#  Fay identities of Pfaffian type for hyperelliptic curves

Gaëtan Borot111Institut für Mathematik und Institut für Physik, Humboldt-
Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. , Thomas
Buc–d’Alché222UMPA UMR 5669, ENS de Lyon, CNRS; 46, allée d’Italie 69007,
Lyon, France

###### Abstract

We establish identities of Pfaffian type for the theta function associated
with twice or half the period matrix of a hyperelliptic curve. They are
implied by the large size asymptotic analysis of exact Pfaffian identities for
expectation values of ratios of characteristic polynomials in ensembles of
orthogonal or quaternionic self-dual random matrices. We show that they amount
to identities for the theta function with the period matrix of a hyperelliptic
curve, and in this form we reprove them by direct geometric methods.

###### Contents

  1. 1 Introduction
  2. 2 $\beta$-ensembles and their properties
    1. 2.1 The unconstrained model
    2. 2.2 The model with fixed filling fractions
    3. 2.3 Equilibrium measures and their Stieltjes transform
    4. 2.4 Determinantal and pfaffian formulae
  3. 3 Geometry of the spectral curves
    1. 3.1 Theta functions
    2. 3.2 Geometry of Riemann surfaces
    3. 3.3 The spectral curve
    4. 3.4 Characterisation of spectral curves of $\beta$-ensembles
    5. 3.5 Deformations of the curve
  4. 4 Asymptotics of the partition function and the kernels
    1. 4.1 Expansion of the partition function and generalised central limit theorem
    2. 4.2 Three explicit formulae
    3. 4.3 Kernel asymptotics: intermediate computations
  5. 5 Derivation of the theta identities
    1. 5.1 The $\beta=2$ formula
    2. 5.2 The $\beta=1$ formula
    3. 5.3 The $\beta=4$ formula
    4. 5.4 Formula for the multi-cut equilibrium energy (Proof of Proposition 4.3)
  6. A Variation of the entropy with respect to filling fractions (Proof of Proposition 4.4)

#### Acknowledgments

G.B. thanks Semyon Klevtsov for his encouragement to carry out this study.

T.B. is supported by ERC Project LDRAM : ERC-2019-ADG Project 884584.

##  1 Introduction

Fay trisecant identity [Fay70] is a property of the Riemann theta function
associated to the period matrix $\bm{\tau}$ of a compact Riemann surface
$\hat{C}$ of genus $g>0$

$\begin{split}0&=\theta(\bm{v}|\bm{\tau})\theta\big{(}\bm{u}(z_{1})-\bm{u}(z_{3})+\bm{c}\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{u}(z_{2})-\bm{u}(z_{4})+\bm{c}\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{v}+\bm{u}(z_{1})-\bm{u}(z_{2})+\bm{u}(z_{3})-\bm{u}(z_{4})\big{|}\bm{\tau}\big{)}\\\
&\quad-\theta(\bm{u}(z_{1})-\bm{u}(z_{4})+\bm{c}\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{u}(z_{3})-\bm{u}(z_{2})+\bm{c}\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{v}+\bm{u}(z_{1})-\bm{u}(z_{2})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{v}+\bm{u}(z_{3})-\bm{u}(z_{4})\big{|}\bm{\tau}\big{)}\\\
&\quad+\theta\big{(}\bm{u}(z_{1})-\bm{u}(z_{2})+c\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{u}(z_{3})-\bm{u}(z_{4})+\bm{c}\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{v}+\bm{u}(z_{1})-\bm{u}(z_{4})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{v}+\bm{u}(z_{3})-\bm{u}(z_{2})\big{|}\bm{\tau}\big{)}\,,\end{split}$
(1.1)

where $\bm{u}(z_{i})$ is the image via the Abel map of a point $z_{i}$ in the
universal cover $\widetilde{C}$ of $C$ and $\bm{v}$ is an arbitrary
$g$-dimensional vector — all notations appearing in this formula will be
reviewed later. Its proof is an application of basic facts about the geometry
of the Jacobian of $C$. This identity admits a generalisation to a
determinantal identity involving $2n$ points on $\widetilde{C}$.

Fay identity is a showcase of the deep relations between the geometry of
Riemann surfaces and integrability. It is responsible for the existence of the
algebro-geometric solutions to the KP hierarchy [Kri77], which are associated
to any fixed Riemann surface $C$; in this context (1.1) is an equivalent form
of the Hirota equation. The real-valued solutions among those give rise to the
finite-gap potentials for the associated linear differential system, which
historically have played an important role in the study of the KdV and KP
hierarchies [Mat08]. Fay identity is also the basis of a solution to the
Schottky problem: as conjectured by Novikov and proved by Shiota [Shi86]
building on earlier work of Mulase [Mul84], (1.1) characterises period
matrices among complex symmetric matrices $\bm{\tau}$ with positive-definite
imaginary part. In the algebro-geometric solutions, coupling the moduli of $C$
to the KP flows does not give anymore an exact solution, but captures the
long-time asymptotics of more general solutions of KP, see e.g. the
conjectures in [BE12].

The $1$-hermitian matrix model exhibits Toda integrability (closely related to
KP integrability), which manifests itself by the existence of determinantal
formulae to compute various observables, valid for any matrix size $N$. In
particular, the average of any ratio of characteristic polynomials (called
$2n$-point functions) can be expressed as a determinant of expectation values
of the ratio of two characteristic polynomials. The large $N$ asymptotic of
these matrix models has been extensively studied, either by Riemann–Hilbert
methods relying on integrability [Pas06, CGM15, CFWW21], or by probabilistic
techniques [Joh98, APS01, BG13, BG24, Shc13, BGK15]. The multi-cut regime,
when the large $N$ spectral density of the random matrix is supported on $g+1$
segments, is particularly interesting. As observed numerically in [Jur91],
explained heuristically in [BDE00] and justified rigorously in [BG24, Shc13],
the asymptotic behavior is of oscillatory nature. In particular, fluctuations
of linear statistics in the macroscopic regime are asymptotically described as
the independent sum of a Gaussian and a discrete Gaussian living on an
$N$-dependence lattice. This can be precisely described through the Riemann
theta function of the underlying spectral curve, which is hyperelliptic of
genus $g$. Given the parallel with integrability, it should not be a surprise
that the exact determinantal formulae in the hermitian matrix model imply, in
the large $N$ limit up to $o(1)$, the Fay identities (1.1). This implication
will be shown in Proposition 5.2.

The purpose of our work is to generalise this to orthogonal and quaternionic
self-dual 1-matrix models. The determinantal formulae of the hermitian case
for $2n$-point functions of ratios of characteristic polynomials are then
replaced with the Pfaffian formulae found by Borodin and Strahov [BS06]. These
models correspond to the $\beta=1$ and $\beta=4$ cases of the
$\beta$-ensembles, whose asymptotic analysis in the $(g+1)$-cut regime has
been established for all $\beta>0$ by probabilistic techniques in [BG24,
Shc13, BGK15]. The large $N$ spectral density is described by a hyperelliptic
curve of genus $g$ independent of $\beta$ and having a period matrix
$\bm{\tau}$. The asymptotics of the partition function and the $2n$-point
functions are governed by the theta function associated with the matrix
$\frac{\beta}{2}\bm{\tau}$. The appearance of the theta function does not have
a geometric origin111The asymptotic analysis carried via the Riemann–Hilbert
method does give a geometric origin to the theta function, but it is only
applied to the hermitian case, i.e. $\beta=2$. but is rather explained by
eigenvalue tunnelling between the different connected components of the
support [BDE00]. Inserting these asymptotics up to $o(1)$ in the Pfaffian
identities for the $2n$-point functions yield identities between these theta
functions, which can be expressed solely in terms of the geometry of the
underlying spectral curve.

The spectral curves arising from the large $N$ limit of the matrix models we
consider must be hyperelliptic, have real Weierstraß points, and have the
Boutroux property. We show that all such curves can be realised as the
spectral curve of an off-critical $\beta$-ensemble with polynomial potential
(Proposition 3.11). By analytic continuation we can extend the validity of the
resulting identities to all hyperelliptic curves. This gives our main results:
Theorem 5.1 for $\beta=2$, Theorem 5.4 for $\beta=1$ and Theorem 5.7 for
$\beta=4$. It turns out that all three identities can be reformulated in terms
of theta functions for the matrix of periods $\bm{\tau}$ (instead of
$\frac{\beta}{2}\bm{\tau}$) and in this form we are able to prove them by
direct algebraic methods. Interestingly, the $\beta=1$ and $\beta=4$
identities are equivalent via the modular properties of theta functions, and
the $\beta=2$ identity implies the Fay identity in the special case of
hyperelliptic curves.

As a byproduct of our proofs, we obtain a seemingly new formula (Proposition
4.3 proved in Section 5.4) for the equilibrium energy of the $\beta$-ensembles
in the multi-cut regime in terms of the geometry of the spectral curve.
Although the ingredients are the same, at first sight it does not have exactly
the same form as the 1-matrix model specialisation of the formula known in the
context of the $2$-matrix model [Ber03]. Independently of our analysis, we
also establish (Proposition 4.4 proved in Appendix A) an explicit formula for
the derivative with respect to filling fractions of the equilibrium entropy.
For $\beta\neq 2$, the equilibrium entropy appears as the order $N$ term in
the free energy of the $\beta$-ensemble, and its derivatives with respect to
filling fractions appear both in the asymptotics of the partition function
(Theorem 4.1) and in the centering in the generalised central limit theorem
(Theorem 4.2).

The strategy of proof via asymptotics in integrable random matrix ensembles is
somehow more interesting than the resulting identities in the particular case
we studied, and constitutes the originality of this study. In principle this
strategy can be applied to any random matrix ensemble:

  * (i)

which is amenable to asymptotic analysis up to $o(1)$ in the large size limit
and in the multi-cut regime;

  * (ii)

in which exact formulae for $2n$-point functions in terms of $k$-point
functions (with $k$ independent of $n$) are available.

In principle, more general algebraic spectral curves can be obtained in two-
matrix models or in linearly coupled chain of matrices. In such models, (ii)
is addressed by the Eynard–Mehta formulae [EM98] but obtaining (i) already for
the two-matrix model away from critical points in the multi-cut regime is a
notoriously hard open problem. More general algebraic spectral curves can also
be obtained in repulsive particle systems with $d$ groups of particles, in
which the repulsion intensity between particles of groups $i$ and $j$ is
$\beta_{i,j}$. The latter appear naturally in various situations (see e.g.
[BEO15, BE17]) and (i) has been addressed by [BGK15]. Their discrete
counterpart appears in models related to random two-dimensional tilings and is
also amenable to asymptotic analysis [BGG]. We expect that in some of these
models, integrability properties (ii) should exist, and will therefore imply
identities between theta functions with a more complicated structure and that
depend on a larger class of algebraic curves. By this we mean the matrix
$\bm{\tau}$ in the theta function will be specified from the geometry of these
curves, although it may not exactly be the matrix of periods. This would in
fact be the interesting situation, as the corresponding theta function is then
associated to an abelian variety which may not be a Jacobian. Whether
genuinely new identities for theta functions of certain abelian varieties —
i.e. for which a proof by direct algebraic methods is not easily available —
as consequences of the integrability of probabilistic models can be obtained
is a question left to future investigations.

##  2 $\beta$-ensembles and their properties

We recall a few facts about the $\beta$-ensembles and review the determinantal
and pfaffian formulae of Borodin and Strahov [BS06].

###  2.1 The unconstrained model

Fix a finite union $A$ of compact intervals of $\mathbb{R}$, a positive
integer $N$, a real number $\beta>0$, and an even-degree polynomial $V$ (the
potential) with real coefficients and positive top coefficient. We consider
the probability measure $\mu_{N}^{V}$ on $A^{N}$ defined by

$\begin{split}\differential\mathbb{P}_{N}^{V}(\bm{\lambda})&=\frac{1}{Z_{N}^{V}}\,|\Delta(\bm{\lambda})|^{\beta}\mathrm{e}^{-\frac{\beta
N}{2}\sum_{i=1}^{N}V(\lambda_{i})}\prod_{i=1}^{N}\mathds{1}_{A}(\lambda_{i})\differential\lambda_{i},\end{split}$
(2.1)

where $\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{N})\in A^{N}$,
$\Delta(\bm{\lambda})=\prod_{i<j}(\lambda_{j}-\lambda_{i})$ is the Vandermonde
determinant, and

$\begin{split}Z_{N}^{V}&=\int_{A^{N}}|\Delta(\bm{\lambda})|^{\beta}\exp\bigg(-\frac{\beta
N}{2}\sum_{i=1}^{N}V(\lambda_{i})\bigg{missing})\prod_{i=1}^{N}\differential\lambda_{i}\end{split}$

is the partition function. Many results that we quote are formulated with
$A=\mathbb{R}$ but their validity trivially extend to the case of $A$ compact.
We choose to work from the start with a compact $A$ as it facilitates the
statement of the asymptotic results we will need and does not lead to any loss
of generality.

When $\beta=1,2\text{ or }4$, $\mathbb{P}_{N}^{V}$ is the distribution of the
$N$ eigenvalues of a random matrix whose law is proportional to
$e^{-\frac{\beta N}{2}V(\bm{M})}\differential\bm{M}$, where $\bm{M}$ is a
matrix which is real symmetric ($\beta=1$), Hermitian ($\beta=2$) or
quaternionic self-dual ($\beta=4$) and which is conditioned to have spectrum
in $A$. The measure $\differential\bm{M}$ is the product of Lebesgue measure
on the $\mathbb{R}$-linearly independent entries of $\bm{M}$. In particular,
when $V(\bm{M})=\frac{1}{2}\bm{M}^{2}$, the entries $M_{ij},i\leq j$ of the
matrix are independent Gaussian random variables. These matrix ensembles are
known under the name of Gaussian Orthogonal Ensemble (GOE) for $\beta=1$,
Gaussian Unitary Ensemble (GUE) for $\beta=2$, and Gaussian Symplectic
Ensemble (GSE) for $\beta=4$, see [Meh04]. The $\beta$-ensembles (2.1)
constitute a generalisation of these models.

###  2.2 The model with fixed filling fractions

Let us write $A=\bigsqcup_{h=0}^{g}A_{h}$ where $A_{h}$ are the connected
components of $A$. In addition to the measure (2.1), we define the
$\beta$-ensemble with fixed filling fractions as follows. Let
$\bm{N}=(N_{h})_{h=1}^{g}\in\mathbb{Z}_{>0}^{g}$ such that
$N_{1}+\cdots+N_{g}<N$ and introduce $N_{0}\in\mathbb{Z}_{>0}$ such that

$\begin{split}N_{0}+\cdots+N_{g}=N.\end{split}$

We call $\epsilon_{h}=N_{h}/N$ the filling fraction of $A_{h}$, and let
$\bm{\epsilon}=(\epsilon_{h})_{h=1}^{g}$. We define the measure with fixed
filling fractions by

$\begin{split}\differential\mathbb{P}_{N,\bm{N}/N}^{V}&=\frac{1}{Z^{V}_{N,\bm{N}/N}}|\Delta(\bm{\lambda})|^{\beta}\exp\bigg(-\frac{\beta
N}{2}\sum_{h=0}^{g}\sum_{i=1}^{N_{h}}V(\lambda_{h,i})\bigg{missing})\prod_{h=0}^{g}\prod_{i=1}^{N_{h}}\mathds{1}_{A_{h}}(\lambda_{h,i})\differential\lambda_{h,i},\end{split}$
(2.2)

where $\bm{\lambda}=(\lambda_{h,1},\ldots,\lambda_{h,N_{h}})_{h=0}^{g}$ is a
$N$-tuple and

$\begin{split}Z^{V}_{N,\bm{N}/N}&=\int_{A^{N}}|\Delta(\bm{\lambda})|^{\beta}\mathrm{e}^{-\frac{\beta
N}{2}\sum_{h=0}^{g}\sum_{i=1}^{N_{h}}V(\lambda_{h,i})}\prod_{h=0}^{g}\prod_{i=1}^{N_{h}}\mathds{1}_{A_{h}}(\lambda_{h,i})\differential\lambda_{h,i}\end{split}$

is the partition function for fixed filling fractions. To distinguish it from
the model with fixed filling fractions, we refer to (2.1) as the unconstrained
model.

###  2.3 Equilibrium measures and their Stieltjes transform

We define the empirical measure as
$L_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$. It belongs to the space
of probability measures on $A$, which we equip with the weak topology. We
first consider $L_{N}$ in the original model. The following result comes from
large deviation arguments.

######  Theorem 2.1.

Assume that $V$ is an even-degree real polynomial with positive top
coefficient. As $N\rightarrow\infty$, $L_{N}$ converges under
$\mathbb{P}_{N}^{V}$ almost surely and in expectation to the unique
probability measure $\mu_{\text{eq}}$ on $A$ maximising

$\begin{split}\mathcal{E}[\mu]&=\frac{\beta}{2}\iint_{A^{2}}\bigg{(}\ln|\xi-\eta|-\frac{V(\xi)+V(\eta)}{2}\bigg{)}\differential\mu(\xi)\differential\mu(\eta).\end{split}$
(2.3)

Furthermore, $\mu_{\text{eq}}$ has compact support $S$ consisting in a finite
union of segments. It is characterised by the existence of a constant $c$ such
that

$\begin{split}\forall x\in A\qquad
2\int_{A}\ln|x-\xi|\differential\mu_{\text{eq}}(\xi)-V(x)\leq c,\end{split}$

with equality $\mu_{\text{eq}}$-almost everywhere.

We will only need to consider $S=\bigsqcup_{h=0}^{g}S_{h}$ where
$S_{h}\subset\mathring{A}_{h}$ is a segment, and in particular we only
consider the case of soft edges. Without loss of generality one can and one
will restrict $A_{h}$ to be a small enlargement of $S_{h}$. The choice of this
enlargement will be irrelevant for our purposes, as it does not change the
equilibrium measure and only affects the model by corrections which are
exponentially small in $N$. In the model with fixed filling fractions, Theorem
2.1 has the following adaptation.

######  Theorem 2.2.

[BG24, Theorem 1.2] Consider a sequence indexed by $N$ of $g$-tuples of
nonnegative integers $\bm{N}=(N_{1},\ldots,N_{g})$ with
$\sum_{h=1}^{g}N_{h}<N$ and assume there exists
$\bm{\epsilon}=(\epsilon_{h})_{1=0}^{g}$ such that $N_{h}/N\to\epsilon_{h}$
for all $h\in[g]$. Then,
$L_{N}=\frac{1}{N}\sum_{h=0}^{g}\sum_{i=1}^{N_{h}}\delta_{\lambda_{h,i}}$
converges almost surely and in expectation under $\mathbb{P}^{V}_{N,\bm{N}/N}$
towards a deterministic probability measure $\mu_{\text{eq},\bm{\epsilon}}$,
which is the maximiser of (2.3) among probability measures giving mass
$\epsilon_{h}$ to the segment $A_{h}$ for each $h\in[0,g]$. It is
characterised by the existence of constants $(c_{h})_{h=0}^{g}$ such that

$\forall h\in[0,g]\quad\forall x\in A_{h}\qquad
2\int_{A}\ln|x-\xi|\differential\mu_{\text{eq},\bm{\epsilon}}(\xi)-V(x)\leq
c_{h}$

with equality $\mu_{\text{eq},\bm{\epsilon}}|_{A_{h}}$-almost everywhere.

The filling fractions at equilibrium
$\bm{\epsilon}^{*}=(\epsilon_{h}^{*})_{h=1}^{g}$ are defined as
$\epsilon_{h}^{*}=\mu_{\text{eq}}(A_{h})$, and one can show that
$\mu_{\text{eq}}=\mu_{\text{eq},\bm{\epsilon}^{*}}$ — see [BG24, Section 1.4].

