting to shuffle permutations removes the redundancy
in permuting variables which have the same $L_{j}$. We have now proven another
lemma:

######  Lemma 5.2.

If all $L_{j}$ are odd, but total charge $N$ is even, then

$Z_{\vec{M}}=\sum_{\tau\in{\rm{Sh}}(M_{1},\ldots,M_{J})}\int\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\,d\varepsilon_{{\rm{vol}}}.$

Recall the example from subsection 4.4 and subsection 4.5. Modify this example
by replacing the even charge 2 particle with an odd charge 3 particle, leaving
the other particle of odd charge 3 and three particles of odd charge 1. The
three columns in variables $a$ and $b$ produce $3\times 3$ Wronskian minors,
while the remaining columns in the variables $c_{1},c_{2},c_{3}$ produce
$1\times 1$ Wronskian minors.

Under the identity permutation, we pair the three columns in variable $a$ with
the three columns in variable $b$ to produce $\eta_{3,3}$. Pairing the one
column in variable $c_{1}$ with the one column in variable $c_{2}$ produces
$\eta_{1,1}$.

Under the permutation $\tau$ which previously swapped the three columns in
variable $b$ with the one column in variable $c_{1}$, we pair variable $a$
(charge 3) with $c_{1}$ (charge 1), and we pair $b$ (charge 3) with $c_{2}$
(charge 1). After integrating out all the variables, the result is two copies
of $\eta_{3,1}$.

The permutation which swaps $a$ with $b$ produces the same $\eta_{3,3}$ as the
identity permutation. To avoid this redundancy, we consider only shuffle
permutations. The permutation which moves $c_{1}$ to the front (ordering the
variables as $c_{1},a,b,c_{2},c_{3}$) produces $\eta_{1,3}$ followed by the
distinct $\eta_{3,1}$.

Note, in this example as stated, the last variable $c_{3}$ is unpaired because
we have an odd number of variables. In the next subsection, we demonstrate how
to fix this. Amending an extra column to the confluent Vandermonde matrix
allows us to pair the last variable with a placeholder. Once integrated, this
last “pair” produces the single Wronskian form $\gamma_{j}$ instead of the
double Wronskian form $\eta_{j,k}$.

###  5.3 Odd number of odd charges

In this subsection, we again assume all $L_{j}$ are odd, but the total number
of particles $K_{o}=\sum_{j=1}^{J}M_{j}=2K-1$ is odd. This happens, for
example, when total charge $N$ is odd. Start by modifying
$H_{\tau}^{\vec{L},\vec{M}}(\vec{w})$ as follows:

$H_{\tau}^{\vec{L},\vec{M},1}(\vec{w})=\begin{bmatrix}H_{\tau}^{\vec{L},\vec{M}}(\vec{w})&0\\\
0&1\end{bmatrix}$

In the previous subsection, we constructed even forms $\alpha_{j,k}$ and
subsequent $\eta_{j,k}$ by pairing an $L_{j}$ charge with an adjacent $L_{k}$
charge. We also showed taking determinants of appropriate minors produces an
antisymmetrized form $\alpha_{j,k}$ (for which our Lemma 4.3 will apply). We
will do this pairing again for the first $2K-2$ variables, which makes $K-1$
pairs. Explicitly, simply define
$Q_{\mathfrak{t}}^{j,k},A_{\mathfrak{t}}^{j,k},\alpha_{j,k},$ and $\eta_{j,k}$
as before.

For convenience, let $L=\lambda^{o}_{\tau^{-1}(K_{o})}$ be the charge
corresponding to the last variable $w_{\tau^{-1}(K_{o})}$. Construct the final
$\alpha_{\tau^{-1}(K_{o})}$ by taking $(L+1)\times(L+1)$ minors from the last
$L+1$ columns of $H_{\tau}^{\vec{L},\vec{M},1}(\vec{w})$. These minors have
non-zero determinant only when the last row is chosen. Thus, valid minors are
entirely determined by a choice of only $L$ many other rows, and

$\alpha_{\tau^{-1}(K_{o})}=\sum_{\mathfrak{t}:\underline{L}\nearrow\underline{N}}\sum_{\pi\in
S_{L}}{\mathrm{sgn}}(\pi)Q_{\mathfrak{t}\circ\pi}^{\tau^{-1}(K_{o})}\,\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{N+1}$

is an antisymmetrized $(L+1)$-form when

$Q_{\mathfrak{t}\circ\pi}^{\tau^{-1}(K_{o})}(x)=\prod_{l=1}^{L}\exp(-U(x))\cdot
D^{l-1}p_{\pi(l)}(x).$

As in subsection 5.1, applying $\langle\cdot\rangle_{1}$ to
$\alpha_{\tau^{-1}(K_{o})}$ produces
$\gamma_{\tau^{-1}(K_{o})}\wedge\varepsilon_{N+1}$. Then the expression for
$Z_{\vec{M}}$ changes only slightly to include this new (unpaired, single
Wronskian) factor:

######  Lemma 5.3.

If all $L_{j}$ are odd, and the total charge $N$ is odd, then

$Z_{\vec{M}}=\sum_{\tau\in{\rm{Sh}}(M_{1},\ldots,M_{J})}\int\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\wedge\gamma_{\tau^{-1}(K_{o})}\,d\varepsilon_{{\rm{vol}}},$

where $K=(K_{o}+1)/2=1+\sum_{j,k}M_{\tau}^{j,k}$ is the total number of
factors in the wedge product.

Recall the example at the end of the previous subsection with two particles of
charge 3 and three particles of charge 1. Under the identity permutation, we
pair a charge 3 with a charge 3, pair a charge 1 with a charge 1, and leave a
charge 1 unpaired. This produces $\eta_{3,3}\wedge\eta_{1,1}\wedge\gamma_{1}$.
Under the permutation $\tau$ which swapped the second charge 3 with the first
charge 1, we got $\eta_{3,1}\wedge\eta_{3,1}\wedge\gamma_{1}$.

Consider instead the permutation which puts all of the charge 1 particles
before the charge 3 particles. We pair a charge 1 with a charge 1, pair the
last charge 1 with a charge 3, and leave a charge 3 unpaired. This produces
$\eta_{1,1}\wedge\eta_{1,3}\wedge\gamma_{3}$.

###  5.4 Arbitrary charge vector

Finally, we allow any mix of odd and even charges. Recall (from the beginning
of section 5),

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{e}}(\sigma)}\int_{\Delta_{K_{o}}(\tau)}\det
H^{\vec{L},\vec{M}}_{\sigma,\tau}({\textbf{x}})\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}}.$

Let $N_{e}=\sum_{j=1}^{r}L_{j}M_{j}$ be the total charge of the even charges,
and let $N_{o}=\sum_{j=r+1}^{J}L_{j}M_{j}$ be the total charge of the odd
charges. By the Laplace expansion of the determinant,

$\det
H^{\vec{L},\vec{M}}_{\sigma,\tau}(\vec{y},\vec{w})\,\varepsilon_{{\rm{vol}}}=\sum_{\mathfrak{t}:\underline{N_{e}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{N_{o}}\nearrow\underline{N}}\det
H^{\vec{L},\vec{M}}_{\sigma,\mathfrak{t}}(\vec{y})\det
H^{\vec{L},\vec{M}}_{\tau,\mathfrak{s}}(\vec{w})\cdot\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}},$

where $\det H^{\vec{L},\vec{M}}_{\sigma,\mathfrak{t}}(\vec{y})$ is an
$N_{e}\times N_{e}$ minor taken only from columns in the (even charge)
variables $\vec{y}$, and $\det
H^{\vec{L},\vec{M}}_{\sigma,\mathfrak{s}}(\vec{w})$ is an $N_{o}\times N_{o}$
minor taken only from columns in the (odd charge) variables $\vec{w}$. Note,
$\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}}=0$ whenever these
minors are not complimentary.

With the variables separated in this way, we can apply Lemma 5.1 to the
determinant in the even charges, and we can apply either Lemma 5.2 or Lemma
5.3 to the determinant in the odd charges. For the even charges, we have

$\frac{1}{M_{1}!\cdots M_{r}!}\sum_{\sigma\in
S_{K_{e}}}\int_{\Delta_{K_{e}}(\sigma)}\det
H^{\vec{L},\vec{M}}_{\sigma,\mathfrak{t}}(\vec{y})\,dy_{1}\cdots
dy_{K_{e}}\cdot\varepsilon_{\mathfrak{t}}=\frac{{\gamma^{\mathfrak{t}}_{1}}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{{\gamma^{\mathfrak{t}}_{r}}^{\wedge
M_{r}}}{M_{r}!},$

where $\gamma_{j}^{\mathfrak{t}}$ is subtly different from $\gamma_{j}$
because $H^{\vec{L},\vec{M}}_{\sigma,\mathfrak{t}}(\vec{y})$ is already an
$N_{e}\times N_{e}$ minor chosen by $\mathfrak{t}$. Explicitly,

$\gamma_{j}^{\mathfrak{t}}=\sum_{\mathfrak{t}_{j}:\underline{L_{j}}\nearrow\underline{N_{e}}}\int_{{\mathbb{R}}}{\rm{Wr}}(\vec{p}_{\mathfrak{t}\circ\mathfrak{t}_{j}},x)\,d\mu_{j}(x)\,\varepsilon_{\mathfrak{t}_{j}}.$

Taking the sum over all $\mathfrak{t}:\underline{N_{e}}\nearrow\underline{N}$
gives us back the original $\gamma_{j}$ forms

$\sum_{\mathfrak{t}:\underline{N_{e}}\nearrow\underline{N}}\frac{{\gamma^{\mathfrak{t}}_{1}}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{{\gamma^{\mathfrak{t}}_{r}}^{\wedge
M_{r}}}{M_{r}!}=\frac{\gamma_{1}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{\gamma_{r}^{\wedge M_{r}}}{M_{r}!}.$

It is straightforward to check an analogous result holds for the determinant
in the odd charges. The following lemma supersedes Lemmas 5.1, 5.2, and 5.3:

######  Lemma 5.4.

Suppose $L_{j}$ is even for $1\leq j\leq r$, then when $N$ is even,

$Z_{\vec{M}}=\int\frac{\gamma_{1}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{\gamma_{r}^{\wedge
M_{r}}}{M_{r}!}\wedge\sum_{\tau\in{\rm{Sh}}(M_{r+1},\ldots,M_{J})}\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\,d\varepsilon_{{\rm{vol}}},$

and when $N$ is odd,

$Z_{\vec{M}}=\int\frac{\gamma_{1}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{\gamma_{r}^{\wedge
M_{r}}}{M_{r}!}\wedge\sum_{\tau\in{\rm{Sh}}(M_{r+1},\ldots,M_{J})}\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\wedge\gamma_{\tau^{-1}(K_{o})}\,d\varepsilon_{{\rm{vol}}}.$

##  6 Isocharge grand canonical ensemble

Recall from subsection 1.3, we want to compute

$Z_{N}=\sum_{\vec{L}\cdot\vec{M}=N}z_{1}^{M_{1}}z_{2}^{M_{2}}\cdots
z_{J}^{M_{J}}Z_{\vec{M}}.$

###  6.1 When all $L_{j}$ are even

Starting from Lemma 5.1, we have

$Z_{N}=\sum_{\vec{L}\cdot\vec{M}=N}z_{1}^{M_{1}}\cdots
z_{J}^{M_{J}}\int\frac{\gamma_{1}^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{\gamma_{J}^{\wedge
M_{J}}}{M_{J}!}\,d\varepsilon_{{\rm{vol}}}.$

Recall from subsection 2.2, the Berezin integral is a projection onto the
highest exterior power $\bigwedge^{N}({\mathbb{R}}^{N})$. If each $\gamma_{j}$
is an $L_{j}$-form, then the wedge product above is an
$\vec{L}\cdot\vec{M}$-form. If we extend the sum over all $\vec{M}$, the
Berezin integral will eliminate any summands for which
$\vec{L}\cdot\vec{M}\neq N$. Thus,

$\displaystyle Z_{N}$
$\displaystyle=\int\sum_{M_{1}=0}^{\infty}\cdots\sum_{M_{J}=0}^{\infty}\frac{(z_{1}\gamma_{1})^{\wedge
M_{1}}}{M_{1}!}\wedge\cdots\wedge\frac{(z_{J}\gamma_{J})^{\wedge
M_{J}}}{M_{J}!}\,d\varepsilon_{{\rm{vol}}}$
$\displaystyle=\int\bigwedge_{j=1}^{J}\sum_{M=1}^{\infty}\frac{(z_{j}\gamma_{j})^{\wedge
M}}{M!}\,d\varepsilon_{{\rm{vol}}}$
$\displaystyle=\int\exp(z_{1}\gamma_{1})\wedge\cdots\wedge\exp(z_{J}\gamma_{J})\,d\varepsilon_{{\rm{vol}}}$
$\displaystyle={\rm{BE}}_{{\rm{vol}}}(z_{1}\gamma_{1}+\cdots+z_{J}\gamma_{J}).$

In the last line, we replace the product of these exponentials with the
exponential of the sum, which we can do because our forms are even and
therefore commute. This completes the proof of Theorem 3.1.

###  6.2 When all $L_{j}$ are odd

Let’s start by assuming there are no even species, and the total charge $N$ is
even. Recall Lemma 5.2 which gives us

$Z_{N}=\sum_{\vec{L}\cdot\vec{M}=N}z_{1}^{M_{1}}\cdots
z_{J}^{M_{J}}\sum_{\tau\in{\rm{Sh}}(M_{1},\ldots,M_{J})}\int\frac{1}{K!}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\eta_{j,k}^{\wedge
M_{\tau}^{j,k}}\,d\varepsilon_{{\rm{vol}}}.$

For $\vec{M}$ fixed and $\tau\in{\rm{Sh}}(M_{1},\ldots,M_{J})$, the number of
other permutations which produce the same pairs $(j,k)$ is

$K!\prod_{j=1}^{J}\prod_{k=1}^{J}\frac{1}{M_{\tau}^{j,k}!}.$

This is just a multinomial coefficient, recalling $K$ is the sum of the
$M_{\tau}^{j,k}$. Of these permutations, there exists a unique representative
which orders the pairs $(j,k)$ lexicographically. Let $\mathcal{L}_{\vec{M}}$
be the set of these representatives, then

$Z_{N}=\int\sum_{M_{1}=0}^{\infty}\cdots\sum_{M_{J}=0}^{\infty}\sum_{\tau\in\mathcal{L}_{\vec{M}}}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\frac{(z_{j}z_{k}\eta_{j,k})^{\wedge
M_{\tau}^{j,k}}}{M_{\tau}^{j,k}!}\,d\varepsilon_{{\rm{vol}}}.$

Next, we condition on population vectors $\vec{M}$ which produce the same $K$,
the number of $(j,k)$ pairs, and so

$Z_{N}=\int\sum_{K=0}^{\infty}\sum_{M_{1}+\cdots+M_{J}=2K}\sum_{\tau\in\mathcal{L}_{\vec{M}}}\bigwedge_{j=1}^{J}\bigwedge_{k=1}^{J}\frac{(z_{j}z_{k}\eta_{j,k})^{\wedge
M_{\tau}^{j,k}}}{M_{\tau}^{j,k}!}\,d\varepsilon_{{\rm{vol}}}.$

Collecting these together, we get the $K^{\rm{th}}$ power of the sum over all
possible pairs $(j,k)$. Thus,