Let us now discuss the properties of the equilibrium measure, both in the
unconstrained case (Theorem 2.1) or fixed filling fraction case (Theorem 2.2)
. We introduce the Stieltjes transform of the equilibrium measure

$\begin{split}W_{1}(x)&=\int_{A}\frac{\differential\mu_{\text{eq}}(\xi)}{x-\xi},\end{split}$

defined for $x\in\mathbb{C}\setminus S$. In [Joh98], Johansson introduces the
polynomial

$\begin{split}P(x)&=\int_{A}\frac{V^{\prime}(x)-V^{\prime}(\xi)}{x-\xi}\differential\mu_{\text{eq}}(\xi),\end{split}$

and derives the equation

$\begin{split}W_{1}(x)^{2}-V^{\prime}(x)W_{1}(x)+P(x)=0,\end{split}$ (2.4)

for all $x\in\mathbb{C}\setminus S$. This equation is the large $N$ limit of
the first Dyson–Schwinger of the model, and its origin can be traced back to
[BIPZ78, Mig83]. In particular, it implies that

$\begin{split}W_{1}(x)&=\frac{V^{\prime}(x)}{2}\pm\frac{\sqrt{V^{\prime}(x)^{2}-4P(x)}}{2}.\end{split}$
(2.5)

The determination of the squareroot should be chosen such that
$W_{1}(x)\sim\frac{1}{x}$ as $x\rightarrow\infty$ and $W_{1}$ is holomorphic
in $\mathbb{C}\setminus S$. As this determination plays an important role in
our discussion, it is worth reviewing in details how this can be achieved. The
standard determination of the squareroot gives a holomorphic function
$x\mapsto\sqrt{x}$ on $\mathbb{C}\setminus\mathbb{R}_{\leq 0}$ such that
$\sqrt{\mathbb{R}_{>0}}=\mathbb{R}_{>0}$ and $(\sqrt{x})^{2}=x$. We decompose
$V^{\prime}(x)^{2}-4P(x)=M(x)^{2}\sigma(x)$, where $\sigma$ is a monic
polynomial with simple real roots and $M$ is a real polynomial with positive
top coefficient. We write further

$\sigma(x)=\prod_{h=0}^{g}(x-a_{h})(x-b_{h}),$

with:

$a_{0}<b_{0}<a_{1}<b_{1}<\cdots<a_{g}<b_{g},$

The locus $\sigma^{-1}(\mathbb{C}\setminus\mathbb{R}_{\leq 0})$ is a union of
$g+2$ connected components, labeled from left to right: $C_{0}$ contains
$a_{0}$ in its closure, $C_{h}$ contains $b_{h}$ and $a_{h+1}$ in its closure
for $h\in[g-1]$, and $C_{g+1}$ contains $b_{g}$ in its closure. For $x\in
C_{h}$ with $h\in[0,g+1]$ we set $s(x)=(-1)^{g+1-h}\sqrt{\sigma(x)}$. This
definition makes $s(x)$ a continuous (thus holomorphic) function of
$x\in\mathbb{C}\setminus S$. It is discontinuous on $(a_{h},b_{h})$ because by
crossing this segment we stay in the same component, so we keep the same
global sign coming from the component we are in while the standard
determination of the squareroot does get a sign change since $\sigma(x)$
crosses $\mathbb{R}_{<0}$. Then, $M(x)s(x)$ is a holomorphic function of
$x\in\mathbb{C} \setminus S$, and $M(x)s(x)\sim tx^{d-1}$ for some $d\geq 2$
and $t>0$. The constraint $W_{1}(x)\sim\frac{1}{x}$ as $x\rightarrow\infty$
leads to the formula:

$W_{1}(x)=\frac{V^{\prime}(x)-M(x)s(x)}{2}.$ (2.6)

The fact that $W_{1}$ is the Stieltjes transform of the equilibrium measure
puts some constraints on the polynomial $M$.

######  Lemma 2.3.

The support of $\mu_{\text{eq}}$ is $S=\bigsqcup_{h=0}^{g}S_{h}$ with
$S_{h}=[a_{h},b_{h}]$ and we have

$\frac{\differential\mu_{\text{eq}}}{\differential
x}=\frac{M(x)\text{Im}(s(x+{\rm i}0))}{2\pi}\mathds{1}_{S}(x).$

For each $h\in[g]$, the number of zeros (with multiplicity) of $M$ in
$[b_{h-1},a_{h}]$ is odd. For each $h\in[0,g]$, the zeros of $M$ in
$(a_{h},b_{h})$ have even multiplicity (if there is any).

###### Proof.

By construction, for any $h\in[0,g]$ and $x\in(a_{h},b_{h})$ we have $s(x+{\rm
i}0)\in(-1)^{g-h}{\rm i}\mathbb{R}_{>0}$ and for any $h\in[0,g+1]$ and
$x\in(b_{h-1},a_{h})$ we have $s(x)\in(-1)^{g+1-h}\mathbb{R}_{>0}$, with the
conventions $b_{-1}=-\infty$ and $a_{g+1}=+\infty$. By definition of the
Stieltjes transform, the function $W_{1}(x)$ has a discontinuity in the
interior of the support of $\mu_{\text{eq}}$. It is identified as the _real_
locus where the polynomial $V^{\prime}(x)^{2}-4P(x)$ takes nonpositive values,
and thus coincides with $S=\bigsqcup_{h=0}^{g}[a_{h},b_{h}]$. The density of
the equilibrium measure is reconstructed from the jump:

$\forall
x\in\mathbb{R},\qquad\frac{\differential\mu_{\text{eq}}}{\differential
x}=\frac{W_{1}(x-{\rm i}0)-W_{1}(x+{\rm i}0)}{2{\rm i}\pi}=\frac{M(x)s(x+{\rm
i}0)}{2{\rm i}\pi}\mathds{1}_{S}(x)=\frac{M(x)\text{Im}(s(x+{\rm
i}0))}{2\pi}\mathds{1}_{S}(x).$

Since $\mu_{\text{eq}}$ is a positive measure and since $\text{Im}(s(x+{\rm
i}0))$ has a constant sign in each $(a_{h},b_{h})$, $M$ should have constant
sign in $S_{h}$ and thus have zeros of even multiplicity there (if there is
any). Likewise, since the sign of $\text{Im}(s(x+{\rm i}0))$ changes between
two consecutive segments in the support, $M$ should have at least a sign
change in the closure of the interval between these two segments, hence an odd
number of zeros. ∎

Lemma 2.3 allows us to give an expression for the density $\rho$ of the
equilibrium measure. Indeed,

$\rho(x)=\frac{W_{1}(x-i0)-W_{1}(x+i0)}{2i\pi}=M(x)\frac{s(x+i0)-s(x-i0)}{4\pi}\mathds{1}_{S}(x)=\frac{\sqrt{-M(x)\sigma(x)}}{2\pi}\mathds{1}_{S}(x)\,.$
(2.7)

######  Definition 2.4.

The effective potential is defined for $x\in A$ by
$U(x):=V(x)-2\int_{A}\ln|x-\xi|\differential\mu_{\text{eq}}(\xi)$. It
satisfies

$\begin{split}\forall x\in A\setminus S&\qquad
U^{\prime}(x)=V^{\prime}(x)-2W_{1}(x)=M(x)s(x),\\\ \forall
x\in\mathring{S}&\qquad U^{\prime}(x)=V^{\prime}(x)-W_{1}(x+{\rm
i}0)-W_{1}(x-{\rm i}0)=0.\end{split}$ (2.8)

######  Remark 2.5.

For the equilibrium measure in the unconstrained case, Theorem 2.1 says there
exists a constant $c$ such that $U(x)=c$ for all $x\in S$. In view of (2.8)
the latter property is equivalent to

$\forall h\in[g]\qquad\int_{b_{h-1}}^{a_{h}}M(x)s(x)\differential x=0.$ (2.9)

###  2.4 Determinantal and pfaffian formulae

Expectation values of ratios of characteristic polynomials, also called
kernels, are quantities of interest in random matrix theory. Let us introduce
the notation $\left\langle{\cdot}\right\rangle^{V}_{N}$ for the expectation
value with respect to $\mathbb{P}_{N}^{V}$ (the value of $\beta$ will be
specified in each case), and $\Lambda=\text{diag}(\bm{\lambda})$. Given
$c_{1},\ldots,c_{m}\in\mathbb{Z}$, and $x_{1},\ldots,x_{m}\in\mathbb{C}$ with
the condition $x_{j}\notin A$ if $c_{j}<0$, the $m$-point kernel is defined as

$\bigg{\langle}\prod_{j=1}^{m}\det(x_{j}-\Lambda)^{c_{j}}\bigg{\rangle}_{N}^{V}=\bigg{\langle}\prod_{j=1}^{m}\prod_{i=1}^{N}(x_{j}-\lambda_{i})^{c_{j}}\bigg{\rangle}_{N}^{V}.$

In [BS06], Borodin and Strahov derive formulae to compute the kernels. In what
follows, we will always consider the “balanced” case, that is, when there are
as many characteristic polynomials in the numerator as in the denominator.
Given two tuples of complex numbers $\bm{x}=(x_{1},\ldots,x_{m_{1}})$ and
$\bm{\tilde{x}}=(\tilde{x}_{1},\ldots,\tilde{x}_{m_{2}})$, we write

$\begin{split}\Delta(\bm{x},\bm{\tilde{x}})=\prod_{i=1}^{m_{1}}\prod_{j=1}^{m_{2}}(x_{i}-\tilde{x}_{j}).\end{split}$

####  2.4.1 The determinantal case: $\beta=2$

In that case, we have the following formulae.

######  Theorem 2.6.

[BS06, Theorem 4.1.1] Let $N,m_{1},m_{2}$ be positive integers, and sets of
complex numbers

$\begin{split}\bm{x}=\\{x_{1},\ldots,x_{m_{1}}\\}&\qquad\bm{x}^{\prime}=\\{x^{\prime}_{1},\dots,x^{\prime}_{m_{1}}\\}\\\
\tilde{\bm{x}}=\\{\tilde{x}_{1},\ldots,\tilde{x}_{m_{2}}\\}&\qquad\tilde{\bm{x}}^{\prime}=\\{\tilde{x}^{\prime}_{1},\dots,\tilde{x}^{\prime}_{m_{2}}\\},\end{split}$

such that

$\begin{split}\bm{x}\cap\bm{x}^{\prime}=\varnothing,\qquad\bm{y}\cap\bm{y}^{\prime}=\varnothing,\qquad\bm{x}^{\prime}\cap
A=\varnothing,\qquad\bm{y}^{\prime}\cap A=\varnothing.\end{split}$

We have:

$\begin{split}\left\langle{\prod_{j=1}^{m_{1}}\frac{\det(x_{j}-\Lambda)}{\det(x^{\prime}_{j}-\Lambda)}\prod_{j=1}^{m_{2}}\frac{\det(\tilde{x}_{j}-\Lambda)}{\det(\tilde{x}^{\prime}_{j}-\Lambda)}}\right\rangle_{N}^{V}&=(-1)^{\frac{1}{2}((m_{1}+m_{2})^{2}+m_{2}-m_{1})}\frac{\Delta(\bm{x},\bm{x}^{\prime})}{\Delta(\bm{x})\Delta(\bm{x}^{\prime})}\frac{\Delta(\tilde{\bm{x}},\tilde{\bm{x}}^{\prime})}{\Delta(\tilde{\bm{x}})\Delta(\tilde{\bm{x}}^{\prime})}\\\
&\times\det\big(\mathcal{M}^{(2)}(\bm{x},\bm{x}^{\prime};\tilde{\bm{x}},\tilde{\bm{x}}^{\prime})\big{missing}),\end{split}$
(2.10)

in terms of the block matrix of size $(m_{1}+m_{2})$:

$\mathcal{M}^{(2)}(\bm{x},\bm{x}^{\prime};\tilde{\bm{x}},\tilde{\bm{x}}^{\prime})=\left(\begin{array}[]{cc}\mathcal{M}^{(2)}_{++}(x_{i},\tilde{x}_{j})&\mathcal{M}^{(2)}_{+-}(x_{i},x_{j}^{\prime})\\\
\mathcal{M}_{-+}^{(2)}(\tilde{x}_{i}^{\prime},\tilde{x}_{j})&\mathcal{M}^{(2)}_{--}(\tilde{x}_{i}^{\prime},x_{j}^{\prime})\end{array}\right),$

where $i$ is a row index, $j$ a column index, and the entries are:

$\begin{split}\mathcal{M}_{++}^{(2)}(x,\tilde{x})&=N\frac{Z_{N-1}^{\frac{N}{N-1}V}}{Z_{N}^{V}}\left\langle{\det(x-\Lambda)\det(\tilde{x}-\Lambda)}\right\rangle_{N-1}^{\frac{N}{N-1}V},\\\
\mathcal{M}^{(2)}_{+-}(x,x^{\prime})&=\frac{1}{x-x^{\prime}}\left\langle{\frac{\det(x-\Lambda)}{\det(x^{\prime}-\Lambda)}}\right\rangle_{N}^{V},\\\
\mathcal{M}^{(2)}_{-+}(\tilde{x}^{\prime},\tilde{x})&=\frac{1}{\tilde{x}^{\prime}-\tilde{x}}\left\langle{\frac{\det(\tilde{x}-\Lambda)}{\det(\tilde{x}^{\prime}-\Lambda)}}\right\rangle_{N}^{V},\\\
\mathcal{M}^{(2)}_{--}(\tilde{x}^{\prime},x^{\prime})&=\frac{1}{N+1}\frac{Z_{N+1}^{\frac{N}{N+1}V}}{Z_{N}^{V}}\left\langle{\frac{1}{\det(\tilde{x}^{\prime}-\Lambda)\det(x^{\prime}-\Lambda)}}\right\rangle_{N+1}^{\frac{N}{N+1}V}.\end{split}$

####  2.4.2 For orthogonal matrices: $\beta=1$

######  Theorem 2.7.

[BS06, Theorem 1.2.1] Let $N,m$ be positive integers and set of complex
numbers

$\bm{x}=\\{x_{1},\ldots,x_{m}\\},\qquad\bm{x}^{\prime}=\\{x^{\prime}_{1},\ldots,x_{m}^{\prime}\\},$

such that $\bm{x}^{\prime}\cap A=\varnothing$. We have:

$\begin{split}\left\langle{\prod_{j=1}^{m}\frac{\det(x_{j}-\Lambda)}{\det(x_{j}^{\prime}-\Lambda)}}\right\rangle_{2N}^{V}&=\frac{\Delta(\bm{x},\bm{x}^{\prime})}{\Delta(\bm{x})\Delta(\bm{x}^{\prime})}\operatorname{Pf}\big{(}\mathcal{M}^{(1)}(\bm{x},\bm{x}^{\prime})\big{)},\end{split}$
(2.11)

in terms of the antisymmetric matrix of size $2m$:

$\mathcal{M}^{(1)}(\bm{x},\bm{x}^{\prime})=\left(\begin{array}[]{cc}\mathcal{M}^{(1)}_{++}(x_{i},x_{j})&\mathcal{M}^{(1)}_{+-}(x_{i},x_{j}^{\prime})\\\
\mathcal{M}^{(1)}_{-+}(x_{i}^{\prime},x_{j})&\mathcal{M}^{(1)}_{--}(x_{i}^{\prime},x_{j}^{\prime})\end{array}\right),$
(2.12)

with entries

$\begin{split}\mathcal{M}^{(1)}_{++}(x,\tilde{x})&=(2N-1)2N(x-\tilde{x})\frac{Z^{\frac{2N}{2N-2}V}_{2N-2}}{Z^{V}_{2N}}\left\langle{\det(x-\Lambda)\det(\tilde{x}-\Lambda)}\right\rangle^{\frac{2N}{2N-2}V}_{2N-2},\\\
\mathcal{M}^{(1)}_{+-}(x,x^{\prime})&=\frac{1}{x-x^{\prime}}\left\langle{\frac{\det(x-\Lambda)}{\det(x^{\prime}-\Lambda)}}\right\rangle^{V}_{2N}=-\mathcal{M}^{(1)}_{-+}(x^{\prime},x),\\\
\mathcal{M}_{--}^{(1)}(x^{\prime},\tilde{x}^{\prime})&=\frac{x^{\prime}-\tilde{x}^{\prime}}{(2N+1)(2N+2)}\frac{Z^{\frac{2N}{2N+2}V}_{2N+2}}{Z^{V}_{2N}}\left\langle{\frac{1}{\det(x^{\prime}-\Lambda)\det(\tilde{x}^{\prime}-\Lambda)}}\right\rangle^{\frac{2N}{2N+2}V}_{2N+2}.\\\
\end{split}$ (2.13)

####  2.4.3 For symplectic matrices: $\beta=4$

The case $\beta=4$ is very similar to the case $\beta=1$.

######  Theorem 2.8.

[BS06, Theorem 1.2.1] Let $N,m$ be positive integers and two sets of complex
numbers

$\bm{x}=\\{x_{1},\ldots,x_{m}\\},\qquad\bm{x}^{\prime}=\\{x^{\prime}_{1},\ldots,x^{\prime}_{m}\\},$

such that $\bm{x}^{\prime}\cap A=\varnothing$. We have.

$\begin{split}\left\langle{\prod_{j=1}^{m}\frac{\det(x_{j}-\Lambda)^{2}}{\det(x^{\prime}_{j}-\Lambda)^{2}}}\right\rangle_{N}^{V}&=\frac{\Delta(\bm{x},\bm{x}^{\prime})}{\Delta(\bm{x})\Delta(\bm{x}^{\prime})}\operatorname{Pf}\big{(}\mathcal{M}^{(4)}(\bm{x},\bm{x}^{\prime})\big{)},\end{split}$
(2.14)

with the same block structure as (2.12) but entries

$\begin{split}\mathcal{M}^{(4)}_{++}(x,\tilde{x})&=N\frac{Z^{\frac{N}{N-1}V}_{N-1}}{Z^{V}_{N}}(x-\tilde{x})\left\langle{\det(x-\Lambda)^{2}\det(\tilde{x}-\Lambda)^{2}}\right\rangle^{\frac{N}{N-1}V}_{N-1},\\\
\mathcal{M}^{(4)}_{+-}(x,x^{\prime})&=\frac{1}{x-x^{\prime}}\left\langle{\frac{\det(x-\Lambda)^{2}}{\det(x^{\prime}-\Lambda)^{2}}}\right\rangle^{V}_{N}=-\mathcal{M}^{(4)}_{-+}(x^{\prime},x),\\\
\mathcal{M}^{(4)}_{--}(x^{\prime},\tilde{x}^{\prime})&=\frac{1}{N+1}\frac{Z^{\frac{N}{N+1}V}_{N+1}}{Z^{V}_{N}}(x^{\prime}-\tilde{x}^{\prime})\left\langle{\frac{1}{\det(x^{\prime}-\Lambda)^{2}\det(\tilde{x}^{\prime}-\Lambda)^{2}}}\right\rangle^{\frac{N}{N+1}V}_{N}.\\\
\end{split}$ (2.15)

##  3 Geometry of the spectral curves

This section collects the information on theta functions and geometry of the
spectral curve that will be needed later to present the large $N$ asymptotics
in the $\beta$-ensembles. We only give the details necessary to understand the
formulae of Section 4 and 5 in a self-contained way, their derivation is
classical and omitted.

###  3.1 Theta functions

Let us recall the definition and properties of the theta function.

######  Definition 3.1.

Let $\bm{\tau}$ be a complex $g\times g$ symmetric matrix such that
$\operatorname{Im}\bm{\tau}$ is definite positive. The theta function with
characteristics $\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$ is the function defined
by

$\forall\bm{z}\in\mathbb{C}^{g}\qquad\vartheta_{\bm{\mu},\bm{\nu}}(\bm{z}|\bm{\tau})=\sum_{\bm{n}\in\mathbb{Z}^{g}}\exp\big({\rm
i}\pi(\bm{n}+\bm{\mu})\cdot\bm{\tau}(\bm{n}+\bm{\mu})+2{\rm
i}\pi(\bm{n}+\bm{\mu})\cdot(\bm{z}+\bm{\nu})\big{missing}).$

We set $\theta:=\vartheta_{\bm{0},\bm{0}}$.

The condition $\operatorname{Im}\bm{\tau}>0$ ensures that the function is well
defined. Let us define the period lattice associated to $\bm{\tau}$ as
$\mathbb{L}=\mathbb{Z}^{g}\oplus\bm{\tau}(\mathbb{Z}^{g})$. The theta function
is quasi-periodic: for $\bm{m},\bm{n}\in\mathbb{Z}^{g}$ we have
$\bm{m}+\bm{\tau}(\bm{n})\in\mathbb{L}$, and for any $\bm{z}\in\mathbb{C}^{g}$

$\begin{split}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{z}+\bm{m}+\bm{\tau}(\bm{n})\big{|}\bm{\tau}\big{)}=e^{2{\rm
i}\pi\bm{m}\cdot\bm{\mu}-{\rm
i}\pi\bm{n}\cdot(\bm{\tau}(\bm{n})+2\bm{z}+2\bm{\nu})}\vartheta_{\bm{\mu},\bm{\nu}}(\bm{z}).\end{split}$

######  Definition 3.2.

An odd half-integer characteristic is
$\bm{c}=\frac{1}{2}\bm{e}+\frac{1}{2}\bm{e}^{\prime}$, with
$\bm{e},\bm{e}^{\prime}\in\mathbb{Z}^{g}$ such that
$\bm{e}\cdot\bm{e}^{\prime}\in 2\mathbb{Z}+1$.

By direct computation, if $\bm{c}$ is a odd half-integer characteristic, then
$\bm{z}\mapsto\theta(\bm{c}+\bm{z}|\bm{\tau})$ is odd, in particular
$\theta(\bm{c})=0$.

###  3.2 Geometry of Riemann surfaces

####  3.2.1 Basis of cycles and forms

Let $\hat{C}$ be a compact Riemann surface of genus $g$. The first homology
group $H_{1}(\hat{C};\mathbb{Z})$ has a basis
$(\mathcal{A}_{h},\mathcal{B}_{h})_{h=1}^{g}$ which can be chosen such that

$\begin{split}\forall
h,k\in[g]\qquad\mathcal{A}_{h}\cap\mathcal{A}_{k}=0,\qquad\mathcal{B}_{h}\cap\mathcal{B}_{k}=0,\qquad\mathcal{A}_{h}\cap\mathcal{B}_{k}=\delta_{h,k}.\end{split}$

Such a basis is called a symplectic basis of homology, and $\hat{C}$ equipped
with such a basis is called a marked Riemann surface. We can for instance
choose a point $p_{0}$ and simple closed curves on $\hat{C}$ representing the
$2g$ classes $(\mathcal{A}_{h},\mathcal{B}_{h})_{h=1}^{g}$ such that all the
curves intersect each other at $p_{0}$ only. For spectral curves of
$\beta$-ensembles, we will later work with another set of representatives
(Section 3.3.1). We keep the same notation for homology classes and their
representatives. The surface
$\hat{C}^{0}=\hat{C}\setminus\bigcup_{h=1}^{g}(\mathcal{A}_{h}\cup\mathcal{B}_{h})$
is then simply-connected. The $\mathcal{A}$-cycles determine a dual basis of
holomorphic 1-forms $(\differential u_{h})_{h=1}^{g}$, such that

$\begin{split}\forall h,k\in[g]\qquad\oint_{\mathcal{A}_{h}}\differential
u_{k}=\delta_{h,k}.\end{split}$

The matrix of periods $\bm{\tau}$ is then defined by

$\forall h,k\in[g]\qquad\tau_{h,k}=\oint_{\mathcal{B}_{h}}\differential
u_{k}.$ (3.1)

It is symmetric and definite positive, in particular we can consider the theta
function with matrix $\frac{\beta}{2}\bm{\tau}$ for any $\beta>0$. The theta
function with matrix equal to (3.1) is called the Riemann theta function.

####  3.2.2 Abel map

With the 1-forms $(\differential u_{h})_{h=1}^{g}$ defined in Section 3.2.1 we
can introduce the Abel map.

######  Definition 3.3.

Choose a base point $p_{0}$ in $\hat{C}$. The Abel map
$\bm{u}\colon\hat{C}^{0}\to\mathbb{C}^{g}$ is defined by

$\begin{split}u_{i}(z)&=\int_{p_{0}}^{z}\differential u_{i},\end{split}$

where the path of integration is in $C^{0}$.

The definition of the Abel map depends on a choice of base point $p_{0}$.
However, we will often consider differences $\bm{u}(z)-\bm{u}(w)$ of Abel
maps, which are independent of $p_{0}$. Depending on the context, we may also
consider the Abel map as a map
$\bm{u}\colon\widetilde{C}\rightarrow\mathbb{C}^{g}$ defined on the universal
cover $\widetilde{C}$ of $\hat{C}$ based at $p_{0}$. We say that $\bm{c}$ is
nonsingular if
$\theta\big{(}\bm{c}+\bm{u}(z)-\bm{u}(w)\big{|}\bm{\tau}\big{)}$ is not
identically $0$ when $z,w\in\widetilde{C}$. Non singular odd half-integer
characteristics exist, and in what follows we fix one.

####  3.2.3 Prime form

Given a half-integer characteristic $\bm{c}$, we introduce the holomorphic
1-form

$\begin{split}\omega_{\bm{c}}=\sum_{h=1}^{g}\partial_{z_{h}}\theta(\bm{z}|\bm{\tau})\big{|}_{\bm{z}=\bm{c}}\,\differential
u_{h}.\end{split}$

The prime form is:

$E(z_{1},z_{2})=\frac{\theta(\bm{c}+\bm{u}(z_{1})-\bm{u}(z_{2})|\bm{\tau})}{\sqrt{\omega_{\bm{c}}(z_{1})}\sqrt{\omega_{\bm{c}}(z_{2})}}.$
(3.2)

It is defined as a holomorphic bispinor on $\widetilde{C}\times\widetilde{C}$,
i.e. a $(-\frac{1}{2})\otimes(-\frac{1}{2})$ form. It has zeros only at
$z_{1}=z_{2}$, and in local coordinates $\zeta$, we have

$E(z_{1},z_{2})\mathop{\sim}_{z_{1}\rightarrow
z_{2}}\frac{\zeta(z_{1})-\zeta(z_{2})}{\sqrt{\differential\zeta(z_{1})\differential\zeta(z_{2})}}.$

The prime form on the Riemann sphere $\hat{\mathbb{C}}$ reads

$E_{0}(x_{1},x_{2})=\frac{x_{1}-x_{2}}{\sqrt{\differential x_{1}\differential
x_{2}}}.$

Given a meromorphic function $X\colon\hat{C}\rightarrow\hat{\mathbb{C}}$, we
can define the relative prime form:

$\tilde{E}(z_{1},z_{2})=\frac{E(z_{1},z_{2})}{E_{0}(X(z_{1}),X(z_{2}))}.$
(3.3)

We observe that $\tilde{E}(z_{1},z_{2})$ is a function on
$\widetilde{C}\times\widetilde{C}$, such that

$\lim_{z_{2}\rightarrow z_{1}}\tilde{E}(z_{1},z_{2})=1.$ (3.4)

####  3.2.4 Fundamental bidifferential

######  Definition 3.4.

The fundamental bidifferential $B(z,w)$ is the unique bidifferential (i.e. a
$1\otimes 1$ form on $C\times C$) such that

  1. 1.

Symmetry: $B(z,w)=B(w,z)$;

  2. 2.

Normalisation: $\forall h\in[g]\quad\oint_{\mathcal{A}_{h}}B(\cdot,w)=0$;

  3. 3.

Singularities: $B(z,w)$ is meromorphic with only a double pole at $z=w$, and
if $\zeta$ is a local coordinate, we have

$B(z,w)\mathop{=}_{z\rightarrow
w}\left(\frac{1}{(\zeta(z)-\zeta(w))^{2}}+S_{B,\zeta}(w)+\order{\zeta(z)-\zeta(w)}\right)\differential\zeta(z)\differential\zeta(w).$

The fundamental bidifferential can be expressed as

$\begin{split}B(z,w)&=\differential_{z}\differential_{w}\ln\theta(\bm{c}+\bm{u}(z)-\bm{u}(w)).\end{split}$

and this expression is independent on the choice of a nonsingular odd half-
integer characteristics $\bm{c}$. Given $p,q\in C$ and a choice of path
$\gamma_{p,q}$ from $p$ to $q$, we define the meromorphic form

$\begin{split}\differential
S_{p,q}(z)=\int_{\gamma_{p,q}}B(z,\cdot),\end{split}$

It has two poles of order 1 in $p$ and $q$, with respective residue $-1$ and
$+1$. The prime form appears in the following computation.

######  Lemma 3.5.

For $i=1,2$, let $z_{i},\tilde{z}_{i}\in\widetilde{C}$ and $\gamma_{i}$ a path
from $\tilde{z}_{i}$ to $z_{i}$. Then:

$\int_{\gamma_{1}}\int_{\gamma_{2}}B=\int_{\gamma_{1}}\differential
S_{\tilde{z}_{2},z_{2}}=\ln\bigg(\frac{E(z_{1},z_{2})E(\tilde{z}_{1},\tilde{z}_{2})}{E(z_{1},\tilde{z}_{2})E(\tilde{z}_{1},z_{2})}\bigg{missing}).$

####  3.2.5 Decomposition of meromorphic forms

One distinguishes between three kinds of meromorphic forms:

  * •

holomorphic 1-forms (first kind);

  * •

meromorphic 1-forms with vanishing residues (second kind);

  * •

meromorphic 1-form with non-vanishing residues (third kind).

The space of first kind differentials has for basis $(\differential
u_{h})_{h=1}^{g}$, which are dual to the $\mathcal{A}$-cycles.

Assume that a choice of local coordinate $\zeta_{p}$ near each point $p\in C$
has been made. A basis of the space of second kind differentials is then given
by

$\begin{split}\differential
B_{p,k}(z)&=\mathop{\mathrm{Res}}_{z^{\prime}=p}\zeta_{p}(z^{\prime})^{-k}B(z^{\prime},z).\end{split}$

for $p\in C$ and $k\in\mathbb{Z}_{>0}$. Given the properties of the
fundamental bidifferential, its only pole is at $p$ with order $(k+1)$, where
it behaves like:

$\differential B_{p,k}(z)\mathop{=}_{z\rightarrow
p}-\differential(\zeta_{p}(z)^{-k})+\order{\differential\zeta_{p}(z)}.$

Besides, we have $\oint_{\mathcal{A}_{h}}\differential B_{p,k}=0$ for any
$h\in[g]$.

Assume that for each $p\in C^{0}$, a choice of path $\gamma_{p_{0},p}$ from
$p_{0}$ to $p$ in $C^{0}$ has been made. An example of third kind differential
is

$\differential S_{p_{0},p}(z)=\int_{\gamma_{p_{0},p}}B(z,\cdot),$ (3.5)

It has two poles of order 1 in $p_{0}$ and $p$, respectively of residue $-1$
and $+1$, and has zero $\mathcal{A}$-periods.

Every meromorphic $1$-form $\phi$ can be decomposed uniquely as a sum of first
kind, second kind and third kind differentials222Given $\phi$, we can always
perturb the representatives of $\mathcal{A}$ and $\mathcal{B}$-cycles so that
all poles are contained in $C^{0}$ and we can use (3.5).:

$\phi(z)=\bigg{(}\sum_{h=1}^{g}\oint_{\mathcal{A}_{h}}\phi\bigg{)}\differential
u_{h}(z)+\sum_{\begin{subarray}{c}p\,\,\text{simple pole}\\\
\text{of}\,\,\phi\end{subarray}}\big{(}\mathop{\text{Res}}_{p}\phi\big{)}\differential
S_{p_{0},p}(z)+\sum_{j\geq 1}\sum_{\begin{subarray}{c}p\,\,\text{pole of}\\\
\text{order}\,\,(k+1)\,\,{\rm
of}\,\,\phi\end{subarray}}\big{(}\mathop{\mathrm{Res}}_{p}\zeta_{p}^{k}\phi\big{)}\,\frac{\differential
B_{p,k}(z)}{k}$

###  3.3 The spectral curve

We elaborate on Section 2.3 and construct the spectral curve associated to the
equilibrium measure of ${\beta}$-ensembles, for the moment indifferently in
the unconstrained case or the fixed filling fraction case. This prepares us
for Section 4 where the asymptotics in the $\beta$-ensembles is described
solely in terms of the geometry of this spectral curve.

####  3.3.1 Construction of the marked Riemann surface

The equation (2.4) satisfied by the Stieltjes transform of the equilibrium
measure has two solutions:

$\begin{split}F_{+}(x)&=\frac{V^{\prime}(x)}{2}+y(x)=W_{1}(x),\\\
F_{-}(x)&=\frac{V^{\prime}(x)}{2}-y(x)=V^{\prime}(x)-W_{1}(x).\end{split}$
(3.6)

where $y(x)=-\frac{1}{2}M(x)s(x)$ and
$y(x)^{2}=\frac{1}{4}V^{\prime}(x)^{2}-P(x)$. After the birational
transformation $(x,y)\mapsto(x,s=-\frac{2y}{M(x)})$ we have the equation of an
hyperelliptic curve

$\begin{split}s^{2}=\sigma(x)=\prod_{h=0}^{g}(x-a_{h})(x-b_{h})\end{split}$

where the Weierstraß points $a_{0},b_{0},\ldots,a_{g},b_{g}$ are real. This
curve is constructed from two sheets homeomorphic to $\mathbb{C}\setminus S$,
which are glued together along $S$. The two sheets are embedded into the curve
by

$\begin{split}\iota_{\pm}\colon\begin{array}[]{ccc}\mathbb{C}\setminus
S&\longrightarrow&C\subset\mathbb{C}\times\mathbb{C}\\\ x&\longmapsto&(x,\pm
s(x))\end{array}.\end{split}$

We denote the sheets by $C_{\pm}=\iota_{\pm}(\mathbb{C}\setminus S)$, and
$\hat{C}_{\pm}$ are the sheets including their point at infinity. Adding these
points at infinity $\infty_{\pm}$ to $C$, we get a compact Riemann surface
$\hat{C}$. We define the projection map as the meromorphic function

$\begin{split}X\colon\begin{array}[]{ccc}\hat{C}&\longrightarrow&\hat{\mathbb{C}}\\\
(x,s)&\longmapsto&x\end{array}.\end{split}$

This function has simple poles at $\infty_{\pm}$ and it defines a degree $2$
branched covering of the Riemann sphere $\hat{\mathbb{C}}$, whose branch
points are the zeros of $\sigma$. It allows us to define local coordinates
$\zeta_{p}$ around any point $p\in\hat{C}$, which is used in Section 3.2.5 to
define a basis of meromorphic differentials:

  * •

If $p$ is a ramification point, we take $\zeta_{p}(z)=\sqrt{X(z)-X(p)}$ for
some choice of sign for the squareroot.

  * •

If $p=\infty_{\pm}$, then $\zeta_{\infty_{\pm}}=X(z)^{-1}$.

  * •

In all the other cases, $\zeta_{p}(z)=X(z)-X(p)$.

We shall take $p_{0}=\infty_{+}$ as reference point for the definition of the
Abel map (Section 3.2.2). For later use we analyse the prime form over
$\hat{\mathbb{C}}$ near $\infty_{\pm}$.

######  Lemma 3.6.

We have:

$E_{0}(z,\tilde{z})\sqrt{\differential\zeta_{\infty_{\pm}}(\tilde{z})}\Big{|}_{\tilde{z}=\infty_{\pm}}=\frac{-1}{\sqrt{-\differential
X(z)}}.$

###### Proof.

Since $\differential
X(\tilde{z})=-X(\tilde{z})^{2}\differential\zeta_{\infty_{\pm}}(\tilde{z})$,
we have

$\lim_{\tilde{z}\rightarrow\infty_{\pm}}E_{0}(z,\tilde{z})\sqrt{\differential\zeta_{\infty_{\pm}}(\tilde{z})}=\lim_{\tilde{z}\rightarrow\infty_{\pm}}(X(z)-X(\tilde{z}))\frac{\sqrt{-X(\tilde{z})^{-2}\differential
X(\tilde{z})}}{\sqrt{\differential X(z)\differential
X(\tilde{z})}}=\frac{-1}{\sqrt{-\differential X(z)}}.$

∎

We choose representatives for a symplectic basis of homology on $\hat{C}$ like
in Figure 1. Namely, we take $\mathcal{A}_{h}$ representing a counterclockwise
loop in $C_{+}$ going around the cut $S_{h}$, for $h\in[g]$. For convenience
we fix a representative $\mathcal{A}_{0}$ of a counterclockwise loop
surrounding $S_{0}$ in $C_{+}$, whose homology class is
$-(\mathcal{A}_{1}+\cdots+\mathcal{A}_{g})$. We take $\mathcal{B}_{h}$
representing a loop in $\hat{C}$ travelling from $S_{0}$ to $S_{h}$ in $C_{+}$
and in the opposite direction in $C_{-}$.

Figure 1:  Two cycles $\mathcal{A}_{h}$ and $\mathcal{B}_{h}$.

####  3.3.2 The $1$-form $\phi$ and the Stieltjes transform

The branched covering $X:\hat{C}\rightarrow\hat{\mathbb{C}}$ alone does not
determine the equilibrium measure: we also need to specify the meromorphic
function $Y:(x,s)\mapsto-\frac{1}{2}M(x)s$ on $\hat{C}$. It is such that
$Y(x)=\mp\frac{1}{2}M(x)s(x)$ for $x\in\hat{C}_{\pm}$. The advantage to work
with the Riemann surface $\hat{C}$ is that the function $W_{1}$, originally
defined on $\mathbb{C}\setminus S$ (Section 2.3) can be analytically continued
to a meromorphic function on the whole $\hat{C}$. Indeed, the meromorphic
function $\mathcal{W}_{1}(z)=\frac{V^{\prime}(X(z))}{2}+Y(z)$ coincides with
$W_{1}(X(z))$ for $z\in\hat{C}_{+}$. We can also see this by noticing that
$F_{\pm}$ in (3.6) are two solutions of the Riemann–Hilbert problem

$\begin{split}\forall x\in\mathring{S}\qquad F(x+{\rm i}0)+F(x-{\rm
i}0)=V^{\prime}(x).\end{split}$ (3.7)

######  Definition 3.7.

We equip $\hat{C}$ with the meromorphic 1-form
$\phi(z)=\mathcal{W}_{1}(z)\differential X(z)$.

The previous discussion shows that $\phi$ has a simple pole at $\infty_{+}$
with residue $-1$, and a higher order pole at $\infty_{-}$ with:

$\begin{split}\phi(z)&\mathop{=}_{z\rightarrow\infty_{-}}\differential
V(X(z))-\frac{\differential X(z)}{X(z)}+\order{\frac{\differential
X(z)}{X(z)^{2}}},\end{split}$

where we recall that $\zeta_{\infty_{\pm}}=1/X$ is a local coordinate near
$\infty_{\pm}$. As $X$ has two simple poles at $\infty_{\pm}$, the form
$\differential X$ thus has double poles at these points with

$\differential X=-\zeta_{\infty_{\pm}}^{2}\differential\zeta_{\infty_{\pm}}.$

The $1$-form $\phi$ therefore decomposes as in Section 3.2.5:

$\begin{split}\phi=\sum_{h=1}^{g}2{\rm i}\pi\,\epsilon^{*}_{h}\differential
u_{h}+\differential
S_{\infty_{+},\infty_{-}}-\sum_{k=1}^{d}\frac{t_{k}}{k}\differential
B_{\infty_{-},k}.\end{split}$ (3.8)

where the potential is $V(x)=\sum_{k=1}^{d}\frac{t_{k}x^{k}}{k}$. The path
from $\infty_{+}$ to $\infty_{-}$ used in the definition of the $1$-form
$\differential S_{\infty_{+},\infty_{-}}$ is chosen so that it does not
intersect $(\mathcal{A}_{h},\mathcal{B}_{h})_{h=1}^{g}$. For instance, one can
take it to be $\iota_{+}(a_{0}+{\rm i}\mathbb{R}_{\geq
0})\cup\iota_{-}(a_{0}-{\rm i}\mathbb{R}_{\leq 0})$.

######  Remark 3.8.

For the equilibrium measure in the unconstrained case, the property noticed in
Remark 2.5 can be equivalently rewritten as

$\forall h\in[g]\qquad\oint_{\mathcal{B}_{h}}\phi=0.$

As the filling fractions are real, this implies that for any $\gamma\in
H_{1}(\hat{C},\mathbb{Z})$, we have
$\text{Re}\big{(}\oint_{\gamma}\phi\big{)}=0$. Pairs $(\hat{C},\phi)$
satisfying this property are called Boutroux curves, see [Ber11].