$\displaystyle Z_{N}$
$\displaystyle=\int\sum_{K=0}^{\infty}\frac{1}{K!}\left(\sum_{j=1}^{J}\sum_{k=1}^{J}z_{j}z_{k}\eta_{j,k}\right)^{\wedge
K}\,d\varepsilon_{{\rm{vol}}}$
$\displaystyle={\rm{BE}}_{{\rm{vol}}}\left(\sum_{j=1}^{J}\sum_{k=1}^{J}z_{j}z_{k}\eta_{j,k}\right).$

In the case with total charge $N$ odd, we can go through the same steps
starting from Lemma 5.3. This produces

$Z_{N}={\rm{BE}}_{{\rm{vol}}_{1}}\left(\sum_{j=1}^{J}\sum_{k=1}^{J}z_{j}z_{k}\eta_{j,k}+\sum_{j=1}^{J}z_{j}\gamma_{j}\wedge\varepsilon_{N+1}\right).$

Note, when $N$ is odd, every $Z_{\vec{M}}$ has exactly one $\gamma_{j}$ for
each $\tau$ (see Lemma 5.3). The $\varepsilon_{N+1}$ shown attached to each of
the $\gamma_{j}$ above ensures this is the case when we exponentiate and take
the Berezin integral. First, $\varepsilon_{N+1}\wedge\varepsilon_{N+1}=0$, so
$\gamma_{j}\wedge\varepsilon_{N+1}\wedge\gamma_{k}\wedge\varepsilon_{N+1}=0$.
Thus, terms in the expansion of the exponential with more than one
$\gamma_{j}$ are annihilated. Because the Berezin integral with respect to
$d\varepsilon_{{\rm{vol}}_{1}}$ projects onto the highest exterior power
$\bigwedge^{N+1}({\mathbb{R}}^{N+1})$, terms in the expansion of the
exponential with no $\gamma_{j}$ are missing the basis vector
$\varepsilon_{N+1}$ and are annihilated by the Berezin integral (with respect
to $d\varepsilon_{{\rm{vol}}_{1}}$). Thus, we only get summands (in the
expansion of the exponential) with exactly one $\gamma_{j}$, as in Lemma 5.3.

###  6.3 Full generalization

Recall Lemma 5.4, in which the $\gamma_{1},\ldots,\gamma_{r}$ corresponding to
the even charges are already factored out. Summing over all possible
$M_{1},\ldots,M_{r}$, we can factor out an $\exp(z_{1}\gamma_{1}+\cdots
z_{r}\gamma_{r})$ as in subsection 6.1. From what remains, we obtain the
exponential of the sum of the $\eta_{j,k}$, possibly with an extra set of
$\gamma_{j}\wedge\varepsilon_{N+1}$ forms. For $N$ even,

$\displaystyle Z_{N}$
$\displaystyle=\int\left[\bigwedge_{j=1}^{r}\sum_{M=1}^{\infty}\frac{(z_{j}\gamma_{j})^{\wedge
M}}{M!}\right]\wedge\sum_{K=0}^{\infty}\frac{1}{K!}\left(\sum_{j=1}^{J}\sum_{k=1}^{J}z_{j}z_{k}\eta_{j,k}\right)^{\wedge
K}\,d\varepsilon_{{\rm{vol}}}$
$\displaystyle=\int\exp\left(\sum_{j=1}^{r}z_{j}\gamma_{j}\right)\wedge\exp\left(\sum_{j=r+1}^{J}\sum_{k=r+1}^{J}z_{j}z_{k}\eta_{j,k}\right)\,\varepsilon_{{\rm{vol}}}$
$\displaystyle={\rm{BE}}_{{\rm{vol}}}\left(\sum_{j=1}^{r}z_{j}\gamma_{j}+\sum_{j=r+1}^{J}\sum_{k=r+1}^{J}z_{j}z_{k}\eta_{j,k}\right).$

This concludes the proof of Theorem 3.2 and, with a slight modification,
Theorem 3.3.

##  7 Circular ensembles

Returning to the original setup in subsection 1.3, consider instead charged
particles on the unit circle. Substitute all instances of ${\mathbb{R}}$ with
$[0,2\pi)$ so that

${\textbf{x}}=(\textbf{x}^{1},\textbf{x}^{2},\ldots,\textbf{x}^{J})\in[0,2\pi)^{M_{1}}\times[0,2\pi)^{M_{2}}\times\cdots\times[0,2\pi)^{M_{J}}$

gives the locations of the particles around the unit circle with each
$x_{n}^{j}\in[0,2\pi)$ corresponding to an angle. Assuming logarithmic
interaction between the particles, the energy contributed by interaction
between two particles of charge $L_{j}$ and $L_{k}$ at angles $x_{n}^{j}$ and
$x_{m}^{k}$ respectively is given by
$-L_{j}L_{k}\log\left|e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right|$. Thus, at inverse
temperature $\beta$, the total potential energy of the system is given by