####  3.3.3 The fundamental bidifferential and the second correlator

Under the assumptions discussed in Section 4.1, [BG24] shows that the
$N\to\infty$ limit of the second correlator in the model with fixed filling
fractions $\bm{N}/N\rightarrow\bm{\epsilon}$ exists:

$\begin{split}W_{2}(x_{1},x_{2})&=\lim_{N\to\infty}\frac{\beta}{2}\bigg{(}\left\langle{\Tr\Big(\frac{1}{x_{1}-\Lambda}\Big{missing})\Tr\Big(\frac{1}{x_{2}-\Lambda}\Big{missing})}\right\rangle^{V}_{N,\bm{N}/N}-W_{1}(x_{1})W_{1}(x_{2})\bigg{)}\,.\end{split}$

It can be shown to satisfy the Riemann–Hilbert problem

$\begin{split}\forall(x_{1},x_{2})\in(\mathbb{C}\setminus
S)\times\mathring{S},\qquad W_{2}(x_{1},x_{2}+{\rm i}0)+W_{2}(x_{1},x_{2}-{\rm
i}0)=-\frac{1}{(x_{1}-x_{2})^{2}},\end{split}$ (3.9)

for $x\in C\setminus S$ and $y\in S$, see for instance [EKR15, Chapter 3] or
[BEO15]. Besides, $W_{2}(x_{1},x_{2})=\order{1/x_{i}^{2}}$ as
$x_{i}\rightarrow\infty$ since the total number of particles is deterministic,
and

$\forall h\in[g]\qquad\oint_{\mathcal{A}_{h}}W_{2}(x_{1},x_{2})\differential
x_{1}=0,$ (3.10)

since the filling fraction of the segment $A_{h}$ is fixed.

The Riemann–Hilbert problem (3.9) implies that we can define a meromorphic
function $\mathcal{W}_{2}(z_{1},z_{2})$ on $\hat{C}\times\hat{C}$ such that
$\mathcal{W}_{2}(z_{1},z_{2})=W_{2}(X(z_{1}),X(z_{2}))$ when
$z_{1},z_{2}\in\hat{C}_{+}$. By examining the behavior of $\mathcal{W}_{2}$ at
the poles and considering the $\mathcal{A}$-period conditions (3.10), one can
identify it in terms of the fundamental bidifferential:

$B(z_{1},z_{2})=\mathcal{W}_{2}(z_{1},z_{2})\differential
X(z_{1})\differential X(z_{2})+\frac{\differential X(z_{1})\differential
X(z_{2})}{(X(z_{1})-X(z_{2}))^{2}}.$ (3.11)

######  Definition 3.9.

The spectral curve of a $\beta$-ensemble is the marked compact Riemann surface
$(\hat{C},\mathcal{A},\mathcal{B})$ equipped with the meromorphic functions
$X,Y$ and the bidifferential $B$.

###  3.4 Characterisation of spectral curves of $\beta$-ensembles

In Section 3.3.1 we explained that the Riemann surface $\hat{C}$ underlying
the spectral curve of a $\beta$-ensemble is hyperelliptic with real Weierstraß
points. We now prove the converse, namely that all such Riemann surfaces can
be realised (non uniquely) as the underlying Riemann surface of the spectral
curve of an unconstrained $\beta$-ensemble.

######  Definition 3.10.

If $G$ is a meromorphic function in a neighborhood of $\infty$ in
$\hat{\mathbb{C}}$, we define its polynomial part $\mathcal{V}[G](x)$, which
is the unique polynomial such that
$G(x)=\mathcal{V}[G](x)+\order{\frac{1}{x}}$ as $x\rightarrow\infty$.

######  Proposition 3.11.

For any $a_{0}<b_{0}<\cdots<a_{g}<b_{g}$, there exists a polynomial $V$ of
degree $(2g+2)$ with top coefficient $\frac{t_{2g+2}}{2g+2}>0$ and there exist
for each $h\in[0,g]$ a segment $A_{h}$ which is a neighborhood of
$[a_{h},b_{h}]$, such that the unconstrained $\beta$-ensemble with potential
$V$ on $A=\bigsqcup_{h=0}^{g}A_{h}$ admits an equilibrium measure with support
$S=\bigsqcup_{h=0}^{g}[a_{h},b_{h}]$ and in (2.6) we have
$M(x)=t_{2g+2}\prod_{h=1}^{g}(x-z_{h})$ having roots outside $A$ and such that
$b_{h-1}<z_{h}<a_{h}$ for any $h\in[g]$.

###### Proof.

Take $2g$ real points $a_{0}<b_{0}<\cdots<a_{g}<b_{g}$ and introduce
polynomials $\sigma(x)=\prod_{h=0}^{g}(x-a_{h})(x-b_{h})$. We have seen in
Section 2.3 that there exists a unique holomorphic function $s(x)$ on
$\mathbb{C}\setminus\bigsqcup_{h=0}^{g}[a_{h},b_{h}]$ such that $s(x)\sim
x^{g+1}$ as $x\rightarrow\infty$ in the complex plane. Take $h\in[g]$ and
introduce the continuous function

$\forall\bm{\lambda}\in[0,1]^{g}\qquad
J_{h}(\bm{\lambda})=\int_{b_{h-1}}^{a_{h}}s(x)\prod_{k=1}^{g}\big{(}x-(\lambda_{k}b_{k-1}+(1-\lambda_{k})a_{k})\big{)}\differential
x.$

By continuity, $s(x)$ has constant sign for $x\in(b_{h-1},a_{h})$. Besides,
for $\lambda_{h}\in\\{0,1\\}$ we have
$\text{sgn}(J_{h}(\bm{\lambda}))=(-1)^{g-h+1-\lambda_{h}}$. The
Poincaré–Miranda theorem [Mir41] then implies the existence of
$\bm{\lambda}^{*}\in[0,1]^{g}$ such that $J_{h}(\bm{\lambda}^{*})$ vanishes
for any $h\in[g]$. Since $J_{h}(\bm{\lambda})$ does not vanish when
$\lambda_{h}\in\\{0,1\\}$, this $\bm{\lambda}^{*}$ must be in $(0,1)^{g}$. We
let $z_{h}=\lambda_{h}^{*}b_{h-1}+(1-\lambda_{h}^{*})a_{h}$ for $h\in[g]$ and
introduce the polynomial $M(x)=t_{2g+2}\prod_{h=1}^{g}(x-z_{h})$, where the
constant $t_{2g+2}$ is chosen such that

$\sum_{h=0}^{g}\int_{a_{h}}^{b_{h}}\frac{M(x)\text{Im}(s(x+{\rm
i}0))}{2\pi}=1.$ (3.12)

The sign discussion for $s$ in Section 2.3 reveals that all terms in (3.12)
are positive, thus $t_{2g+2}>0$. Then, $V(x)=\int_{0}^{x}\mathcal{V}[M\cdot
s](\xi)\differential\xi$ is a polynomial of degree $2g+2$ with top coefficient
$\frac{t_{2g+2}}{2g+2}$, and

$\mathop{\text{Res}}_{x=\infty}\frac{s(x)M(x)}{2}\differential
x=-\frac{1}{2{\rm
i}\pi}\sum_{h=0}^{g}\oint_{\mathcal{A}_{h}}\frac{s(x)M(x)}{2}\differential
x=\sum_{h=0}^{g}\int_{a_{h}}^{b_{h}}\frac{M(x)\text{Im}(s(x+{\rm
i}0))}{2\pi}=1.$

where $\mathcal{A}_{h}$ is a counterclockwise loop around $[a_{h},b_{h}]$.
Therefore,

$\frac{V^{\prime}(x)}{2}\mathop{\mathop{=}}_{x\rightarrow\infty}\frac{M(x)s(x)}{2}+\frac{1}{x}+\mathcal{O}\bigg{(}\frac{1}{x^{2}}\bigg{)}.$

This $V$ defines the potential in a $\beta$-ensemble which we consider over
the domain $A=\bigsqcup_{h=0}^{g}A_{h}$, where
$A_{h}=[a_{h}^{\prime},b_{h}^{\prime}]$ and $z_{h}<a_{h}^{\prime}<a_{h}$ and
$b_{h}<b_{h}^{\prime}<z_{h+1}$ for any $h\in[0,g]$, with the conventions
$z_{0}=-\infty$ and $z_{g+1}=+\infty$. It remains to check that
$W(x):=\frac{1}{2}(V^{\prime}(x)-M(x)s(x))$ is the Stieltjes transform of the
equilibrium measure of this (unconstrained) $\beta$-ensemble.

We define the measure $\mu$ with support $S=\bigsqcup_{h=0}^{g}[a_{h},b_{h}]$
and density

$\frac{\differential\mu}{\differential x}=\frac{W(x-{\rm i}0)-W(x+{\rm
i}0)}{2{\rm i}\pi}=\frac{M(x)s(x)}{2\pi}\mathds{1}_{S}(x).$

By construction, $W$ is the Stieltjes transform of $\mu$. Since $M$ has a
single zero between each components of the support, $\mu$ is a positive
measure — see the sign discussion in Section 2.3. Define
$U(x)=V(x)-2\int_{S}\ln|x-\xi|\differential\mu(\xi)$. We clearly have

$\forall x\in S\qquad U^{\prime}(x)=V^{\prime}(x)-W_{1}(x+{\rm
i}0)-W_{1}(x-{\rm i}0)=0$

Integrating this from $a_{h}$ to $x\in(a_{h},b_{h})$, we find a constant
$c_{h}$ such that $U(x)=c_{h}$ for any $x\in(a_{h},b_{h})$. Besides, for any
$h\in[g]$ we compute

$c_{h}-c_{h-1}=\int_{b_{h-1}}^{a_{h}}U^{\prime}(x)\differential
x=\int_{b_{h-1}}^{a_{h}}\big{(}V^{\prime}(x)-2W_{1}(x)\big{)}\differential
x=\int_{b_{h-1}}^{a_{h}}M(x)s(x)\differential x.$

Since we have chosen $(z_{1},\ldots,z_{g})$ so that this integral vanishes,
$c_{h}$ is independent of $h$. As a result, $\mu$ satisfies the
characterisation of the equilibrium measure from Theorem 2.1 (unconstrained
case). By uniqueness, this must be the equilibrium measure:
$\mu=\mu_{\text{eq}}$. ∎

###  3.5 Deformations of the curve

We consider real and complex deformations of the complex curves, that will be
used in Section 4 to extend the validity of our formulae beyond their
realisation for spectral curves of $\beta$-ensembles. We first show that
within the class of spectral curves of $\beta$-ensembles, we can always
realise any vector of filling fractions in a small neighborhood of a given one
by perturbation of the support.

######  Lemma 3.12.

Let $a_{0}<b_{0}<\cdots<a_{g}<b_{g}$ and take a corresponding
$M(x)=t_{2g+2}\prod_{h=1}^{g}(x-z_{h})$ with $z_{h}\in(b_{h-1},a_{h})$ as in
Proposition 3.11. There exists a small neighborhood
$\Omega\subset\mathbb{R}^{2g}$ of $(a_{h},z_{h})_{h=1}^{g}$ such that the map
$\Pi:\Omega\rightarrow\mathbb{R}^{2g}$ given by

$\Pi(\tilde{a}_{1},\tilde{z}_{1},\ldots,\tilde{a}_{g},\tilde{z}_{g})=\bigg{(}\int_{\tilde{a}_{h}}^{b_{h}}\tilde{M}(x)\sqrt{-\tilde{\sigma}(x)}\differential
x\,,\,\int_{b_{h-1}}^{\tilde{a}_{h}}\tilde{M}(x)\sqrt{\tilde{\sigma}(x)}\differential
x\bigg{)}_{h=1}^{g}\,,$

is a diffeomorphism onto its image, where we have set:

$\tilde{M}(x)=t_{2g+2}\prod_{k=1}^{g}(x-\tilde{z}_{k})\qquad\text{and}\qquad\tilde{\sigma}(x)=(x-a_{0})(x-b_{0})\prod_{k=1}^{g}(x-\tilde{a}_{k})(x-b_{k}).$

This will be used in the following form.

######  Corollary 3.13.

There is a dense set of $a_{0}<b_{0}<\cdots<a_{g}<b_{g}$ for which there
exists a $\beta$-ensemble whose associated equilibrium measure of Theorem 2.1
has filling fractions $\bm{\epsilon}^{*}$ whose components
$\epsilon_{1}^{*},\ldots,\epsilon_{g}^{*}$ are $\mathbb{Q}$-linearly
independent.

###### Proof of Lemma 3.12.

$\Pi$ is a smooth function of $(\tilde{a}_{h},\tilde{z}_{h})_{h=1}^{g}$ in the
range
$b_{0}<\tilde{z}_{1}<\tilde{a}_{1}<b_{1}<\tilde{z}_{2}<\tilde{a}_{2}<\cdots<\tilde{z}_{g}<\tilde{a}_{g}<b_{g}$.
We compute its Jacobian

$\begin{split}&\quad\text{det}\begin{pmatrix}\int_{\tilde{a}_{h}}^{b_{h}}\frac{-\tilde{M}(x)\sqrt{-\tilde{\sigma}(x)}}{2(x-\tilde{a}_{k})}\differential
x&\int_{\tilde{a}_{h}}^{b_{h}}\frac{-\tilde{M}(x)\sqrt{-\tilde{\sigma}(x)}}{(x-\tilde{z}_{k})}\differential
x\\\
\int_{b_{h-1}}^{\tilde{a}_{h}}\frac{-\tilde{M}(x)\sqrt{\tilde{\sigma}(x)}}{2(x-\tilde{a}_{k})}\differential
x&\int_{b_{h-1}}^{\tilde{a}_{h}}\frac{-\tilde{M}(x)\sqrt{\tilde{\sigma}(x)}}{(x-\tilde{z}_{k})}\differential
x\end{pmatrix}_{1\leq h,k\leq g}\\\
&=\frac{1}{2^{g}}\int_{\tilde{a}_{1}}^{b_{1}}\\!\\!\cdots\int_{\tilde{a}_{g}}^{b_{g}}\prod_{h=1}^{g}\differential
x_{h}\tilde{M}(x_{h})\sqrt{-\tilde{\sigma}(x_{h})}\int_{b_{0}}^{\tilde{a}_{1}}\\!\\!\cdots\int_{b_{g-1}}^{\tilde{a}_{g}}\differential\xi_{h}\tilde{M}(\xi_{h})\sqrt{\tilde{\sigma}(\xi_{h})}\cdot\det\begin{pmatrix}\frac{1}{x_{h}-\tilde{a}_{k}}&\frac{1}{x_{h}-\tilde{z}_{k}}\\\
\frac{1}{\xi_{h}-\tilde{a}_{k}}&\frac{1}{\xi_{h}-\tilde{z}_{k}}\end{pmatrix}_{1\leq
h,k\leq g}.\end{split}$ (3.13)

where we used the fact that $\sqrt{\pm\tilde{\sigma}(x)}$ vanishes at the
endpoints of the integration intervals. The determinant in the integrand can
be readily evaluated

$\begin{split}\text{det}\begin{pmatrix}\frac{1}{x_{h}-\tilde{a}_{k}}&\frac{1}{x_{h}-\tilde{z}_{k}}\\\
\frac{1}{\xi_{h}-\tilde{a}_{k}}&\frac{1}{\xi_{h}-\tilde{z}_{k}}\\\
\end{pmatrix}_{1\leq h,k\leq
g}&=\Delta(\bm{\tilde{a}})\Delta(\bm{\tilde{z}})\Delta(\bm{x})\Delta(\bm{\xi})\prod_{h,k=1}^{g}\frac{(\tilde{z}_{h}-\tilde{a}_{k})(\xi_{h}-x_{k})}{(x_{h}-\tilde{a}_{k})(\xi_{h}-\tilde{a}_{k})(x_{h}-\tilde{z}_{k})(\xi_{h}-\tilde{z}_{k})}\\\
&=\frac{t_{2g+2}^{2g}\Delta(\bm{\tilde{a}})\Delta(\bm{\tilde{z}})\Delta(\bm{x})\Delta(\bm{\xi})}{\prod_{h=1}^{g}\tilde{M}(x_{h})\tilde{M}(\xi_{h})}\prod_{h,k=1}^{g}\frac{(\tilde{z}_{h}-\tilde{a}_{k})(\xi_{h}-x_{k})}{(x_{h}-\tilde{a}_{k})(\xi_{h}-\tilde{a}_{k})}.\end{split}$
(3.14)

For $(\tilde{a}_{h},\tilde{z}_{h})_{h=1}^{g}$ close enough to
$(a_{h},z_{h})_{h=1}^{g}$, the zeros of $\tilde{M}$ are outside
$\bigsqcup_{h=1}^{g}[\tilde{a}_{h},\tilde{b}_{h}]$, so that the sign of the
integrand in (3.13) remains constant in the whole integration range. The
determinant of the Jacobian of $\Pi$ is thus nonzero, and $\Pi$ is a local
diffeomorphism. ∎

###### Proof of Corollary 3.13.

If $(\tilde{a}_{h},\tilde{z}_{h})_{h=1}^{g}\in\Omega$, call $\tilde{\mu}$ the
measure supported on
$\tilde{S}=[a_{0},b_{0}]\cup\bigsqcup_{h=1}^{g}[\tilde{a}_{h},b_{h}]$ with
density $\frac{1}{2\pi}\tilde{M}(x)\sqrt{-\tilde{\sigma}(x)}$. At
$(a_{h},z_{h})_{h=1}^{g}$ this $\tilde{\mu}$ is by construction the
equilibrium measure of a $\beta$-ensemble, which we simply denote $\mu$: it is
in particular a probability measure with vector of filling fractions
$(\epsilon_{h}^{*})_{h=1}^{g}$ and the $h$-th second component of
$\Pi(a_{1},z_{1},\ldots,a_{g},z_{g})$ is equal to
$(-1)^{g-h}(U(a_{h})-U(b_{h-1}))=0$ for $h\in[g]$. So, $\Pi$ induces a
homeomorphism from $\Omega$ to a neighborhood
$\Omega^{\prime}\subset\mathbb{R}^{2g}$ of
$\big{(}(-1)^{g-h}2\pi\epsilon_{h}^{*}\,,\,0\big{)}_{h=1}^{g}$. By continuity
with respect to the parameters, $\tilde{\mu}$ remains a positive measure on
each component of $\tilde{S}$ for all parameters in a (possibly smaller)
$\Omega$, and that the total mass of $\tilde{\mu}$ defines a positive
continuous function on $\Omega$. In particular,
$\tilde{\mu}^{\prime}=\tilde{\mu}/\tilde{\mu}(\tilde{S})$ is a probability
measure on $\tilde{S}$.

If $\epsilon_{1},\ldots,\epsilon_{g}$ are $\mathbb{Q}$-linearly dependent, we
can approximate $\big{(}(-1)^{g-h}2\pi\epsilon_{h}^{*}\,,\,0\big{)}_{h=1}^{g}$
to arbitrary precision by $2g$-tuples
$\big{(}(-1)^{g-h}2\pi\tilde{\epsilon}_{h}\,,\,0\big{)}_{h=1}^{g}\in\Omega^{\prime}$
such that $\tilde{\epsilon}_{1},\ldots,\tilde{\epsilon}_{g}$ are
$\mathbb{Q}$-linearly independent. Applying $\Pi^{-1}$, we get an
approximation $(\tilde{a}_{h},\tilde{z}_{h})_{h=1}^{g}$ of
$(a_{h},z_{h})_{h=1}^{g}$ at arbitrary precision whose associated probability
measure $\tilde{\mu}^{\prime}$ is by construction (follow the proof of
Proposition 3.11) the equilibrium measure of the $\beta$-ensemble with
potential

$\tilde{V}(x)=\frac{1}{\tilde{\mu}(\tilde{S})}\int_{0}^{x}\mathcal{V}[\tilde{M}\cdot\tilde{s}](x),$

with $\tilde{s}$ like $s$ of Section 2.3 but with $\tilde{a}$s instead of
$a$s. This equilibrium measure has vector of filling fractions
$(\tilde{\epsilon}_{h}/\tilde{\mu}(\tilde{S}))_{h=1}^{g}$, whose components
remain $\mathbb{Q}$-linearly independent. ∎

In a second step, we will leave the realm of spectral curves of
$\beta$-ensembles and rather consider their complex deformations. Here it
becomes important to keep track of the marking. The equation of a
hyperelliptic curve $s^{2}=\prod_{h=0}^{g}(x-a_{h})(x-b_{h})$ is parameterised
by the set $\bm{\Delta}_{2g+2}$ of $(2g+2)$-tuple $(a_{h},b_{h})_{h=0}^{g}$ of
pairwise distinct complex numbers. Its universal cover
$\widetilde{\bm{\Delta}}_{2g+2}$ based at a tuple of strictly increasing real
numbers parametrises the equation of the hyperelliptic curve together with a
choice of marking: at the base point it is the one described in Section 3.3.1,
and there is a unique way to get from there a marking for any other point in
$\widetilde{\bm{\Delta}}_{2g+2}$ by performing continuous deformations of the
representatives of the homology cycles. The outcome is an analytic family
$\hat{\bm{C}} \rightarrow\widetilde{\bm{\Delta}}_{2g+2}$ of marked
hyperelliptic curves, which coincide with the one described in Section 3.3.1
above the connected component of the base point in the real locus of
$\widetilde{\bm{\Delta}}_{2g+2}$. Concretely, in other real connected
components, the symplectic basis of homology has changed by an
$\text{Sp}_{2g}(\mathbb{Z})$-transformation compared to Section 3.3.1, and so
must have the matrix of periods. We will rely on the following basic fact in
complex geometry — see e.g. [CMSP17, Chapter 1].

######  Lemma 3.14.

The period matrix (3.1) is a holomorphic function on
$\widetilde{\bm{\Delta}}_{2g+2}$. The Abel map based at $\infty_{+}$ is a
holomorphic function $\widetilde{\bm{C}}\rightarrow\mathbb{C}^{g}$.

##  4 Asymptotics of the partition function and the kernels

###  4.1 Expansion of the partition function and generalised central limit
theorem

The large $N$ asymptotic expansion of the partition function of the
$\beta$-ensembles in the multi-cut regime was established in [BG24], under
assumptions which are satisfied for the potentials that we consider in Theorem
2.1. In particular, the off-criticality assumption on $A$ corresponds to $M$
of Lemma 2.3 having no zeros on $A$. We reproduce here the formulae for these
asymptotics, which will be our starting point.

######  Theorem 4.1.

[BG24, Theorem 1.5] Let $g\geq 1$, let $V$ as in Theorem 2.1 and assume $M$
from Lemma 2.3 has no zeros on $A$. The partition function has the following
expansion as $N\to\infty$

$Z^{V}_{N}\sim
N^{\frac{\beta}{2}N+\varkappa}e^{N^{2}\mathcal{E}[\mu_{\text{eq}}]+N\mathcal{S}[\mu_{\text{eq}}]+\mathcal{G}[\mu_{\text{eq}}]}\,\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}.$

Here

$\mathcal{S}[\mu]=\bigg{(}1-\frac{\beta}{2}\bigg{)}\big{(}{\rm
Ent}[\mu]-\ln(\beta/2)\big{)}+(\beta/2)\ln(2\pi/e)-\ln\Gamma(\beta/2),$

where ${\rm Ent}[\mu]$ is the von Neumann entropy of the probability measure
$\mu$, $\varkappa$ is a known universal constant depending only on $g$ and
$\beta$, $\mathcal{G}[\mu]$ is a continuous functional whose expression is
irrelevant for our purposes, and

$\bm{v}_{\text{eq}}=\frac{\nabla_{\epsilon}\mathcal{S}[\mu_{\text{eq},\bm{\epsilon}}]}{2{\rm
i}\pi}\bigg{|}_{\bm{\epsilon}=\bm{\epsilon}^{*}}.$ (4.1)

In (4.1) one differentiates with respect to
$\bm{\epsilon}=(\epsilon_{1},\ldots,\epsilon_{g})$, keeping in mind that
$\epsilon_{0}=1-(\epsilon_{1}+\cdots+\epsilon_{g})$ depends on them. The
kernels appearing in the determinantal and pfaffian formulae of Borodin and
Strahov (Theorems 2.6–2.8) can be estimated using the following generalised
central limit theorem.

######  Theorem 4.2 ([BG24, Theorem 1.6]).

Let $f$ be a holomorphic function in a complex neighborhood of $A$. Under the
same assumptions as Theorem 4.1, we have as $N\to\infty$:

$\left\langle{\mathrm{e}^{\sum_{i=1}^{N}f(\lambda_{i})}}\right\rangle^{V}_{N}\sim
e^{N\mathcal{L}[f]+\mathcal{H}[f]+\frac{1}{2}\mathcal{Q}[f,f]}\,\frac{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{\mathcal{U}}[f]\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}.$

Here

$\mathcal{L}[f]=\oint_{S}W_{1}(\xi)\frac{f(\xi)\differential\xi}{2{\rm
i}\pi},\qquad\mathcal{Q}[f,f]=\frac{2}{\beta}\oint_{S^{2}}W_{2}(\xi_{1},\xi_{2})\frac{f(\xi_{1})\differential\xi_{1}}{2{\rm
i}\pi}\frac{f(\xi_{2})\differential\xi_{2}}{2{\rm i}\pi},$

where $W_{1}$ and $W_{2}$ are calculated in a model with fixed filling
fraction tending to $\bm{\epsilon}^{*}$, $\mathcal{H}[f]$ is a linear form
whose expression is irrelevant, and

$\bm{\mathcal{U}}[f]=\oint_{S}f(X(z))\frac{\differential\bm{u}(z)}{2{\rm
i}\pi}$

###  4.2 Three explicit formulae

We will establish in Section 5.4 the following expression for the equilibrium
energy in terms of the geometry of the spectral curve.

######  Proposition 4.3.

The equilibrium energy is

$-\mathcal{E}[\mu_{\text{eq}}]=\frac{\beta}{2}\mathcal{L}[V]+\frac{\beta^{2}}{8}\mathcal{Q}[V,V]+{\rm
i}\pi\beta\bm{\epsilon}^{*}\cdot\big{(}\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-})\big{)}.$

The right-hand side is proportional to $\frac{\beta}{2}$, since $W_{2}$
contains a factor $\frac{2}{\beta}$, and so is the left-hand side. Accordingly
this is an identity between $\beta$-independent quantities. There is a
classical link between random matrix theory and the theory of Frobenius
manifolds: the free energy at leading order, namely
$\mathcal{E}[\mu_{\text{eq}}]$ coincides with the prepotential of the
Hurwitz–Frobenius manifold associated to the spectral curve of the random
matrix ensemble. A formula for this free energy was established in the more
general context of the two matrix model in [Ber03], involving only the
geometry of the spectral curve. It involves the same ingredient but does not
have exactly the same form as Proposition 4.3, which is the formula we need.

Before going further, we give two extra formulae. The first one evaluates the
argument $\bm{v}_{\text{eq}}$ of the theta functions in Theorems 4.1-4.2; it
is not necessary for Section 5, but we include it for completeness. The second
one will be used in the proof of Lemma 4.8.

######  Proposition 4.4.

Assume $M$ in (2.6) is of the form $M(x)=t_{2g+2}\prod_{h=1}^{g}(x-z_{h})$
with roots $z_{h}\notin A$ satisfying $b_{h-1}<z_{h}<a_{h}$ for any $h\in[g]$,
and denote by $(\bm{e}_{1},\ldots,\bm{e}_{g})$ the canonical basis of
$\mathbb{C}^{g}$.

The function $\text{Im}\,\bm{u}(z)=\int_{\infty_{+}}^{z}\differential\bm{u}$
is a single-valued function of $z\in\hat{C}_{+}$, and we have

$\bm{v}_{\text{eq}}=2\pi\bigg{(}1-\frac{\beta}{2}\bigg{)}\bigg{[}\frac{g+1}{2{\rm
i}}\bm{u}(\infty_{-})+\sum_{k=1}^{g}\Big{(}\text{Im}\,\bm{u}(z_{k})+\frac{g+1-k}{2{\rm
i}}\bm{\tau}(e_{k})\Big{)}\bigg{]}.$ (4.2)

###### Proof.

See Appendix A. As we explain in Section 5.4, $\bm{u}(\infty_{-})$ and
$\bm{\tau}$ are purely imaginary, so all terms in the right-hand side are
real, as it should be. The domain $\hat{C}_{+}$ is homeomorphic to the non
simply-connected domain $\hat{\mathbb{C}}\setminus S$, so
$\int_{\infty_{+}}^{z}\differential\bm{u}$ is multi-valued. However, the
ambiguities are $\mathcal{A}$-periods of $\differential\bm{u}$ which are real,
so $\text{Im}\,\bm{u}(z)$ is single-valued. ∎

######  Remark 4.5.

If we remove the assumption on the roots of $M$ in Lemma 4.4, we can still
compute $\bm{v}_{\text{eq}}$ and obtain a formula of a similar form.

######  Lemma 4.6.

$\mathcal{U}[V]=-\Res_{\infty_{+}}V\differential\bm{u}=\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-}).$

###### Proof.

The first equality is obtained by moving the contour around $S$ to
$\infty_{+}$. We focus on the second equality. Since $\differential
V=2(\phi-Y\differential X)$, we have

$\begin{split}\Res_{\infty_{+}}V\differential\bm{u}&=-\Res_{\infty_{+}}\bm{u}\differential
V=-2\Res_{\infty_{+}}\bm{u}(\phi-Y\differential X)\,.\end{split}$

Since $\infty_{+}$ is the base point for the Abel map, we have
$\bm{u}(\infty_{+})=0$ and since $\phi$ has only a simple pole at
$\infty_{+}$, the first term gives a vanishing residue. The hyperelliptic
involution preserves $X$, sends $Y$ and $\differential\bm{u}$ to their
opposite, and $\infty_{+}$ to $\infty_{-}$. Hence, it sends $\bm{u}$ to
$\bm{u}(\infty_{-})-\bm{u}$, where $\infty_{-}\in C^{0}$. Using the involution
as a change of variables, we get

$\begin{split}\Res_{\infty_{+}}\bm{u}Y\differential
X=\Res_{\infty_{-}}(\bm{u}-\bm{u}(\infty_{-}))Y\differential
X=\Res_{\infty_{-}}\bm{u}Y\differential X-\bm{u}(\infty_{-}).\end{split}$

For the last equality, since the only poles of $Y\differential X$ are
$\infty_{\pm}$ we could evaluate

$\Res_{\infty_{-}}Y\differential X=-\Res_{\infty_{+}}Y\differential
X=-\Res_{\infty_{+}}\phi=1.$

We then write

$\Res_{\infty_{+}}V\differential\bm{u}=\big{(}\Res_{\infty_{+}}+\Res_{\infty_{-}}\big{)}Y\differential
X-\bm{u}(\infty_{-})$

The first term can be computed with the Riemann bilinear identity [FK92,
Equation 3.0. 2]

$\big{(}\Res_{\infty_{+}}+\Res_{\infty_{-}}\big{)}\bm{u}Y\differential
X=\frac{1}{2{\rm
i}\pi}\sum_{h=1}^{g}\bigg{(}\oint_{\mathcal{A}_{h}}\differential\bm{u}\cdot\oint_{\mathcal{B}_{h}}Y\differential
X-\oint_{\mathcal{B}_{h}}\differential\bm{u}\cdot\oint_{\mathcal{A}_{h}}Y\differential
X\bigg{)}.$ (4.3)

Taking into account that for any $h,k\in[g]$, we have
$\oint_{\mathcal{B}_{h}}Y\differential X=0$ (Remark 3.8), and
$\oint_{\mathcal{B}_{h}}\differential u_{k}=\tau_{h,k}$ and
$\oint_{\mathcal{A}_{h}}Y\differential X=2{\rm i}\pi\,\epsilon^{*}_{h}$, we
find that (4.3) is equal to $-\bm{\tau}(\bm{\epsilon}^{*})$. ∎

###  4.3 Kernel asymptotics: intermediate computations

We need to compute an asymptotic equivalent as $K\rightarrow\infty$ of kernels
of the form:

$\begin{split}\frac{Z_{M}^{\frac{K}{M}V}}{Z_{K}^{V}}\left\langle{\prod_{j=1}^{m}\det(x_{j}-\Lambda)^{c_{j}}}\right\rangle^{\frac{K}{M}V}_{M},\end{split}$

where $x_{1},\ldots,x_{m}\notin A$, $c_{1},\ldots,c_{m}\in\mathbb{Z}$, and
$(M-K)=p$ is a fixed integer. Notice that

$\begin{split}\frac{Z_{M}^{\frac{K}{M}V}}{Z_{K}^{V}}\left\langle{\prod_{j=1}^{m}\det(x_{j}-\Lambda)^{c_{j}}}\right\rangle^{\frac{K}{M}V}_{M}&=\frac{Z_{M}^{V}}{Z_{K}^{V}}\left\langle{e^{\sum_{i=1}^{M}f_{\bm{c}}(\lambda_{i})+\frac{\beta}{2}pV(\lambda_{i})}}\right\rangle^{V}_{M},\end{split}$
(4.4)

where we used the holomorphic function on a neighborhood of $A$

$\begin{split}f_{\bm{c}}(\lambda)&=\sum_{j=1}^{m}c_{j}\ln(x_{j}-\lambda),\end{split}$

and for $x_{j}\notin A$ we choose the cut of the logarithm away from $A$. In
order to access the asymptotics of (4.4) via Theorem 4.2, we first have to
evaluate the following quantities

######  Lemma 4.7.

Let $z,z_{1},z_{2}\in C_{+}$. We have:

$\begin{split}\mathcal{L}\big{[}\ln(X(z)-\bullet)\big{]}&=2{\rm
i}\pi\bm{\epsilon}^{*}\cdot\bm{u}(z)-\ln\big(\eta(z)E(z,\infty_{+})^{2}\differential\zeta_{\infty_{+}}(\infty_{+})\big{missing})-\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}\differential
B_{\infty_{-},k},\\\
\mathcal{Q}\big{[}\ln(X(z)-\bullet),V\big{]}&=\frac{2}{\beta}\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}\differential
B_{\infty_{+},k},\\\
\mathcal{Q}\big{[}\ln(X(z_{1})-\bullet),\ln(X(z_{2})-\bullet)\big{]}&=\frac{2}{\beta}\ln\bigg(\frac{E(z_{1},z_{2})}{E(z_{1},\infty_{+})E(z_{2},\infty_{+})(X(z_{2})-X(z_{1}))\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{missing}),\\\
\mathcal{Q}\big{[}\ln(X(z)-\bullet),\ln(X(z)-\bullet)\big{]}&=\frac{2}{\beta}\ln\bigg(\frac{1}{E(z,\infty_{+})^{2}\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{missing}),\end{split}$

where $\eta(z)$ is defined in (4.7). In particular, we observe the
simplification

$\mathcal{L}\big{[}\ln(X(z)-\bullet)\big{]}+\frac{\beta}{2}\mathcal{Q}\ln[\ln(X(z)-\bullet),V\big{]}=2{\rm
i}\pi\bm{\epsilon}^{*}\cdot\bm{u}(z)-\ln\big(\eta(z)E(z,\infty_{+})^{2}\differential\zeta_{\infty_{+}}(\infty_{+})\big{missing}).$
(4.5)

###### Proof.

Let $x=X(z)$. The first two formulas are a consequence of the decomposition
(3.8) of $\phi$. . We have:

$\begin{split}\mathcal{L}\big{[}\ln(x-\bullet)\big{]}&=\oint_{S}\frac{\differential\xi}{2{\rm
i}\pi}\,\ln(x-\xi)W_{1}(\xi)\\\ &=\oint_{S}\frac{\differential\xi}{2{\rm
i}\pi}\left[\int_{\infty}^{x}\bigg{(}\frac{1}{\xi^{\prime}-\xi}-\frac{1}{\xi^{\prime}}\bigg{)}\differential\xi^{\prime}+\ln
x\right]W_{1}(\xi)\\\
&=\ln(x)+\int_{\infty}^{x}\differential\xi^{\prime}\bigg{(}\oint_{S}\frac{\differential\xi}{2{\rm
i}\pi}\,\frac{W_{1}(\xi)}{\xi^{\prime}-\xi}-\frac{1}{\xi^{\prime}}\bigg{)}\\\
&=\ln(x)+\int_{\infty}^{x}\differential\xi^{\prime}\bigg{(}W_{1}(\xi^{\prime})-\frac{1}{\xi^{\prime}}\bigg{)}\\\
&=\ln(x)+\int_{\infty_{+}}^{z}\bigg{(}\phi-\frac{\differential
X}{X}\bigg{)}.\end{split}$

We can expand this as

$\begin{split}\mathcal{L}\big{[}\ln(x-\bullet)\big{]}&=\ln x+2{\rm
i}\pi\bm{\epsilon}^{*}\cdot\bm{u}(z)+\int_{\infty_{+}}^{z}\bigg{(}\differential
S_{\infty_{+},\infty_{-}}+\frac{\differential\zeta_{\infty_{+}}}{\zeta_{\infty_{+}}}\bigg{)}-\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}\differential
B_{\infty_{-},k}\\\ &=\ln x+2{\rm
i}\pi\bm{\epsilon}^{*}\cdot\bm{u}(z)+\int_{\infty_{+}}^{z}\differential_{z^{\prime}}\ln\bigg(\frac{E(z^{\prime},\infty_{-})\zeta_{\infty_{+}}(z^{\prime})}{E(z^{\prime},\infty_{+})}\bigg{missing})-\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}B_{\infty_{-},k}\\\
&=\ln x+2{\rm
i}\pi\bm{\epsilon}^{*}\cdot\bm{u}(z)+\ln\bigg(\frac{E(z,\infty_{-})\zeta_{\infty_{+}}(z)}{E(z,\infty_{+})E(\infty_{+},\infty_{-})\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{missing})-\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}B_{\infty_{-},k}.\end{split}$
(4.6)

and $\ln x$ cancels with $\ln\zeta_{\infty_{+}}(z)$. We introduce the $1$-form
on $\widetilde{C}$

$\eta(z)=\frac{E(\infty_{+},\infty_{-})}{E(z,\infty_{+})E(z,\infty_{-})}$
(4.7)

and use it to get rid of $\infty_{-}$ in (4.6). This leads to the claimed
formula.

For the second formula

$\begin{split}\mathcal{Q}\big{[}\ln(x-\bullet),V\big{]}=\frac{2}{\beta}\oint_{S^{2}}\frac{\differential\xi_{1}}{2{\rm
i}\pi}\frac{\differential\xi_{2}}{2{\rm i}\pi}\,V(\xi_{1})\bigg{[}\ln
x+\int_{\infty_{+}}^{x}\bigg{(}\frac{1}{\xi^{\prime}-\xi_{2}}-\frac{1}{\xi^{\prime}}\bigg{)}\differential\xi^{\prime}\bigg{]}W_{2}(\xi_{1},\xi_{2}).\end{split}$

We move the $\xi_{2}$-contour to $\infty$. Since $W_{2}$ has no residue at
$\infty$ the first and third term disappear and we get:

$\begin{split}\mathcal{Q}\big{[}\ln(x-\bullet),V]&=\frac{2}{\beta}\oint_{S}\frac{\differential\xi_{1}}{2{\rm
i}\pi}V(\xi_{1})\int_{\infty_{+}}^{x}W_{2}(\xi_{1},\xi^{\prime})\differential\xi^{\prime}\\\
&=\frac{2}{\beta}\oint_{S}\frac{\differential\xi_{1}}{2{\rm
i}\pi}V(\xi_{1})\int_{\infty_{+}}^{x}\bigg{(}W_{2}(\xi_{1},\xi^{\prime})+\frac{1}{(\xi_{1}-\xi^{\prime})^{2}}\bigg{)}\differential\xi^{\prime},\end{split}$

where in the second line the shift does not affect the integral around $S$ as
it is holomorphic near $S$. Then, we write $X(w)=\xi_{1}$ and
$X(w^{\prime})=\xi^{\prime}$, consider these integrals as integrals on
$\widetilde{C}$, and recognise via (3.11) the fundamental bidifferential $B$.
Since the path on which we integrate $w^{\prime}$ remains in the first sheet
away from the cut, we can move the integral over $w$ to surround the pole
$\infty_{-}$ of $V(X(w))$. We get:

$\begin{split}\mathcal{Q}\big{[}\ln(x-\bullet),V]&=\frac{2}{\beta}\int_{w^{\prime}=\infty_{+}}^{z}\mathop{{\rm
Res}}_{w=\infty_{-}}\differential X(w)\,V(X(w))\,B(w,w^{\prime})\\\
&=\frac{2}{\beta}\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{w^{\prime}=\infty_{+}}^{z}\mathop{{\rm\Res}}_{w=\infty_{-}}\zeta_{\infty_{-}}(w)^{-k}B(w,w^{\prime})\\\
&=\frac{2}{\beta}\sum_{k=1}^{d}\frac{t_{k}}{k}\int_{\infty_{+}}^{z}\differential
B_{\infty_{-},k},\end{split}$

as desired.