$E_{\vec{M}}({\textbf{x}})=-\beta\sum_{j=1}^{J}L_{j}^{2}\sum_{m<n}\log\left|e^{ix_{n}^{j}}-e^{ix_{m}^{j}}\right|-\beta\sum_{j<k}L_{j}L_{k}\sum_{m=1}^{M_{j}}\sum_{n=1}^{M_{k}}\log\left|e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right|,$

with Boltzmann factor

$\Omega_{\vec{M}}({\textbf{x}})=\exp(-E_{\vec{M}}({\textbf{x}}))=\prod_{j=1}^{J}\prod_{m<n}\left|e^{ix_{n}^{j}}-e^{ix_{m}^{j}}\right|^{\beta
L_{j}^{2}}\times\prod_{j<k}\prod_{m=1}^{M_{j}}\prod_{n=1}^{M_{k}}\left|e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right|^{\beta
L_{j}L_{k}}.$

We will write ${\textbf{y}}=\exp(i{\textbf{x}})$ to mean the vector with
entries of the form $e^{ix_{m}^{j}}$. Then the probability of finding the
system in a state corresponding to a location vector x is given by the joint
probability density function

$\rho_{\vec{M}}({\textbf{x}})=\frac{\Omega_{\vec{M}}({\textbf{x}})}{Z_{\vec{M}}M_{1}!M_{2}!\cdots
M_{J}!}=\frac{\left|\det
V^{\vec{L},\vec{M}}({\textbf{y}})\right|}{Z_{\vec{M}}M_{1}!M_{2}!\cdots
M_{J}!},$

with partition function

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\int_{[0,2\pi)^{M_{1}}}\cdots\int_{[0,2\pi)^{M_{J}}}\left|\det
V^{\vec{L},\vec{M}}({\textbf{y}})\right|\,d\nu^{M_{1}}(\textbf{x}^{1})\,d\nu^{M_{2}}(\textbf{x}^{2})\cdots
d\nu^{M_{J}}(\textbf{x}^{J}),$

where $\nu^{M_{j}}$ is Lebesque measure on $[0,2\pi)^{M_{j}}$. Using the same
modifications as before in the linear case, we can assume $\beta=1$ for
computational purposes. Note, the same confluent Vandermonde determinant gives
us the same product of differences (with exponents) as before, even with the
new complex variables y in place of the real variables x.

###  7.1 Complex modulus

We will be able to apply our same algebraic lemmas to our determinantal
integrand once we resolve the absolute value. As observed in [14], the
absolute difference can be decomposed as

$\displaystyle\left|e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right|$
$\displaystyle=-ie^{-i\left(x_{n}^{k}+x_{m}^{j}\right)/2}\left(e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right){\mathrm{sgn}}(x_{n}^{k}-x_{m}^{j})$
$\displaystyle=-ie^{-i\left(x_{n}^{k}+x_{m}^{j}\right)/2}\left(e^{ix_{n}^{k}}-e^{ix_{m}^{j}}\right)\frac{\left(x_{n}^{k}-x_{m}^{j}\right)}{\left|x_{n}^{k}-x_{m}^{j}\right|}.$

Next, we define

$d\mu_{j}(x)=\left(-ie^{-ix}\right)^{L_{j}T/2}\,dx,$

where

$T=-L_{j}+\sum_{k=1}^{J}L_{k}M_{k},$

so that we can bring the (complex valued) weight functions
$(-ie^{-ix_{m}^{j}})^{T/2}$ inside the matrix
$V^{\vec{L},\vec{M}}({\textbf{y}})$ the same way we did in subsection 4.4.
Explicitly, construct $H^{\vec{L},\vec{H}}({\textbf{y}})$ by multiplying each
column with the variable $x_{m}^{j}$ by $(-ie^{-ix_{m}^{j}})^{T/2}$. Note,
there will be $L_{j}$ many columns for each $x_{m}^{j}$. Finally, we can write
the $(x_{n}^{k}-x_{m}^{j})^{L_{j}L_{k}}/|x_{n}^{k}-x_{m}^{j}|^{L_{j}L_{k}}$
factors as separate confluent Vandermonde determinants in x.

We can use the same procedure of separating the odd species from the even
species and decomposing the integral over ordered subsets as in subsection
4.5. Thus,

$\displaystyle Z_{\vec{M}}$ $\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots
M_{J}!}\int_{[0,2\pi)^{K_{e}}}\int_{[0,2\pi)^{K_{o}}}\left|\det
V^{\vec{L},\vec{M}}({\textbf{y}})\right|\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}}$
$\displaystyle=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{e}}(\sigma)}\int_{\Delta_{K_{o}}(\tau)}\det
H^{\vec{L},\vec{M}}({\textbf{y}})\frac{\det
V^{\vec{L},\vec{M}}({\textbf{x}})}{\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|}dy_{1}\cdots dy_{K_{e}}\,dw_{1}\cdots
dw_{K_{o}}.$

Observe

$\det H^{\vec{L},\vec{M}}(\textbf{y})\frac{\det
V^{\vec{L},\vec{M}}({\textbf{x}})}{\left|\det
V^{\vec{L},\vec{M}}({\textbf{x}})\right|}={\mathrm{sgn}}(\tau)\det
H_{\sigma,\tau}^{\vec{L},\vec{M}}({\textbf{y}})\frac{{\mathrm{sgn}}(\tau)\det
V_{\sigma,\tau}^{\vec{L},\vec{M}}({\textbf{x}})}{\left|{\mathrm{sgn}}(\tau)\det
V_{\sigma,\tau}^{\vec{L},\vec{M}}({\textbf{x}})\right|}={\mathrm{sgn}}(\tau)^{2}\det
H_{\sigma,\tau}^{\vec{L},\vec{M}}({\textbf{y}})$

for ${\textbf{y}}\in\Delta_{K_{e}}(\sigma)\times\Delta_{K_{o}}(\tau)$. Thus,

$Z_{\vec{M}}=\frac{1}{M_{1}!M_{2}!\cdots M_{J}!}\sum_{\sigma\in
S_{K_{e}}}\sum_{\tau\in
S_{K_{o}}}\int_{\Delta_{K_{e}}(\sigma)}\int_{\Delta_{K_{o}}(\tau)}\det
H_{\sigma,\tau}^{\vec{L},\vec{M}}({\textbf{y}})\,dy_{1}\cdots
dy_{K_{e}}\,dw_{1}\cdots dw_{K_{o}}.$

###  7.2 Statement of results

Proceeding through the same methods presented in section 5 and section 6, we
get the same theorems from section 3 with slight modification to our forms
$\gamma_{j}$ and $\eta_{j,k}$. Given a complete $N$-family of monic
polynomials, define

$\gamma_{j}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\left[\int_{0}^{2\pi}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},e^{ix})\,d\mu_{j}(x)\right]\,\varepsilon_{\mathfrak{t}},$

and define

$\eta_{j,k}=\sum_{\mathfrak{t}:\underline{L_{j}}\nearrow\underline{N}}\sum_{\mathfrak{s}:\underline{L_{k}}\nearrow\underline{N}}\left[\int\int_{0<x<y<2\pi}{\rm{Wr}}(\vec{p}_{\mathfrak{t}},e^{ix}){\rm{Wr}}(\vec{p}_{\mathfrak{s}},e^{iy})\,d\mu_{j}(x)d\mu_{k}(y)\right]\varepsilon_{\mathfrak{t}}\wedge\varepsilon_{\mathfrak{s}},$

where $d\mu_{j}(x)=(-ie^{-ix})^{L_{j}T/2}\,dx$.

##  Appendix A Appendix

Let $\Lambda=(\lambda_{1},\ldots,\lambda_{K})$ be a partition of $N$. Recall
from subsection 4.4 the definitions of the _Young subgroup_
${\rm{H}}(\Lambda)\subseteq S_{N}$, the subset of _block permutations_
${\rm{Bl}}(\Lambda)\subset S_{N}$, the subset of _shuffle permutations_
${\rm{Sh}}(\Lambda)$, and the subset of _ordered shuffle permutations_
${\rm{Sh}}^{\circ}(\Lambda)$.

Here we provide a proof of Lemma 4.1.

######  Lemma 4.1.

Let $\Lambda=(\lambda_{1},\ldots,\lambda_{K})$ be a partition of $N$. Given
any $\varphi\in S_{N}$, there exists unique permutations
$\tau\in{\rm{H}}(\Lambda)$, $\pi\in{\rm{Bl}}(\Lambda)$, and
$\sigma\in{\rm{Sh}}^{\circ}(\Lambda^{\pi})$ so that
$\varphi=\sigma\circ\pi\circ\tau$.

Conducive to this proof, it will be convenient to give alternate definitions
for the different subsets of permutations. First, define
$s_{j}=\sum_{k=1}^{j-1}\lambda_{k}$ to be the partial sums of the
$\lambda_{k}$, up to but not including $\lambda_{j}$ so that $s_{1}=0$,
$s_{2}=\lambda_{1}$, $s_{3}=\lambda_{1}+\lambda_{2}$, and so on. We
alternatively define the Young subgroup ${\rm{H}}(\Lambda)$ to be the
$\sigma\in S_{N}$ such that

$s_{k}+1\leq\sigma(s_{k}+j)\leq s_{k+1}$

for all $1\leq k\leq K$ and $1\leq j\leq\lambda_{k}$. We define the block
permutations ${\rm{Bl}}(\Lambda)$ to be the $\sigma\in S_{N}$ such that

$\sigma(s_{k}+j)+1=\sigma(s_{k}+j+1)$

for all $1\leq k\leq K$ and $1\leq j\leq\lambda_{k}$. Recall the definition of
$\theta_{\sigma}\in S_{K}$ from subsection 4.4. This is the unique permutation
such that

$\sigma(s_{\theta_{\sigma}^{-1}(1)}+1)<\sigma(s_{\theta_{\sigma}^{-1}(2)}+1)<\cdots<\sigma(s_{\theta_{\sigma}^{-1}(K)}+1).$

The shuffle permutations ${\rm{Sh}}(\Lambda)$ are the $\sigma\in S_{N}$ such
that

$\sigma(s_{k}+i)<\sigma(s_{k}+j)$

for all $1\leq k\leq K$ and $1\leq i<j\leq\lambda_{k}$. The ordered shuffle
permutations ${\rm{Sh}}^{\circ}(\Lambda)$ additionally satisfy

$\sigma(s_{j}+1)<\sigma(s_{k}+1)$

for all $1\leq j<k\leq K$.

Demonstrably,
$|{\rm{H}}(\Lambda)|=\prod_{k=1}^{K}|S_{\lambda_{k}}|=\lambda_{1}!\cdots\lambda_{K}!$,
and $|{\rm{Bl}}(\Lambda)|=|S_{K}|=K!$. Heuristically, a shuffle permutation
$\sigma\in{\rm{Sh}}(\Lambda)$ is constructed by choosing from $N$ positions
the location of the first $\lambda_{1}$ elements, then the next $\lambda_{2}$
elements, and so on until all elements are exhausted. It is straightforward to
see $|{\rm{Sh}}(\Lambda)|$ is the multinomial coefficient

$\left|{\rm{Sh}}(\Lambda)\right|=\binom{N}{\lambda_{1},\ldots,\lambda_{K}}=\frac{N!}{\lambda_{1}!\cdots\lambda_{K}!}.$

We prove Lemma 4.1 in two steps. First, we show the decomposition of an
arbitrary permutation into a product of a shuffle permutation after a
permutation belonging to the Young subgroup. Second, we show this shuffle
permutation can be further decomposed into a product of an ordered shuffle
permutation after a block permutation.

######  Lemma A.1.

Given any $\varphi\in S_{N}$, there exists unique $\tau\in{\rm{H}}(\Lambda)$
and $\rho\in{\rm{Sh}}(\Lambda)$ so that $\varphi=\rho\circ\tau$.

###### Proof.

Consider the collection of right cosets $S_{N}/{\rm{H}}(\Lambda)$. For each
$1\leq k\leq K$ and $1\leq j\leq\lambda_{k}$, let $a_{j}^{k}$ be the
$j^{\rm{th}}$ smallest element of the set
$\\{\varphi(s_{k}+1),\varphi(s_{k}+2),\ldots,\varphi(s_{k}+\lambda_{k})\\}$,
and define a permutation $\tau\in S_{N}$ by

$\tau(s_{k}+j)=\varphi^{-1}(a_{j}^{k})$

for $1\leq k\leq K$ and $1\leq j\leq\lambda_{k}$. Then
$\tau\in{\rm{H}}(\Lambda)$, and

$\varphi\circ\tau(s_{k}+i)<\varphi\circ\tau(s_{k}+j)$

whenever $1\leq i<j\leq\lambda_{k}$. Thus, every coset $T\in
S_{N}/{\rm{H}}(\Lambda)$ contains at least one shuffle permutation. Note,

$|S_{N}/{\rm{H}}(\Lambda)|=\frac{N!}{\lambda_{1}!\cdots\lambda_{K}!}=|{\rm{Sh}}(\Lambda)|.$

Thus, each coset contains a unique shuffle permutation. Define
$\rho\in{\rm{Sh}}(\Lambda)$ to be this unique shuffle permutation in the coset
to which $\varphi$ belongs. Then $\varphi=\rho\circ\tau$ as desired. ∎

######  Lemma A.2.

Given any $\rho\in{\rm{Sh}}(\Lambda)$, there exists unique permutations
$\pi\in{\rm{Bl}}(\Lambda)$ and $\sigma\in{\rm{Sh}}^{\circ}(\Lambda^{\pi})$ so
that $\rho=\sigma\circ\pi$.

###### Proof.

For each $1\leq k\leq K$, define $b_{k}$ to be the $k^{\rm{th}}$ smallest
element of the set

$\\{\rho(s_{1}+1),\rho(s_{2}+1),\ldots,\rho(s_{K}+1)\\}.$

Let $\alpha\in S_{K}$ be the permutation satisfying

$\rho(s_{k}+1)=b_{\alpha(k)}$

for $1\leq k\leq K$. Define $\pi\in{\rm{Bl}}(\Lambda)$ by

$\pi(s_{k}+j)=\rho^{-1}(b_{\alpha(k)})+j-1$

for $1\leq k\leq K$ and $1\leq j\leq\lambda_{k}$. Then $\alpha=\theta_{\pi}$.
Let $\sigma=\rho\circ\pi^{-1}$. We want to show
$\sigma\in{\rm{Sh}}^{\circ}(\Lambda^{\pi})$.

Let $\mu=\Lambda^{\pi}$, and let $t_{j}=\sum_{k=1}^{j-1}\mu_{k}$. Suppose
$1\leq k\leq K$ and $1\leq j\leq\mu_{k}$. Observe that

$\displaystyle\sigma(t_{k}+j)$ $\displaystyle=\rho\circ\pi^{-1}(t_{k}+1+j-1)$
$\displaystyle=\rho\circ\pi^{-1}(\pi(s_{\alpha^{-1}(k)}+1)+j-1)$
$\displaystyle=\rho\circ\pi^{-1}(\pi(s_{\alpha^{-1}(k)}+j))$
$\displaystyle=\rho(s_{\alpha^{-1}(k)}+j).$

Thus,

$\sigma(t_{k}+j)=\rho(s_{\alpha^{-1}(k)}+j).$

If $1\leq i<j\leq\mu_{k}$, then
$\rho(s_{\alpha^{-1}(k)}+i)<\rho(s_{\alpha^{-1}(k)}+j)$ since
$\rho\in{\rm{Sh}}(\Lambda)$. Thus, $\sigma(t_{k}+i)<\sigma(t_{k}+j)$, and
$\sigma\in{\rm{Sh}}(\Lambda^{\pi})$.

Next, if $i<j$, then

$\rho(s_{\alpha^{-1}(i)}+1)=b_{i}<b_{j}=\rho(s_{\alpha^{-1}(j)}+1).$

Thus, $\sigma(t_{i}+1)<\sigma(t_{j}+1)$, and
$\sigma\in{\rm{Sh}}^{\circ}(\Lambda^{\pi})$.

It remains to show that this decomposition $\rho=\sigma\circ\pi$ is unique.
Suppose $\rho=\sigma^{\prime}\circ\pi^{\prime}$ for some
$\pi^{\prime}\in{\rm{Bl}}(\Lambda)$ and
$\sigma^{\prime}\in{\rm{Sh}}^{\circ}(\Lambda^{\pi^{\prime}})$. As before, let
$\alpha\in S_{K}$ be the permutation satisfying

$\rho(s_{k}+1)=b_{\alpha(k)}$

for $1\leq k\leq K$. Define $c_{k}$ to be the $k^{\rm{th}}$ smallest element
of the set
$\\{\pi^{\prime}(s_{1}+1),\pi^{\prime}(s_{2}+1),\ldots,\pi^{\prime}(s_{K}+1)\\}$.
Since $\sigma^{\prime}\in{\rm{Sh}}^{\circ}(\Lambda^{\pi^{\prime}})$, we have
$\sigma^{\prime}(c_{k})=b_{k}$ for each $1\leq k\leq K$, and thus
$\theta_{\pi^{\prime}}=\alpha=\theta_{\pi}$. Since $\pi$ is completely
determined by $\theta_{\pi}$, we have $\pi=\pi^{\prime}$. Thus,
$\sigma^{\prime}=\pi^{-1}\circ\rho=\sigma$ as desired. ∎

Finally, Lemma 4.1 follows immediately from applying Lemma A.2 after Lemma
A.1.

## References

  * [1] M. Adler, P. J. Forrester, T. Nagao, and P. Van Moerbeke.  Classical skew orthogonal polynomials and random matrices.  Journal of Statistical Physics, 99(1-2):141–170, 2000.
  * [2] C. A. H. Andreief.  Note sur une relation entre les intégrales définies des produits des fonctions, par M.C. Andréief.  G. Gounouilhou, 1884.
  * [3] F. A. Berezin.  The method of second quantization.  Pure and applied physics. Academic Press, 1966.
  * [4] A. Borodin and C. Sinclair.  The ginibre ensemble of real random matrices and its scaling limits.  Communications in Mathematical Physics, 291(1):177–224, 2009.
  * [5] K. T. Chen.  Iterated integrals and exponential homomorphisms.  Proceedings of the London Mathematical Society, s3-4(1):502–512, 1953.
  * [6] N.G. de Bruijn.  On some multiple integrals involving determinants.  Journal of the Indian Mathematical Society. New Series, 19:133–151, 1955.
  * [7] I. Dumitriu and A. Edelman.  Matrix models for beta ensembles.  Journal of Mathematical Physics, 43(11):5830–5847, 2002.
  * [8] Freeman J. Dyson.  Statistical theory of the energy levels of complex systems. i-iii.  Journal of Mathematical Physics, 3:140–175, 1962.
  * [9] Freeman J. Dyson and M. L. Mehta.  Statistical theory of the energy levels of complex systems. iv.  Journal of Mathematical Physics, 3:701–712, 1963.
  * [10] S. Eilenberg and S. Mac Lane.  On the groups of H(pi,n).  Annals of Mathematics, 1953.
  * [11] P. Forrester.  Log-Gases and Random Matrices.  Princeton University Press, 2010.
  * [12] J. Luque and J. Thibon.  Pfaffian and hafnian identities in shuffle algebras.  Advances in Applied Mathematics, 2002.
  * [13] G. Mahoux and M. L. Mehta.  A method of integration over matrix variables.  Journal de Physique I, 1991.
  * [14] M. L. Mehta.  Random matrices.  Elsevier/Academic Press, 2004.
  * [15] C. Meray.  Sur un determinant dont celui de vandermonde n’est qu’un particulier.  Revue de Mathématiques Spéciales, 9:217–219, 1899.
  * [16] J. Ramírez, B. Rider, and B Virág.  Beta ensembles, stochastic airy spectrum, and a diffusion.  Journal of the American Mathematical Society, 24(4):919–944, 2011\.
  * [17] B. Rider, C. Sinclair, and Y. Xu.  A solvable mixed charge ensemble on the line: Global results.  Probability Theory and Related Fields, 155, 2010.
  * [18] C. Shum and C. Sinclair.  A solvable two-charge ensemble on the circle, 2014.
  * [19] C. Sinclair.  Ensemble averages when beta is a square integer.  Monatshefte fur Mathematik, 2011.
  * [20] C. Sinclair.  The partition function of multicomponent log-gases.  Journal of Physics A: Mathematical and Theoretical, 45(16), 2012\.
  * [21] C. A. Tracy and H. Widom.  Correlation functions, cluster functions, and spacing distributions for random matrices.  Journal of Statistical Physics, 1988.
  * [22] E. Wigner.  Characteristic vectors of bordered matrices with infinite dimension.  Annals of Mathematics, 62, 1955.
  * [23] J. Wishart.  The generalized product moment distribution in samples from a normal multivariate population.  Biometrika, 20 A, 1928.