For the third formula, let $x_{i}=X(z_{i})$. Using Section 3.3.3, we find

$\begin{split}\mathcal{Q}\big{[}\ln(x_{1}-\bullet),\ln(x_{2}-\bullet)\big{]}&=\frac{2}{\beta}\oint_{S^{2}}\frac{\differential\xi_{1}\differential\xi_{2}}{(2{\rm
i}\pi)^{2}}\,W_{2}(\xi_{1},\xi_{2})\ln(x_{1}-\xi_{1})\ln(x_{2}-\xi_{2})\\\
&=\frac{2}{\beta}\int_{\infty_{+}}^{z_{1}}\int_{\infty_{+}}^{z_{2}}\mathcal{W}_{2}(w_{1},w_{2})\differential
X(w_{1})\differential X(w_{2})\end{split}$

after we handle the logarithms like in the previous proofs. Then:

$\begin{split}\mathcal{Q}\big{[}\ln(x_{1}-\bullet),\ln(x_{2}-\bullet)\big{]}&=\frac{2}{\beta}\int_{\infty_{+}}^{z_{1}}\int_{\infty_{+}}^{z_{2}}\bigg{(}B(w_{1},w_{2})-\frac{\differential
X(w_{1})\differential X(w_{2})}{(X(w_{1})-X(w_{2}))^{2}}\bigg{)}\\\
&=\frac{2}{\beta}\ln\bigg(\frac{E(z_{1},z_{2})E_{0}(z_{1},\infty_{+})E_{0}(\infty_{+},z_{2})}{E(z_{1},\infty_{+})E(\infty_{+},z_{2})E_{0}(z_{1},z_{2})}\bigg{missing}).\end{split}$

where we used Lemma 3.5 both for $\widetilde{C}$ and $\hat{\mathbb{C}}$, and
(3.4) to get rid of the ratio of the relative prime form with two arguments
$\infty_{+}$. Since the prime forms are antisymmetric in their two variables,
we can arrange the formula to have $\infty_{+}$ always in the second argument.
The presence of $\infty_{+}$ in $E_{0}$ factors can be understood by first
replacing it by a point $\tilde{z}$, and then letting
$\tilde{z}\rightarrow\infty_{+}$. Due to Lemma 3.6 the product of the two
$E_{0}$-factors involving $\infty_{+}$ only gives a sign when we use the local
coordinate $\zeta_{\infty_{+}}$, and using
$E_{0}(z_{1},z_{2})=(X(z_{1})-X(z_{2}))/\sqrt{\differential
X(z_{1})\differential X(z_{2})}$ leads to the claim. ∎

We are now in position to evaluate the asymptotics of the kernels. We will
mainly be interested in a situation with only two variables:

$\begin{split}\mathcal{K}_{M}^{\frac{K}{M}V}\big{(}\begin{smallmatrix}c&\tilde{c}\\\
x&\tilde{x}\end{smallmatrix}\big{)}&:=\left\langle{\det(x-\Lambda)^{c}\det(\tilde{x}-\Lambda)^{\tilde{c}}e^{\frac{\beta}{2}p{\rm
Tr}V(\Lambda)}}\right\rangle_{M}^{V}\\\
&=\left\langle{e^{\sum_{i=1}^{M}c\ln(x-\lambda_{i})+\tilde{c}\ln(\tilde{x}-\lambda_{i})+\frac{\beta}{2}pV(\lambda_{i})}}\right\rangle_{M}^{V}.\end{split}$
(4.8)

where $p=M-K$. It will be used in the form:

$\frac{Z_{M}^{\frac{K}{M}V}}{Z_{K}^{V}}\left\langle{\det(x-\Lambda)^{c}\det(\tilde{x}-\Lambda)^{\tilde{c}}}\right\rangle_{M}^{\frac{K}{M}V}=\frac{Z_{M}^{V}}{Z_{K}^{V}}\,\mathcal{K}_{M}^{\frac{K}{M}V}\big{(}\begin{smallmatrix}c&\tilde{c}\\\
x&\tilde{x}\end{smallmatrix}\big{)}.$ (4.9)

######  Lemma 4.8.

Let $z,\tilde{z}\in C_{+}$ and $c,\tilde{c}\in\mathbb{Z}$. We have as
$K\rightarrow\infty$ and $p$ is a fixed integer:

$\begin{split}&\quad\mathcal{K}^{\frac{K}{K+p}V}_{K+p}\big{(}\begin{smallmatrix}c&\tilde{c}\\\
X(z)&X(\tilde{z})\end{smallmatrix}\big{)}\\\ &\sim
e^{Kc\mathcal{L}[\ln(X(z)-\bullet)]+K\tilde{c}\mathcal{L}[\ln(X(\tilde{z})-\bullet)]+c\mathcal{H}[\ln(X(z)-\bullet)]+\tilde{c}\mathcal{H}[\ln(X(\tilde{z})-\bullet)]+K\frac{\beta}{2}p\mathcal{L}[V]+\frac{\beta}{2}p^{2}\mathcal{L}[V]+\frac{\beta}{2}p\mathcal{H}[V]+\frac{\beta^{2}}{8}p^{2}\mathcal{Q}[V,V]}\\\
&\quad\times e^{2{\rm i}\pi
p\bm{\epsilon}^{*}\cdot(c\bm{u}(z)+\tilde{c}\bm{u}(\tilde{z}))}\Big{(}\eta(z)E(z,\infty_{+})^{2}\differential\zeta_{\infty_{+}}(\infty_{+})\Big{)}^{-pc}\Big{(}\eta(\tilde{z})E(\tilde{z},\infty_{+})^{2}\differential\zeta_{\infty_{+}}(\infty_{+})\Big{)}^{-p\tilde{c}}\\\
&\quad\times
e^{\frac{1}{2}c^{2}\mathcal{Q}[\ln(X(z)-\bullet),\ln(X(z)-\bullet)]+\frac{1}{2}\tilde{c}^{2}\mathcal{Q}[\ln(X(\tilde{z})-\bullet),\ln(X(\tilde{z})-\bullet)]}\bigg{(}\frac{E(z,\tilde{z})}{E(z,\infty_{+})E(\tilde{z},\infty_{+})(X(\tilde{z})-X(z))\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{\frac{2}{\beta}c\tilde{c}}\\\
&\quad\times\frac{\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+c\bm{u}(z)+\tilde{c}\bm{u}(\tilde{z})+\frac{\beta}{2}p(\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-}))\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-M\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}.\end{split}$

In particular if $p=0$, we have as $M\rightarrow\infty$

$\begin{split}\mathcal{K}^{V}_{M}\big{(}\begin{smallmatrix}c&\tilde{c}\\\
X(z)&X(\tilde{z})\end{smallmatrix}\big{)}&\sim
e^{Mc\mathcal{L}[\ln(X(z)-\bullet)]+M\tilde{c}\mathcal{L}[\ln(X(\tilde{z})-\bullet)]+c\mathcal{H}[\ln(X(z)-\bullet)]+\tilde{c}\mathcal{H}[\ln(X(\tilde{z})-\bullet)]}\\\
&\quad\times
e^{\frac{1}{2}c^{2}\mathcal{Q}[\ln(X(z)-\bullet),\ln(X(z)-\bullet)]+\frac{1}{2}\tilde{c}^{2}\mathcal{Q}[\ln(X(\tilde{z})-\bullet),\ln(X(\tilde{z})-\bullet)]}\\\
&\quad\times\bigg{(}\frac{E(z,\tilde{z})}{(X(\tilde{z})-X(z))E(z,\infty_{+})E(\tilde{z},\infty_{+})\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{\frac{2}{\beta}c\tilde{c}}\,\frac{\vartheta_{-M\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+c\bm{u}(z)+\tilde{c}\bm{u}(\tilde{z})\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}.\end{split}$

###### Proof.

Let $x=X(z)$ and $\tilde{x}=X(\tilde{z})$. Applying Theorem 4.2 to the
definition (4.8), we get as $M,K\rightarrow\infty$ while $M-K=p$ is fixed:

$\begin{split}\mathcal{K}^{\frac{K}{M}V}_{M}\big{(}\begin{smallmatrix}c&\tilde{c}\\\
x&\tilde{x}\end{smallmatrix}\big{)}&\sim\frac{Z^{V}_{M}}{Z^{\frac{K}{M}V}_{M}}e^{Mc\mathcal{L}[\ln(x-\bullet)]+M\tilde{c}\mathcal{L}[\ln(\tilde{x}-\bullet)]+c\mathcal{H}[\ln(x-\bullet)]+\tilde{c}\mathcal{H}[\ln(\tilde{x}-\bullet)]+M\frac{\beta}{2}p\mathcal{L}[V]+\frac{\beta}{2}\mathcal{H}[V]+\frac{\beta^{2}}{8}p^{2}\mathcal{Q}[V,V]}\\\
&\quad\times
e^{\frac{1}{2}c^{2}\mathcal{Q}[\ln(x-\bullet),\ln(x-\bullet)]+c\tilde{c}\mathcal{Q}[\ln(x-\bullet),\ln(\tilde{x}-\bullet)]+\frac{1}{2}\tilde{c}^{2}\mathcal{Q}[\ln(\tilde{x}-\bullet),\ln(\tilde{x}-\bullet)]+\frac{\beta}{2}pc\mathcal{Q}[\ln(x-\bullet),V]+\frac{\beta}{2}p\tilde{c}\mathcal{Q}[\ln(\tilde{x}-\bullet),V]}\\\
&\quad\times\
\frac{\vartheta_{-M\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\frac{1}{2{\rm
i}\pi}\oint_{S}\big{[}c\ln(x-\bullet)+\tilde{c}\ln(\tilde{x}-\bullet)+\frac{\beta}{2}pV\big{]}\differential\bm{u}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-M\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}.\end{split}$

We split $M=K+p$ in the exponential and combine the newly created $p$-terms
with the $\frac{\beta}{2}\mathcal{Q}[\ln,V]$ terms which is evaluated thanks
to (4.5). We also replace the term $c\tilde{c}\mathcal{Q}[\ln,\ln]$ by its
evaluation from Lemma 4.7, but refrain from doing so for the $c^{2}$ and the
$\tilde{c}^{2}$ terms. Finally, writing the logarithm in the arguments of the
theta function as a primitive, we get an expression in terms of the Abel map
(a similar manipulation was carried out in the proof of Lemma 4.4), and Lemma
4.6 tells us $\frac{1}{2{\rm
i}\pi}\oint_{S}V\differential\bm{u}=\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-})$
which we need to multiply by $\frac{\beta p}{2}$ in the last argument of the
theta function. Together with Lemma 4.6 this implies the claimed formula. ∎

The ratio of partition functions appearing in (4.9) will only be needed
through the following combination.

######  Lemma 4.9.

For fixed $p\in\mathbb{Z}$, we have as $K\rightarrow\infty$

$\begin{split}\frac{Z_{K+p}^{V}Z_{K-p}^{V}}{(Z_{K}^{V})^{2}}\sim
e^{2p^{2}\mathcal{E}[\mu_{\text{eq}}]+{\rm i}\pi\beta
p^{2}\bm{\epsilon}^{*}\cdot\bm{\tau}(\bm{\epsilon}^{*})}\frac{\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}-\tfrac{\beta}{2}p\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\tfrac{\beta}{2}p\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}^{2}}\,.\end{split}$

###### Proof.

Using Theorem 4.1 we find:

$\frac{Z_{K+p}^{V}}{Z_{K}^{V}}\sim
e^{(2Kp+p^{2})\mathcal{E}[\mu_{\text{eq}}]+p\mathcal{S}[\mu_{\text{eq}}]+\frac{\beta}{2}p(\ln
K+1)}\,\frac{\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}\,.$

Multiplying this expression and the same with $p\rightarrow-p$ yields:

$\begin{split}\frac{Z_{K+p}^{V}Z_{K-p}^{V}}{(Z_{K}^{V})^{2}}\sim
e^{2p^{2}\mathcal{E}[\mu_{\text{eq}}]}\,\frac{\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}\vartheta_{-(K-p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}{\vartheta^{2}_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\beta}{2}\bm{\tau}\big{)}}\,.\end{split}$
(4.10)

We would like to rewrite all theta functions with a characteristic
$-K\bm{\epsilon}^{*}$ instead of $-(K\pm p)\bm{\epsilon}^{*}$. For this, we
come back to the definitions in Section 3.1 and find:

$\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}=e^{{\rm
i}\pi\frac{\beta}{2}p^{2}\bm{\epsilon}^{*}\cdot\bm{\tau}(\bm{\epsilon}^{*})-2{\rm
i}\pi
p\bm{\epsilon}^{*}\cdot\bm{v}}\,\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}-p\tfrac{\beta}{2}\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)},$
(4.11)

which holds for any $\bm{v}\in\mathbb{C}^{g}$. We multiply the outcome with
the same factor for $p\rightarrow-p$, we obtain:

$\vartheta_{-(K+p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}\vartheta_{-(K-p)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}=e^{{\rm
i}\pi\beta
p^{2}\bm{\epsilon}^{*}\cdot\bm{\tau}(\bm{\epsilon}^{*})}\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}-\tfrac{\beta}{2}p\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}\vartheta_{-K\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}+\tfrac{\beta}{2}p\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\beta}{2}\bm{\tau}\big{)}.$

Inserting in (4.10) yields the claim. ∎

##  5 Derivation of the theta identities

We shall now state and prove the main theorems. For each case
$\beta\in\\{1,2,4\\}$ we give both a proof based on the analysis in the
previous sections, and a second, direct proof based on geometric arguments. We
recall the definition of the meromorphic $1$-form $\eta$ on $\widetilde{C}$
that appeared in (4.7)

$\eta(z)=\frac{E(\infty_{+},\infty_{-})}{E(z,\infty_{+})E(z,\infty_{-})}.$

Throughout this section, the Abel map will always be based at $\infty_{+}$,
and $\bm{u}(\infty_{-})$ is computed using a path from $\infty_{+}$ to
$\infty_{-}$ which do not cross any of the representatives
$(\mathcal{A}_{h},\mathcal{B}_{h})_{h=1}^{g}$ obtained by analytic
continuation from the ones in Section 3.3.1, see the discussion above Lemma
3.14.

###  5.1 The $\beta=2$ formula

######  Theorem 5.1.

Consider a hyperelliptic curve $\hat{C}$, and let
$z,z^{\prime},w,w^{\prime}\in\widetilde{C}$, and
$\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$. Then, we have:

$\begin{split}&\quad(X(w)-X(z^{\prime}))(X(z)-X(w^{\prime}))\frac{E(z,w)E(z^{\prime},w^{\prime})}{E(w,z^{\prime})E(z,w^{\prime})}\frac{\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,z^{\prime})E(w,w^{\prime})}\vartheta_{\bm{\mu},\bm{\nu}}(\bm{0}\big{|}\bm{\tau})\\\
&\quad-(X(z)-X(w))(X(z^{\prime})-X(w^{\prime}))\frac{\vartheta_{\bm{\mu},\bm{\nu}}(\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,z^{\prime})}\frac{\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(w,w^{\prime})}\\\
&=E(z,w)E(z^{\prime},w^{\prime})\left(\prod_{p\in\\{z,z^{\prime},w,w^{\prime}\\}}\eta(p)\right)\,\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}.\end{split}$
(5.1)

Notice that each of the three terms is actually a $\frac{1}{2}$-form on the
universal cover $\widetilde{C}$ in each of the variable
$z,z^{\prime},w,w^{\prime}$. Even if $\bm{\mu}=0$, it does not descend to the
curve itself. As the proof will show, the formula holds equally well for
arbitrary $\bm{\mu},\bm{\nu}$ complex, in particular, if we shift all
arguments of the theta functions by the same arbitrary but common
$\bm{\nu}\in\mathbb{C}^{g}$.

###### Proof.

Consider first the case where the Weierstraß points of $\hat{C}$ are real. By
Proposition 3.11, we can find a $\beta$-ensemble whose spectral curve has
$\hat{C}$ as underlying Riemann surface and for which the results of Section 4
apply (by construction the potential is off-critical). We then express the
identity of Theorem 2.6 for $m_{1}=m_{2}=1$, for $x=X(z)$,
$x^{\prime}=X(z^{\prime})$, $\tilde{x}=X(w)$ and
$\tilde{x}^{\prime}=X(w^{\prime})$ pairwise distinct points in
$\mathbb{C}\setminus A$, that determine unique points
$z,z^{\prime},w,w^{\prime}\in C_{+}$. Taking into account the definition of
$\mathcal{K}$ in (4.9), we get:

$\begin{split}&\quad\left\langle{\frac{\det(x-\Lambda)\det(\tilde{x}-\Lambda)}{\det(x^{\prime}-\Lambda)\det(\tilde{x}^{\prime}-\Lambda)}}\right\rangle_{N}^{V}\\\
&=(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})\frac{N}{N+1}\frac{Z_{N-1}^{V}Z_{N+1}^{V}}{(Z_{N}^{V})^{2}}\mathcal{K}^{\frac{N}{N-1}V}_{N-1}\big{(}\begin{smallmatrix}1&1\\\
x&\tilde{x}\end{smallmatrix}\big{)}\mathcal{K}^{\frac{N}{N+1}V}_{N+1}\big{(}\begin{smallmatrix}-1&-1\\\
\tilde{x}^{\prime}&x^{\prime}\end{smallmatrix}\big{)}+\mathcal{K}_{N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x&x^{\prime}\end{smallmatrix}\big{)}\mathcal{K}_{N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
\tilde{x}&\tilde{x}^{\prime}\end{smallmatrix}\big{)}.\end{split}$ (5.2)

Let us first consider the asymptotics of left-hand side as
$N\rightarrow\infty$. We have $\bm{v}_{\text{eq}}=0$ since $\beta=2$. Coming
back to Theorem 4.2 and using Lemma 4.7 for the $\mathcal{Q}$-terms involving
two different variables:

$\begin{split}&\quad\left\langle{\frac{\det(x-\Lambda)\det(\tilde{x}-\Lambda)}{\det(x^{\prime}-\Lambda)\det(\tilde{x}^{\prime}-\Lambda)}}\right\rangle_{N}^{V}\\\
&\sim
e^{N\mathcal{L}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\mathcal{H}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\frac{1}{2}\sum_{\xi\in\\{x,\tilde{x},x^{\prime},\tilde{x}^{\prime}\\}}\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi-\bullet)]}\\\
&\quad\times\frac{E(z,w)E(z^{\prime},w^{\prime})E(z,\infty_{+})E(z^{\prime},\infty_{+})E(w,\infty_{+})E(w^{\prime},\infty_{+})\big{(}\differential\zeta_{\infty_{+}}(\infty_{+})\big{)}^{2}}{E(z,z^{\prime})E(z,w^{\prime})E(w,z^{\prime})E(w,w^{\prime})}\,\frac{(x-x^{\prime})(x-\tilde{x}^{\prime})(\tilde{x}-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})}{(x-\tilde{x})(x^{\prime}-\tilde{x}^{\prime})}\\\
&\quad\times\frac{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{0}\big{|}\bm{\tau}\big{)}}.\end{split}$
(5.3)

For the asymptotics of the first term of the right-hand side of
(LABEL:smallrelb2), we use the $\beta=2$ specialisation of Lemma 4.9 with
$p=1$ for the ratio of partition functions, and Lemma 4.8 for the
$\mathcal{K}$-factors with $K=N$, $M=N\mp 1$ and $(c,\tilde{c})=(\pm 1,\pm
1)$, that is $p=\mp 1$. The outcome is

$\begin{split}&\quad(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})\frac{N}{N+1}\frac{Z_{N-1}^{V}Z_{N+1}^{V}}{(Z_{N}^{V})^{2}}\mathcal{K}^{\frac{N}{N-1}V}_{N-1}\big{(}\begin{smallmatrix}1&1\\\
x&\tilde{x}\end{smallmatrix}\big{)}\mathcal{K}^{\frac{N}{N+1}V}_{N+1}\big{(}\begin{smallmatrix}-1&-1\\\
\tilde{x}^{\prime}&x^{\prime}\end{smallmatrix}\big{)}\\\ &\sim
e^{2\mathcal{E}[\mu_{\text{eq}}]+2\mathcal{L}[V]+\mathcal{Q}[V,V]-2{\rm
i}\pi\bm{\epsilon}^{*}\cdot(\bm{u}(z)+\bm{u}(z^{\prime})+\bm{u}(w)+\bm{u}(w^{\prime}))}\\\
&\quad\times
e^{N\mathcal{L}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\mathcal{H}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\frac{1}{2}\sum_{\xi\in\\{x,x^{\prime},\tilde{x},\tilde{x}^{\prime}\\}}\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi-\bullet)]}\\\
&\quad\times\frac{E(z,w)E(z^{\prime},w^{\prime})(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})}{(x-\tilde{x})(x^{\prime}-\tilde{x}^{\prime})}\prod_{p\in\\{z,z^{\prime},w,w^{\prime}\\}}\eta(p)E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\\\
&\quad\times\frac{\vartheta_{-(N-1)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})-\bm{\tau}(\bm{\epsilon}^{*})\big{|}\bm{\tau}\big{)}\vartheta_{-(N+1)\bm{\epsilon}^{*},\bm{0}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})+\bm{\tau}(\bm{\epsilon}^{*})\big{|}\bm{\tau}\big{)}}{\vartheta^{2}_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{0}\big{|}\bm{\tau}\big{)}}.\end{split}$
(5.4)

In the last line we can restore a characteristic $-N\bm{\epsilon}^{*}$ in the
theta functions thanks to (4.11), which we have to use respectively with
$\bm{v}=\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})-\bm{\tau}(\bm{\epsilon}^{*})$
and
$\bm{v}=-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})+\bm{\tau}(\bm{\epsilon}^{*})$.
The outcome is

$\begin{split}&\quad\vartheta_{-(N-1)\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})-\bm{\tau}(\bm{\epsilon}^{*})\big{|}\bm{\tau}\big{)}\vartheta_{-(N+1)\bm{\epsilon}^{*},\bm{0}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+
\bm{u}(\infty_{-})+\bm{\tau}(\bm{\epsilon}^{*})\big{|}\bm{\tau}\big{)}\\\
&=e^{2{\rm i}\pi\bm{\epsilon}^{*}\cdot\bm{\tau}(\bm{\epsilon}^{*})+2{\rm
i}\pi\bm{\epsilon}^{*}\cdot(\bm{u}(z)+\bm{u}(z^{\prime})+\bm{u}(w)+\bm{u}(w^{\prime})-2\bm{u}(\infty_{-})-2\bm{\tau}(\bm{\epsilon}^{*}))}\\\
&\quad\times\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}.\end{split}$

Inserting this in (5.4) gives an exponential prefactor

$e^{2\mathcal{E}[\mu_{\text{eq}}]+2\mathcal{L}[V]+\mathcal{Q}[V,V]+4{\rm
i}\pi\bm{\epsilon}^{*}\cdot(\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-}))}$

which would be equal to $1$ if we had Proposition 4.3. We will establish
Proposition 4.3 as a byproduct in Section 5.4; for the moment, we proceed
assuming it holds. We get:

$\begin{split}&\quad(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})\frac{N}{N+1}\frac{Z_{N-1}^{V}Z_{N+1}^{V}}{(Z_{N}^{V})^{2}}\mathcal{K}^{\frac{N}{N-1}V}_{N-1}\big{(}\begin{smallmatrix}1&1\\\
x&\tilde{x}\end{smallmatrix}\big{)}\mathcal{K}^{\frac{N}{N+1}V}_{N+1}\big{(}\begin{smallmatrix}-1&-1\\\
\tilde{x}^{\prime}&x^{\prime}\end{smallmatrix}\big{)}\\\ &\sim
e^{N\mathcal{L}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\mathcal{H}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\frac{1}{2}\sum_{\xi\in\\{x,x^{\prime},\tilde{x},\tilde{x}^{\prime}\\}}\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi-\bullet)]}\\\
&\quad\times\frac{E(z,w)E(z^{\prime},w^{\prime})(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})}{(x-\tilde{x})(x^{\prime}-\tilde{x}^{\prime})}\prod_{p\in\\{z,z^{\prime},w,w^{\prime}\\}}\eta(p)E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\\\
&\quad\times\frac{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{+})\big{|}\bm{\tau}\big{)}}{\vartheta_{-N\bm{\epsilon}^{*}}^{2}\big{(}\bm{0}\big{|}\bm{\tau}\big{)}}.\end{split}$
(5.5)

The asymptotics of the second term is simpler as we just need to use the $p=0$
case of Lemma 4.8. The outcome is:

$\begin{split}\mathcal{K}_{N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x&x^{\prime}\end{smallmatrix}\big{)}\mathcal{K}_{N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
\tilde{x}&\tilde{x}^{\prime}\end{smallmatrix}\big{)}&\sim
e^{N\mathcal{L}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\mathcal{H}\big{[}\ln\big(\frac{(x-\bullet)(\tilde{x}-\bullet)}{(x^{\prime}-\bullet)(\tilde{x}^{\prime}-\bullet)}\big{missing})\big{]}+\frac{1}{2}\sum_{\xi\in\\{x,x^{\prime},\tilde{x},\tilde{x}^{\prime}\\}}\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi-\bullet)]}\\\
&\quad\times\frac{(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})}{E(z,z^{\prime})E(w,w^{\prime})}\prod_{p\in\\{z,z^{\prime},w,w^{\prime}\\}}E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\\\
&\quad\times\frac{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}^{2}\big{(}\bm{0}\big{|}\bm{\tau}\big{)}}.\end{split}$
(5.6)

We also observe that (5.3), (5.5) and (5.6) all have the same exponential
factor involving $\mathcal{L},\mathcal{H}$ and $\mathcal{Q}$, the same factor

$\prod_{p\in\\{z,z^{\prime},w,w^{\prime}\\}}E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})},$

and the same squared $\vartheta$ in the denominator — except for (5.3) where
the latter is not squared. After we cancel those and multiply further by

$\frac{(x-\tilde{x})(x^{\prime}-\tilde{x}^{\prime})}{(x-x^{\prime})(\tilde{x}-\tilde{x}^{\prime})},$

the identity (LABEL:smallrelb2) as $N\rightarrow\infty$ then becomes:

$\begin{split}&\quad\frac{E(z,w)E(z^{\prime},w^{\prime})\,(x-\tilde{x}^{\prime})(\tilde{x}-x^{\prime})}{E(z,z^{\prime})E(z,w^{\prime})E(w,z^{\prime})E(w,w^{\prime})}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{0}\big{|}\bm{\tau}\big{)}\\\
&=E(z,w)E(z^{\prime},w^{\prime})\eta(z)\eta(z^{\prime})\eta(w)\eta(w^{\prime})\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\\\
&\quad+\frac{(x-\tilde{x})(x^{\prime}-\tilde{x}^{\prime})}{E(z,z^{\prime})E(w,w^{\prime})}\,\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}\vartheta_{-N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}+o(1).\end{split}$
(5.7)

Let $\bm{\mu}\in\mathbb{R}^{g}$. Let us assume temporarily that
$\epsilon_{1}^{*},\ldots,\epsilon_{g}^{*}$ are $\mathbb{Q}$-linearly
independent. Then, by Kronecker’s theorem, one can find an increasing sequence
$(N^{(n)})_{n\geq 1}$ such that

$\begin{split}\lim_{n\rightarrow\infty}\big{(}N^{(n)}\bm{\epsilon}^{*}+\lfloor-N^{(n)}\bm{\epsilon}^{*}\rfloor\big{)}=-\bm{\mu},\end{split}$

where the integer part is applied to each component of the vector. Using
$N=N^{(n)}$ in (5.7) and letting $n\rightarrow\infty$ we get the same identity
without the $o(1)$ and with characteristic $\bm{\mu},\bm{0}$ for all theta
functions, which is (5.1) with $\bm{\nu}=\bm{0}$ and pairwise distinct points
$z,z^{\prime},w,w^{\prime}\in C_{+}$.

If $\epsilon_{1}^{*},\ldots,\epsilon_{g}^{*}$ are not $\mathbb{Q}$-linearly
independent, thanks to Corollary 3.13 we can take a sequence of
$\beta$-ensembles whose spectral curve admits as underlying Riemann surface a
hyperelliptic curve $\hat{C}^{(n)}$ with real Weierstraß points converging to
those of $\hat{C}$, and whose filling fractions are $\mathbb{Q}$-linearly
independent. By the previous argument, we know (5.1) with arbitrary
$\bm{\mu}\in\mathbb{R}^{g}$ and $\bm{\nu}=\bm{0}$ and
$z,z^{\prime},w,w^{\prime}\in C^{(n)}_{+}$ pairwise distinct. Since all the
members of this identity are continuous in the real Weierstraß points
$a_{0}<b_{0}<\cdots<a_{g}<b_{g}$ while the values
$X(z),X(z^{\prime}),X(w),X(w^{\prime})\in\mathbb{C}$ are fixed away from them,
taking $n\rightarrow\infty$ yields the formula also holds for $\hat{C}$,
$\bm{\nu}=0$ and pairwise distinct points $z,z^{\prime},w,w^{\prime}\in
C_{+}$. Let us call $(\star)$ this formula. From there, we can derive the
desired identity in full generality by using repeatedly analytic continuation,
as follows.

Firstly, all terms in ($\star$) are holomorphic functions of
$\bm{\mu}\in\mathbb{C}^{g}$, and the identity holds for real $\bm{\mu}$.
Therefore, it must hold as well for complex $\bm{\mu}$. In particular, we can
replace $\bm{\mu}$ with $\bm{\mu}+\bm{\tau}^{-1}(\bm{\nu})$ for arbitrary
$\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$, and rewrite all theta functions as

$\vartheta_{\bm{\mu}+\bm{\tau}(\bm{\nu}),\bm{0}}(\bm{z}|\bm{\tau})=e^{{\rm
i}\pi\bm{\tau}^{-1}(\bm{\nu})\cdot\bm{\nu}+2{\rm
i}\pi\bm{\tau}^{-1}(\bm{\nu})\cdot\bm{z}}\vartheta_{\bm{\mu},\bm{\nu}}(\bm{z}|\bm{\tau})$

The resulting phase is common to the three terms of the identity, therefore
($\star$) is valid for arbitrary $\bm{\mu},\bm{\nu}\in\mathbb{C}^{g}$, and a
fortiori for arbitrary real $\bm{\mu},\bm{\nu}$.

Secondly, fix $R>0$ large enough, and let $\widetilde{\bm{\Delta}}_{2g+2}(R)$
be the subset of points in our parameter space of marked hyperelliptic curves
$\widetilde{\bm{\Delta}}_{2g+2}$ such that the $X$-image of the Weierstraß
points have moduli $\leq R$. Having fixed and pairwise distinct values
$x,x^{\prime},\tilde{x},\tilde{x}^{\prime}\in\mathbb{C}$ such that
$\max\big{(}|x|,|x^{\prime}|,|\tilde{x}|,|\tilde{x}^{\prime}|\big{)}\geq 2R$
determines a unique quadruple of analytic sections
$z,z^{\prime},w,w^{\prime}:\widetilde{\bm{\Delta}}_{2g+2}(R)\rightarrow\hat{\bm{C}}$
such that $x=X(z)$, $x^{\prime}=X(z^{\prime})$, $\tilde{x}=X(w)$ and
$\tilde{x}^{\prime}=X(w^{\prime})$. These sections represent points in the
(moving with parameters) hyperelliptic curve. Due to Lemma 3.14 and the
discussion preceding it, all terms in ($\star$) are holomorphic functions on
$\widetilde{\bm{\Delta}}_{2g+2}(R)$, but since we know that ($\star$) holds in
the connected component of the base point in the real locus of
$\widetilde{\bm{\Delta}}_{2g+2}(R)$, it must also hold over the whole
$\widetilde{\bm{\Delta}}_{2g+2}(R)$ with
$\max\big{(}|x|,|x^{\prime}|,|\tilde{x}|,|\tilde{x}^{\prime}|\big{)}\geq 2R$.
Now rather fixing a marked hyperelliptic curve corresponding to a point in
$\widetilde{\bm{\Delta}}_{2g+2}(R)$, the identity ($\star$) is valid for
points $z,z^{\prime},w,w^{\prime}$ in a neighborhood of $\infty_{+}$, but
since it can be seen as an identity involving only meromorphic functions of
$z,z^{\prime},w,w^{\prime}\in\widetilde{C}$, it must hold for arbitrary
quadruple of points in $\widetilde{C}$. Eventually as $R$ is arbitrary in the
argument we get (5.1) over the whole parameter space
$\widetilde{\bm{\Delta}}_{2g+2}$ and quadruples of points in the universal
cover of the associated hyperelliptic curve. ∎

This formula implies Fay identity, as we now show.

######  Proposition 5.2.

Formula (5.1) implies Fay identity (1.1) for hyperelliptic curves.

###### Proof.

The key remark is that $z$ and $w$ play almost symmetric roles in (5.1). We
write the two equations obtained when exchanging $z$ and $w$, specialised to
$\bm{\mu}=0$ and $\bm{\nu}\in\mathbb{C}^{g}$ arbitrary but rather transferred
to the argument of the theta functions, so that everything is expressed in
terms of $\theta=\vartheta_{\bm{0},\bm{0}}$:

$\begin{split}&\quad(X(w)-X(z^{\prime}))(X(z)-X(w^{\prime}))\frac{E(z,w)E(z^{\prime},w^{\prime})}{E(w,z^{\prime})E(z,w^{\prime})}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,z^{\prime})E(w,w^{\prime})}\theta\big{(}\bm{\nu}\big{|}\bm{\tau}\big{)}\\\
&\quad-(X(z)-X(w))(X(z^{\prime})-X(w^{\prime}))\frac{\theta(\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,z^{\prime})}\frac{\theta\big{(}\bm{\nu}+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(w,w^{\prime})}\\\
&=E(z,w)E(z^{\prime},w^{\prime})\eta(z)\eta(z^{\prime})\eta(w)\eta(w^{\prime})\theta\big{(}\bm{\nu}+\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\\\
&=(X(z)-X(z^{\prime}))(X(w)-X(w^{\prime}))\frac{E(z,w)E(z^{\prime},w^{\prime})}{E(z,z^{\prime})E(w,w^{\prime})}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(w,z^{\prime})E(z,w^{\prime})}\theta\big{(}\bm{\nu}\big{|}\bm{\tau}\big{)}\\\
&\quad-(X(z)-X(w))(X(z^{\prime})-X(w^{\prime}))\frac{\theta\big{(}\bm{\nu}+\bm{u}(w)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}}{E(w,z^{\prime})}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,w^{\prime})}\,.\end{split}$
(5.8)

Subtracting the first member from the third member of the equalities, grouping
the terms together and dividing by $(X(z)-X(w))(X(z^{\prime})-X(w^{\prime}))$
yields the identity

$\begin{split}0&=\frac{E(z,w)E(z^{\prime},w^{\prime})}{E(w,z^{\prime})E(z,w^{\prime})E(z,z^{\prime})E(w,w^{\prime})}\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}\big{|}\bm{\tau}\big{)}\\\
&\quad-\frac{\theta\big{(}\bm{\nu}+\bm{u}(w)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(w,z^{\prime})E(z,w^{\prime})}+\frac{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{E(z,z^{\prime})E(w,w^{\prime})},\end{split}$

which is exactly Fay identity (1.1) after we replace the prime form with its
expression (3.2) and take
$(z_{1},z_{2},z_{3},z_{4})=(z,w,z^{\prime},w^{\prime})$. ∎

Finally, we provide a direct proof of (5.1) based on complex analysis. This
proof is based on a classical theorem of Riemann which we recall for the
convenience of the reader.

######  Theorem 5.3 ([Mum07, Theorem 3.1]).

There is a vector $\bm{k}\in\mathbb{C}^{g}$ such that for all
$\nu^{\prime}\in\mathbb{C}^{g}$, the function
$z\mapsto\theta\big{(}\bm{\nu}^{\prime}+\bm{u}(z)\big{|}\bm{\tau}\big{)}$ of
$z\in\widetilde{C}$ either vanishes identically or has $g$ zeroes
$w_{1},\ldots,w_{g}$ in a fundamental domain, satisfying

$\sum_{h=1}^{g}\bm{u}(w_{h})=-\bm{\nu}+\bm{k}\mod\mathbb{L}$

$\bm{k}$ is called vector of Riemann constants.

###### Direct geometric proof of Theorem 5.1.

It suffices to prove the identity for $\bm{\mu}=0$, since we still have
$\bm{\nu}\in\mathbb{C}^{g}$ arbitrary which allow reconstructing arbitrary
characteristics. Let $\bm{\nu}^{\prime}\in\mathbb{C}^{g}$. Riemann’s theorem
5.3 implies that seen as a function of $z\in\widetilde{C}$, the theta function
$z\mapsto\theta\big{(}\bm{\nu}^{\prime}+\bm{u}(z)\big{|}\bm{\tau}\big{)}$ is
either identically zero of has $g$ zeroes in a fundamental domain. Let
$\mathcal{D}_{\bm{\nu}^{\prime}}$ be its zero divisor. We apply this to
$\bm{\nu}^{\prime}=\bm{\nu}+\bm{u}(w)-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})$
with $\bm{\nu},z^{\prime},w,w^{\prime}$ generic such that it is not in the
theta divisor. A classical consequence of Riemann’s theorem [FK92, Theorem
VI.3.3.] is that meromorphic functions on $C$ with pole divisor at most
$\mathcal{D}_{\bm{\nu}^{\prime}}$ are constant. For convenience, write

$\begin{split}\mathfrak{c}_{1}&=\frac{(X(w)-X(z^{\prime}))(X(z)-X(w^{\prime}))E(z,w)E(z^{\prime},w^{\prime})}{E(w,z^{\prime})E(z,w^{\prime})E(z,z^{\prime})E(w,w^{\prime})}\\\
\mathfrak{c}_{2}&=-\frac{(X(z)-X(w))(X(z^{\prime})-X(w^{\prime}))}{E(z,z^{\prime})E(w,w^{\prime})}\\\
\mathfrak{c}_{3}&=E(z,w)E(z^{\prime},w^{\prime})\eta(z)\eta(z^{\prime})\eta(w)\eta(w^{\prime})\end{split}$

and consider

$\begin{split}\Psi(z)&=\frac{\mathfrak{c}_{2}}{\mathfrak{c}_{1}}\,\frac{\theta(\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}}{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}\big{|}\bm{\tau}\big{)}}\\\
&\quad+\frac{\mathfrak{c}_{3}}{\mathfrak{c}_{1}}\,\frac{\theta\big{(}\bm{\nu}+\bm{u}(z)+\bm{u}(w)-\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}-\bm{u}(z^{\prime})-\bm{u}(w^{\prime})+\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}}{\theta\big{(}\bm{\nu}+\bm{u}(z)-\bm{u}(z^{\prime})+\bm{u}(w)-\bm{u}(w^{\prime})\big{|}\bm{\tau}\big{)}\theta\big{(}\bm{\nu}\big{|}\bm{\tau}\big{)}}.\end{split}$

This is a meromorphic function of $z\in\hat{C}$. We have seen that the theta
function in the denominator has $g$ zeroes – which are thus poles of $\Psi$.
We now consider the poles of the other factors: the zeroes of
$\mathfrak{c}_{1}$ and the poles of $\mathfrak{c}_{2}$ and $\mathfrak{c}_{3}$.
The coefficient $\mathfrak{c}_{1}$ has a simple zero at
$z=\jmath(w^{\prime})$, where $\jmath$ is the hyperelliptic involution, and at
$z=w$. The coefficients $\mathfrak{c}_{2}$ and $\mathfrak{c}_{3}$ have simple
poles at $z=z^{\prime}$. Accordingly, both ratios
$\frac{\mathfrak{c}_{2}}{\mathfrak{c}_{1}}$ and
$\frac{\mathfrak{c}_{3}}{\mathfrak{c}_{1}}$ have a simple pole only at
$z=\jmath(w^{\prime})$. A careful computation of the residues taking into
account $\bm{u}(\jmath(w^{\prime}))=\bm{u}(\infty_{-})-\bm{u}(w^{\prime})$
shows that $\Psi$ does not have a pole when $z=\jmath(w^{\prime})$. Notice
that there are no pole at $\infty_{\pm}$ as poles coming from linear terms are
cancelled by other linear terms or the form $\eta$. We conclude that the
divisor of poles of $\Psi$ is at most $\mathcal{D}_{\bm{\nu}^{\prime}}$. Thus,
it is a constant function of $z$. A similar argument with the other variables
would show that it is a constant function of $z,z^{\prime},w,w^{\prime}$. By
sending the points $z^{\prime},w,w^{\prime}$ to $z$ one after the other, and
we arrive to $\Psi=1$. ∎

###  5.2 The $\beta=1$ formula

######  Theorem 5.4.

Consider a marked hyperelliptic curve $\hat{C}$ and let
$z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}\in\widetilde{C}$ and
$\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$. Writing $x_{i}=X(z_{i})$ and
$x_{i}^{\prime}=X(z_{i}^{\prime})$, we have

$\begin{split}&\quad\bigg{(}\frac{E(z_{1},z_{2})E(z^{\prime}_{1},z_{2}^{\prime})}{E(z_{1},z_{1}^{\prime})E(z_{1},z_{2}^{\prime})E(z_{2},z_{1}^{\prime})E(z_{2},z_{2}^{\prime})}\bigg{)}^{2}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z_{1})-\bm{u}(z^{\prime}_{1})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{0}\big{|}\tfrac{\bm{\tau}}{2}\big{)}\\\
&\quad=\frac{(x_{1}-x_{2})(x^{\prime}_{1}-x_{2}^{\prime})}{(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}\frac{\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z_{1})-\bm{u}(z_{2}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}}{E(z_{1},z_{2}^{\prime})^{2}}\frac{\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z_{2})-\bm{u}(z_{1}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}}{E(z_{1}^{\prime},z_{2})^{2}}\\\
&\quad-\frac{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}{(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})}\frac{\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}}{E(z_{1},z^{\prime}_{1})^{2}}\frac{\vartheta_{\bm{\mu},\bm{\nu}}(\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}}{E(z_{2}^{\prime},z_{2})^{2}}\\\
&\quad+\frac{\big{(}E(z_{1},z_{2})E(z^{\prime}_{1},z^{\prime}_{2})\eta(z_{1})\eta(z_{1}^{\prime})\eta(z_{2})\eta(z_{2}^{\prime})\big{)}^{2}}{(x_{1}-x^{\prime}_{1})(x_{1}-x_{2}^{\prime})(x_{2}-x^{\prime}_{1})(x_{2}-x_{2}^{\prime})}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{u}(z_{1})+\bm{u}(z_{2})-\bm{u}(\infty_{-})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{\bm{\mu},\bm{\nu}}\big{(}-\bm{u}(z^{\prime}_{1})-\bm{u}(z^{\prime}_{2})+\bm{u}(\infty_{-})\big{|}\tfrac{\bm{\tau}}{2}\big{)},\end{split}$
(5.9)

###### Proof.

The strategy is similar to the proof for $\beta=2$ in Theorem 5.1. In
particular, we first prove an asymptotic identity for hyperelliptic curves
arising from $\beta=1$ ensembles, use approximations to get arbitrary
characteristic $\bm{\mu},\bm{0}$, and then analytic continuation to get the
identity for marked hyperelliptic curves with arbitrary complex Weierstraß
points and characteristics $\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$.

The starting point is the exact identity of Theorem 2.7 in the simplest non-
trivial case, i.e. $m=2$ (pfaffian of size $4$). Taking
$x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime}\in\mathbb{C}\setminus A$ pairwise
distinct, this gives

$\begin{split}&\quad\left\langle{\frac{\det(x_{1}-\Lambda)\det(x_{2}-\Lambda)}{\det(x_{1}^{\prime}-\Lambda)\det(x_{2}^{\prime}-\Lambda)}}\right\rangle_{2N}^{V}\\\
&=\frac{2N(2N-1)}{(2N+2)(2N+1)}(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})\frac{Z_{2N-2}^{V}Z_{2N+2}^{V}}{(Z_{2N}^{V})^{2}}\mathcal{K}_{2N-2}^{\frac{2N}{2N-2}V}\big{(}\begin{smallmatrix}1&1\\\
x_{1}&x_{2}\end{smallmatrix}\big{)}\mathcal{K}_{2N+2}^{\frac{2N}{2N+2}V}\big{(}\begin{smallmatrix}-1&-1\\\
x^{\prime}_{1}&x_{2}^{\prime}\end{smallmatrix}\big{)}\\\
&\quad-\frac{(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\mathcal{K}_{2N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x_{1}&x_{1}^{\prime}\end{smallmatrix}\big{)}\mathcal{K}_{2N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x_{2}&x_{2}^{\prime}\end{smallmatrix}\big{)}+\frac{(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\mathcal{K}_{2N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x_{1}&x_{2}^{\prime}\end{smallmatrix}\big{)}\mathcal{K}_{2N}^{V}\big{(}\begin{smallmatrix}1&-1\\\
x_{2}&x_{1}^{\prime}\end{smallmatrix}\big{)}.\end{split}$ (5.10)

As the computation is similar to $\beta=2$ we only streamline it. Let
$z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}\in C_{+}$ be such that
$x_{i}=X(z_{i})$ and $x_{i}^{\prime}=X(z_{i}^{\prime})$. The asymptotic
equivalent of the left-hand side of (5.10) as $N\rightarrow\infty$ is obtained
from Theorem 4.2:

$\begin{split}&\quad
e^{(2N\mathcal{L}+\mathcal{H})\big{[}\ln\big(\frac{(x_{1}-\bullet)(x_{2}-\bullet)}{(x_{1}^{\prime}-\bullet)(x_{2}^{\prime}-\bullet)}\big{missing})\big{]}+\frac{1}{2}\sum_{\xi\in\\{x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime}\\}}\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi-\bullet)]}\\\
&\times\bigg{(}\prod_{p\in\\{z_{1},z^{\prime}_{1},z_{2},z_{2}^{\prime}\\}}E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{2}\\\
&\times\bigg{(}\frac{E(z_{1},z_{2})E(z^{\prime}_{1},z_{2}^{\prime})}{E(z_{1},z_{1}^{\prime})E(z_{1},z_{2}^{\prime})E(z_{2},z^{\prime}_{1})E(z_{2},z_{2}^{\prime})}\,\frac{(x_{1}-x_{1}^{\prime})(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\bigg{)}^{2}\\\
&\times\frac{\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{1})-\bm{u}(z^{\prime}_{1})+\bm{u}(z_{2})-\bm{u}(z^{\prime}_{2})\big{|}\frac{\bm{\tau}}{2}\big{)}}{\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}\big{|}\frac{\bm{\tau}}{2}\big{)}}.\end{split}$
(5.11)

We have used Lemma 4.7 to evaluate the
$\mathcal{Q}[\ln(\xi-\bullet),\ln(\xi^{\prime}-\bullet)]$ terms that appear
with $\xi\neq\xi^{\prime}$. In the right-hand side of (5.10), we use the
asymptotics of the $2$-point kernel from Lemma 4.8 with $K=2N$ and $p=\mp 2$.

Consider the asymptotics of the first term in the right-hand side of (5.10).
It contains a product of theta functions with characteristic $-(2N\pm
2)\bm{\epsilon}^{*}$, which we can replace by two theta functions with same
characteristic $-2N\bm{\epsilon}^{*}$ using (5.11) up to an extra exponential
factor. The latter combines with the asymptotics of the ratio of partition
functions of shifted size from Lemma 4.9 to reproduce a factor

$e^{8\mathcal{E}[\mu_{\text{eq}}]+4\mathcal{L}[V]+\mathcal{Q}[V,V]+8{\rm
i}\pi\bm{\epsilon}^{*}\cdot(\bm{\tau}(\bm{\epsilon}^{*})+\bm{u}(\infty_{-}))}$

which is equal to $1$ due to Proposition 4.3, and to kill the factor of
$e^{-4{\rm
i}\pi\bm{\epsilon}^{*}\cdot(\bm{u}(z_{1})+\bm{u}(z_{2})+\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2}^{\prime}))}$
coming from the use of (4.5). The other factors are the first line of (5.11)
multiplied by

$\begin{split}&\quad\bigg{(}\frac{E(z_{1},z_{2})E(z_{1}^{\prime},z_{2}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\prod_{p\in\\{z_{1},z^{\prime}_{1},z_{2},z_{2}^{\prime}\\}}\eta(p)E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{2}\\\
&\times\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}-\bm{\tau}(\epsilon^{*})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{\tau}(\bm{\epsilon}^{*})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\\\
&\times\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{1})+\bm{u}(z_{2})-\bm{u}(\infty_{-})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}-\bm{u}(z_{1}^{\prime})-\bm{u}(z_{2}^{\prime})+\bm{u}(\infty_{-})\big{|}\tfrac{\bm{\tau}}{2}\big{)},\end{split}$

where the last line was already explained.

The asymptotics of the second and third terms in the right-hand side of (5.10)
are more straightforward to get. They both contain the first line and the
third line of (5.11), and the other asymptotic factors are

$\begin{split}&\quad-\frac{(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\,\bigg{(}\frac{(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}{E(z_{1},z_{1}^{\prime})E(z_{2},z_{2}^{\prime})}\prod_{p\in\\{z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}\\}}E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{2}\\\
&\times\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\end{split}$

for the second term (including its sign), and

$\begin{split}&\quad\frac{(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\,\bigg{(}\frac{(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})}{E(z_{1},z_{2}^{\prime})E(z_{2},z_{1}^{\prime})}\prod_{p\in\\{z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}\\}}E(p,\infty_{+})\sqrt{\differential\zeta_{\infty_{+}}(\infty_{+})}\bigg{)}^{2}\\\
&\times\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{1})-\bm{u}(z_{2}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{-2N\bm{\epsilon}^{*},\bm{0}}\big{(}\bm{v}_{\text{eq}}+\bm{u}(z_{2})-\bm{u}(z_{1}^{\prime})\big{|}\tfrac{\bm{\tau}}{2}\big{)}.\end{split}$

We then divide all terms by the first and second line of (5.11) (common factor
to all terms) and by

$\bigg{(}\frac{(x_{1}-x_{1}^{\prime})(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}\bigg{)}^{2}$

to arrive to (5.9) with $\bm{\mu}=-2N\bm{\epsilon}^{*}$, $\bm{\nu}=0$ and an
extra $\bm{v}_{\text{eq}}$ added to the arguments of all theta functions.
Then, we repeat the end of the proof of Theorem 5.1 to get exactly and in full
generality the claimed (5.1). ∎

Theorem 5.4 can be reformulated as an identity for theta functions with matrix
of periods $\bm{\tau}$ instead of $\frac{\bm{\tau}}{2}$.

######  Lemma 5.5.

There is an equivalence between (5.9) for any
$\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$, and the formula

$\begin{split}0&=\mathfrak{c}_{1}\,\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}+\mathfrak{c}_{2}\,\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{u}(z_{1})+\bm{u}(z_{1}^{\prime})-\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}\\\
&\quad+\mathfrak{c}_{3}\,\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{u}(z_{1})-\bm{u}(z^{\prime}_{1})-\bm{u}(z_{2})+\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}+\mathfrak{c}_{4}\,\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{u}(z_{1})+\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})+\bm{u}(z_{2}^{\prime})-2\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}\,,\end{split}$
(5.12)

for any $\bm{\alpha}\in\mathbb{Z}^{g}/2\mathbb{Z}^{g}$, where

$\begin{split}\mathfrak{c}_{1}&=-\bigg{(}\frac{E(z_{1},z_{2})E(z_{1}^{\prime},z_{2}^{\prime})}{E(z_{1},z_{1}^{\prime})E(z_{1},z_{2}^{\prime})E(z_{2},z_{1}^{\prime})E(z_{2},z_{2}^{\prime})}\bigg{)}^{2},\\\
\mathfrak{c}_{2}&=\frac{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}{(x_{1}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}\frac{1}{\big{(}E(z_{1},z_{2}^{\prime})E(z_{1}^{\prime},z_{2})\big{)}^{2}},\\\
\mathfrak{c}_{3}&=-\frac{(x_{1}-x_{2})(x_{1}^{\prime}-x_{2}^{\prime})}{(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})}\frac{1}{\big{(}E(z_{1},z_{1}^{\prime})E(z_{2}^{\prime},z_{2})\big{)}^{2}},\\\
\mathfrak{c}_{4}&=\frac{\big{(}E(z_{1},z_{2})E(z_{1}^{\prime},z_{2}^{\prime})\eta(z_{1})\eta(z_{1}^{\prime})\eta(z_{2})\eta(z_{2}^{\prime})\big{)}^{2}}{(x_{1}-x_{1}^{\prime})(x_{1}-x_{2}^{\prime})(x_{2}-x_{1}^{\prime})(x_{2}-x_{2}^{\prime})}.\end{split}$

###### Proof.

The trick is to use Riemann’s binary addition theorem — see e.g. [Mum07,
Equation 6.6]. It states that for any
$\bm{\mu},\bm{\nu},\bm{\mu}^{\prime},\bm{\nu}^{\prime}\in\mathbb{R}^{g}$ and
$\bm{z}_{1},\bm{z}_{2}\in\mathbb{C}^{g}$

$\vartheta_{\bm{\mu},\bm{\nu}}\big{(}\bm{z}_{1}+\bm{z}_{2}\big{|}\tfrac{\bm{\tau}}{2}\big{)}\vartheta_{\bm{\mu}^{\prime},\bm{\nu}^{\prime}}\big{(}\bm{z}_{1}-\bm{z}_{2}\big{|}\tfrac{\bm{\tau}}{2}\big{)}=\sum_{\bm{\alpha}\in\mathbf{Z}^{g}/2\mathbf{Z}^{g}}\vartheta_{\frac{\bm{\mu}+\bm{\mu^{\prime}}+\bm{\alpha}}{2},\bm{\nu}+\bm{\nu}^{\prime}}\big{(}2\bm{z}_{1}\big{|}\bm{\tau}\big{)}\vartheta_{\frac{\bm{\mu}-\bm{\mu}^{\prime}+\bm{\alpha}}{2},\bm{\nu}-\bm{\nu}^{\prime}}\big{(}2\bm{z}_{2}\big{|}\bm{\tau}\big{)}.$
(5.13)

We apply the transformation (5.13) with $\bm{\mu}^{\prime}=\bm{\mu}$ and
$\bm{\nu}=\bm{\nu}^{\prime}$ to each term in (5.9), writing it in the
equivalent form

$\sum_{\bm{\alpha}\in\mathbb{Z}^{g}/2\mathbb{Z}^{g}}\vartheta_{\bm{\mu}+\frac{\bm{\alpha}}{2},\bm{\nu}}\big{(}\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}\bigg{(}\sum_{i=1}^{4}\mathfrak{c}_{i}\,\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{w}_{i}(z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime})\big{|}\bm{\tau}\big{)}\bigg{)}$
(5.14)

where the $\bm{w}_{i}$ are exactly the four arguments of the theta functions
appearing in (5.12). In this form, the converse implication is clear. The
direct implication follows from the observation that $\bm{\nu}$ is arbitrary,
and the family of functions
$T_{\bm{\alpha}}(\bm{\nu})=\vartheta_{\frac{\bm{\alpha}}{2},\bm{0}}\big{(}\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}$
indexed by $\bm{\alpha}\in\mathbb{Z}^{g}/2\mathbb{Z}^{g}$ are linearly
independent, forcing the sum inside the bracket to vanish for each individual
$\bm{\alpha}$. ∎

This is an identity involving only Riemann theta functions, for which we can
offer a direct geometric proof, in a slightly more general form.

######  Theorem 5.6.

Equation 5.12 holds for any marked hyperelliptic curve for any characteristic
$\bm{\mu},\bm{\nu}\in\mathbb{R}^{g}$ (instead of just half-integer
characteristics).

###### Direct geometric proof of Theorem 5.4.

The strategy is similar of the direct proof of Theorem 5.1. We start without
lack of generality to set $\bm{\mu}=0$ and for $\bm{\nu}\in\mathbb{C}^{g}$
arbitrary, consider

$\begin{split}\Psi(z_{1})&=\frac{\mathfrak{c}_{2}}{\mathfrak{c}_{1}}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})+\bm{u}(z_{1}^{\prime})-\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}}{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})-\bm{u}(z^{\prime}_{1})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}}\\\
&+\frac{\mathfrak{c}_{3}}{\mathfrak{c}_{1}}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})-\bm{u}(z_{2})+\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}}{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}}\\\
&+\frac{\mathfrak{c}_{4}}{\mathfrak{c}_{1}}\frac{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})+\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})+\bm{u}(z_{2}^{\prime})-2\bm{u}(\infty_{-})\big{|}\bm{\tau}\big{)}}{\theta\big{(}\bm{\nu}+\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}}\,.\end{split}$

This is a meromorphic function of $z_{1}\in\hat{C}$. We first analyse the
poles that may come from the ratios of coefficients. The ratio
$\frac{\mathfrak{c}_{2}}{\mathfrak{c}_{1}}$ has simple poles at $z_{1}=z_{2}$
and $z_{1}=\jmath(z^{\prime}_{1})$, where $\jmath$ is the hyperelliptic
involution. The ratio $\frac{\mathfrak{c}_{3}}{\mathfrak{c}_{1}}$ has simple
poles only at $z_{1}=\jmath(z_{2}^{\prime})$ and $z_{1}=z_{2}$. The ratio
$\frac{\mathfrak{c}_{4}}{\mathfrak{c}_{1}}$ has simple poles only at
$z_{1}=\jmath(z_{2}^{\prime})$ and $z_{1}=\jmath(z_{2}^{\prime})$. However,
careful computation of the residues show that $\Psi$ has none of these poles.
Thus, the only poles of $\Psi$ are the zeros of
$z_{1}\mapsto\theta\big{(}\bm{\nu}+\bm{u}(z_{1})-\bm{u}(z_{1}^{\prime})+\bm{u}(z_{2})-\bm{u}(z_{2}^{\prime})\big{|}\bm{\tau}\big{)}$.
As in the direct proof of Theorem 5.1, Riemann’s Theorem implies that if we
choose the points $z_{1}^{\prime},z_{2},z_{2}^{\prime}$ and the vector
$\bm{\nu}$ generically, there are no nonconstant meromorphic function whose
poles are the zeroes of this theta function. We deduce that $\Psi(z_{1})$ does
not depend on $z_{1}$. A similar argument shows that $\Psi(z_{1})$ is
independent of all points points $z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}$.
Sending $z_{1}^{\prime},z_{2},z_{2}^{\prime}$ successively to $z_{1}$ we find
that the constant is $1$. ∎

###  5.3 The $\beta=4$ formula

The case $\beta=4$ has the same structure as the $\beta=1$ case of Theorem
5.4, except that the argument of the theta functions are doubled while we use
the matrix $2\bm{\tau}$. This similarity is already manifest in the exact
formulae of Theorems 2.7 and 2.8.

######  Theorem 5.7.

Consider a marked hyperelliptic curve $\hat{C}$, and let
$z_{1},z_{1}^{\prime},z_{2},z_{2}^{\prime}\in\widetilde{C}$ and
$\bm{\mu},\bm{\nu}\in\mathbb{C}^{g}$. Writing $x_{i}=X(z_{i})$ and
$x_{i}^{\prime}=X(z_{i}^{\prime})$, we have have: