eqs

$\begin{split}\BODY\end{split}$ (1)

# 2D Fermions and Statistical Mechanics: Critical Dimers and Dirac Fermions in
a background gauge field

Sri Tata1 1Yale University, Department of Mathematics

###### Abstract

In the limit of the lattice spacing going to zero, we consider the dimer model
on isoradial graphs in the presence of singular $U(N,\mathbb{C})$ gauge fields
flat away from a set of punctures. We consider the cluster expansion of this
twisted dimer partition function show it matches an analogous cluster
expansion of the 2D Dirac partition function in the presence of this gauge
field. The latter is often referred to as a tau function. This reproduces and
generalizes various computations of Dubédat [1]. In particular, both sides’
cluster expansion are matched up term-by-term and each term is shown to equal
a sum of a particular holomorphic integral and its conjugate. On the dimer
side, we evaluate the terms in the expansion using various exact lattice-level
identities of discrete exponential functions and the inverse Kasteleyn matrix.
On the fermion side, the cluster expansion leads us to two novel series
expansions of tau functions, one involving the Fuschian representation and one
involving the monodromy representation.

###### Contents

  1. I Introduction
  2. II Preliminaries
    1. II.1 Dimer Model on Isoradial Graphs, $\overline{\partial}$, and $\frac{1}{\overline{\partial}}$
    2. II.2 Edge-Probabilities and Correlation Kernels
    3. II.3 Graph connections and twisting $\overline{\partial}$
    4. II.4 Height Functions of Dimer Matchings
  3. III Two Punctures
    1. III.1 Straight-line Zippers
    2. III.2 Cluster Expansion of the twisted determinant: single straight-line zipper
      1. III.2.1 Real and Imaginary Parts of Expansions
    3. III.3 Exact expressions for zipper traces
      1. III.3.1 Non-Universality of Imaginary Parts of log-determinant
    4. III.4 Asymptotics of Eq. (27) for $\mathcal{I}_{2n}$
      1. III.4.1 Setting up the calculation
      2. III.4.2 The $n=1$ case, $\mathcal{I}_{2}$
      3. III.4.3 The general case
  4. IV Multiple Punctures
    1. IV.1 Setup for well-separated Zippers
    2. IV.2 Cluster Expansion of the twisted determinant: multiple zippers
    3. IV.3 Evaluating $\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]$ for well-separated zippers
      1. IV.3.1 Setup and Notation
      2. IV.3.2 A first example
      3. IV.3.3 More Setup and More Notation
      4. IV.3.4 Another representative example
      5. IV.3.5 Argument for general terms
    4. IV.4 Modifications for computing $\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]$ with zippers sharing endpoints.
  5. V Discussion
  6. A Asymptotics of the functions $g_{v\to w}(z)$
  7. B Spectral Theory of Finite Hilbert Transformations
  8. C Summing over the leading-order contributions, $-\sum_{n=1}^{\infty}\frac{1}{2n}(2\sin(\alpha/2))^{2}\mathcal{J}_{2n}$ for a single zipper
  9. D  2D Free Fermion Determinants and Tau Functions
    1. D.1 Singular Gauge Fields and Isomonodromic Tau functions
    2. D.2 Determinants of 2D Dirac operators, Counterterms, and Gauge Invariance
      1. D.2.1 Dirac Determinants of Abelian connections
    3. D.3 ‘Fuschian’ expansion of $\log\tau$ and the Jimbo-Miwa-Ueno equations
    4. D.4 ‘Zipper’ expansion of $\log\tau$

##  I Introduction

The dimer model is a statistical model that counts perfect matchings on a
planar graph. The Dirac fermion in two dimensions is a simple yet important
quantum field theory. Both of these models occupy special places in the study
of two-dimensional physics, particularly in the setting of critical phenomena.
In this paper, we show that in a wide sense these theories are the same.

On one hand, the dimer model on the square grid was among the first models for
which conformal invariance was rigorously demonstrated [2, 3]. Moreover, exact
‘discrete holomorphic’ structures studying dimers on ‘critical’ graphs [4]
have been essential in establishing conformal invariance in more general
settings, such as in the Ising model [5, 6].

On the other hand, the 2D Dirac fermion is also an important example of a
conformal field theory. Moreover, it is a ‘building block’ of the large class
of rational conformal field theories (e.g. the Ising model, Potts model,…) in
the sense that its current algebra generates the operator algebras of these
theories (c.f. [7]).

These two theories are closely related. First, we have that the dimer model’s
partition function is the determinant of a so-called Kasteleyn matrix
$\overline{\partial}$. This suggestive notation is because the Kasteleyn
matrix can be thought of as a discretization of the holomorphic derivative
(see Sec. II.1). One way to phrase the dimer partition function is thus to use
the language of Grassmann calculus (see e.g. [7] for a review on Grassmann
calculus)

$Z_{\mathrm{dimer}}=\int d\overline{\psi}d\psi
e^{-\int\overline{\psi}\,\overline{\partial}\,\psi}=\det(\overline{\partial}).$

Immediately, this shares an aesthetic similarity 111Similar observations were
made in [8]. with the partition function of the the two dimensional fermion,

$Z_{\mathrm{Dirac,2D}}=\int d\overline{\psi}d\psi
e^{-\int\overline{\psi}\,\not{\partial}\,\psi}=\det(\not{\partial}),$

especially since the two-dimensional Dirac operator $\not{\partial}$ is
comprised of a holomorphic and antiholomorphic derivative.

This similarity is also quantitative. An early indication around 2000 was the
result [3] that studied fluctuations in the ‘height function’, an integer-
valued function on faces of the graph that’s dual to a dimer matching (see
Sec. II.4). These fluctuations were shown (in the sense of a weak limit) to
identify the height function to the Gaussian Free Field, which is the free
boson theory in 2D with action
$S_{\mathrm{boson}}(\phi)=-{\pi}\int(\partial\phi)^{2}$. This duality is
reminiscent of the continuous version of the boson-fermion duality.

On higher genus surfaces, there are multiple $\overline{\partial}$ operators
identified with spin structures [9] and higher-genus bosonization formulas
[10] apply to compute partition functions, although conformal invariance on
the lattice is harder to formulate and establish on higher-genus surfaces.

Later in 2011, Dubédat [11] considered the stronger forms of height
fluctuations and shows that they agree with the Gaussian Free Field / free
boson expectations. He coupled the dimer model to a background
$U(1)$-connection, often taken to be singular, and computed the partition
function with respect to the connection. A key tool he uses is a special
discrete holomorphic function with monodromy. In [1], he extends this analysis
to coupling to certain (singular) $SL(2,\mathbb{C})$ connections. In both of
these analyses, he finds that the dimer partition function with respect to
this connection matches with a $\tau$ function, which is a mathematical
construction of a 2D Dirac determinant in the presence of a singular
connection (see the Appendix D). This can be rephrased as follows. Given a
background lattice connection $U$ corresponding to a continuum gauge field
$A$, these analyses demonstrate asymptotic equalities between the (normalized)
partition functions

$Z_{\mathrm{dimer}}(U)=\det(\overline{\partial}(U))\quad\text{ and }\quad
Z_{\mathrm{Dirac,2D}}(A)=\det(\not{\partial}-\not{A})$

where $\overline{\partial}(U)$ is a twisted Kasteleyn operator (see Sec. II.3)
and $\not{\partial}-\not{A}$ is the twisted Dirac operator (see Appendix D).

More generally, graph connections have applications to studying the scaling
limit of the dimer model. In [12], Kenyon used $SL(2,\mathbb{C})$ connections
to prove that loops in the double-dimer model (formed by superimposing two
independent dimer matchings) are distributed in a conformally invariant way,
which was conjectured [13, 14] to be related to $CLE(4)$, a certain ‘conformal
loop ensemble’ of loop-soups in the plane with a conformally invariant
distribution. Dubédat’s work [1] about $SL(2,\mathbb{C})$-connections
established a quantitative link between the twisted dimer partition function
and certain $CLE(4)$ observables. More recently in [15],
$SL(N,\mathbb{C})$-twisted determinants were related to weighted sums over so-
called ‘webs’, weighted by traces of the web.

In this paper, we establish another quantitative link between the partition
function of ‘critical’ dimer models in the presence of background singular
$U(N)$ connections that are flat away from a finite number of ‘punctures’. In
particular, given a $U(N)$ lattice gauge field $U$ and corresponding continuum
field $\not{A}$ that are flat away from a set of punctures, and a ‘scale’ $R$
corresponding to an inverse ‘mesh size’ of the lattice, we will have that the
normalized partition functions are related as:

$\boxed{\frac{\det(\overline{\partial}(U))}{\det(\overline{\partial})}\xrightarrow{R\to\infty,\text{moments}}\frac{\det(\not{\partial}-\not{A})}{\det(\not{\partial})}\times
e^{i\Phi_{\text{n.u.}}}.}$

Above, ‘$\xrightarrow{R\to\infty}$’ means up to $O(1/R)$ factors. We explain
the meaning of $\xrightarrow{\text{moments}}$ below. And,
$e^{i\Phi_{\text{n.u.}}}$ is a non-universal phase factor that depends on
various microscopic data (see Sec. III.3.1). And, the ratio
$\frac{\det(\not{\partial}-\not{A})}{\det(\not{\partial})}$ is a ‘regulated’
fermion determinant that diverges logarithmically as $\log(R)$, but that we
can relate to the standard $\tau$ function (see Sec. D).

This extends and complements the works of Dubédat. His strategy to equate the
dimer partition functions to tau functions is variational, i.e. by displacing
the punctures and finding matching differential equations for the tau function
and for (logarithms of) the partition functions. In particular, this leads to
various undetermined constants in the equations, related to the initial data
of the differential equation.

In our strategy, we consider the cluster expansion of the dimer partition
function. We are able to explicitly relate the series in the cluster expansion
to a series of holomorphic integrals; the heart of these calculations is
various exact lattice-level identities of discrete exponentials on isoradial
graphs (see Sec. III.3). In certain cases, this expansion can be resummed and
lead to determinations of the previously undetermined constants (see Sec.
III.4). A quantum field theory-like calculation (see Appendix D.4) confirms
that the dimer cluster expansion matches the free fermion expansion after a
certain resummation of the Dirac partition function. On the Dirac side,
certain integrals in the cluster expansion are divergent. This is not an issue
for the dimer model, since the lattice scale naturally regulates the integrals
to produce finite numbers that diverge with the lattice scale (see Sec. IV.4).

One disadvantage of our method is that while we can rigorously compute the
terms in the cluster expansion, this method alone cannot yet rigorously show
that their sum converges, whereas in the variational analysis of Dubédat this
convergence is manifest. Combining his results with ours do however lead to
rigorous results about the correlations by showing moments with respect to the
connection converge. As an example, Dubédat shows that (let $-\pi<\alpha<\pi$)
height correlations of the form $\braket{e^{i\alpha(h(F)-h(F^{\prime}))}}$
converge as

$\log|\braket{e^{i\alpha(h(F)-h(F^{\prime}))}}|\xrightarrow{|F-F^{\prime}|\to\infty}-\frac{1}{2\pi^{2}}\alpha^{2}\log(|F-F^{\prime}|)+C(\alpha)$

for some constant $C(\alpha)$, which agrees with classical Coulomb gas
heuristics [16]. For us, we can rigorously compute all moments with respect to
$\alpha$ allowing us to determine $C(\alpha)$ (see Sec. III). Our methods
alone can’t demonstrate that the sum of moments converges, due to the presence
of subleading corrections, but our moment calculations rigorously fix
$C(\alpha)$. Similarly, we can rigorously demonstrate moment generating
functions for twisted Kasteleyn determinants corresponding to other singular
gauge fields (see Sec. IV).

The organization of the paper is as follows. In Sec. II, we review the
preliminaries of isoradial graphs, the dimer model, and graph connections. In
Sec. III, we compute the partition function with respect to connections flat
away from two punctures. In Sec. IV, we compute the partition function with
respect to multiple punctures. The details of the specific calculations, the
expectations, and the results are highlighted in the beginning of each of
those sections. In Sec. V, we conclude with discussion and further questions.

Various technical calculations are left in the appendix. However, of
independent interest in the Appendix D, we give a (non-rigorous) overview of
computing the partition function of Dirac fermions with respect to background
connections in two dimensions. There, we employ the series expansions of the
Dirac fermion in the background connections, demonstrating (after a
‘counterterm correction’) the gauge invariance of the series. Representing the
gauge field in two different ways leads to two different expansions of the
determinant, one related to the Fuschian representation of the connection and
one related to its monodromy representation. The first can be shown to satisfy
the defining equations of the tau function and the second is closely related
to the dimer model partition function. We haven’t seen either of these
expansions in the literature.

##  II Preliminaries

In Sec. II.1, we discuss the basic definitions of isoradial graphs,
$\overline{\partial}$ operator, and its inverse following [4] (see also [17]).
Then, we discuss its relation to the dimer model. See [18] for an introduction
to the dimer model. In Sec. II.3, we discuss graph connections and their
implementation in twisting the dimer model. In Sec. II.4, we discuss the
definition of height functions associated to dimer matchings and the
application of graph connections to the study of height functions.

###  II.1 Dimer Model on Isoradial Graphs, $\overline{\partial}$, and
$\frac{1}{\overline{\partial}}$

An isoradial embedding of a graph is a planar embedding of the graph for which
every edge is drawn as a straight line and every face is circumscribed in a
circle of radius 1; a graph is isoradial if it has an isoradial embedding.
Equivalently, one can think of an isoradial embedding as related to an
underlying rhombus graph as follows. The rhombus graph is a planar graph whose
edges are drawn as straight lines and whose faces are all rhombi of side-
lengths 1. Then the isoradial graph can be formed by drawing edges between
opposite faces of the rhombi. Note that a rhombus graph is bipartite since all
faces have an even number of sides, so this procedure will give rise to two
distinct graphs that are dual to each other and that are both isoradial.

Figure 1:  Example portion of a bipartite isoradial graph. (Bottom-Left)
Legend labeling vertices corresponding to dual faces and black/white vertices,
as well as the edges for the isoradial graph and oriented edges of the rhombus
graph. (Bottom-Right) Depiction of the Kasteleyn operator
$\overline{\partial}$ for the different directions
$\overline{\partial}^{\bullet\to\circ}(b,w)$ and
$\overline{\partial}^{\circ\to\bullet}(w,b)$.

Figure 2:  The discrete exponential function. (Left) Rules for relating
$g_{\cdots\to v^{\prime}}$ and $g_{\cdots\to v}$ for $v,v^{\prime}$ adjacent
on the rhombus graph. Consistency of the products around a rhombus implies
each $g_{x\to y}(z)$ is well-defined and independent of the path $x\to y$.
(Right) Example of discrete exponential between a white and black vertex.

In this paper, we will consider exclusively cases where the isoradial graph
itself is bipartite with a decomposition into black and white vertices. Given
a bipartite isoradial graph, one can give an orientation to each edge of the
rhombus graph by having them point away from white vertices and towards black
ones. As such, every edge of the rhombus graph depicts a vector which is
associated to a unit complex number. See Fig. 2 for a depiction.

Now, we can suggestively define the $\overline{\partial}$ operator on the
graph as follows. Given a black vertex $b$ connected by an edge to a white
vertex $w$, the above-stated orientations of the rhombus graph edges define
rhombus angles $e^{i\vartheta}$,$e^{i\varphi}$ (say
$\vartheta-\pi<\varphi<\vartheta$) coming out from $w$ on the right,left side
of $w$ respectively. Then, we define

$\overline{\partial}^{\bullet\to\circ}(b,w)=-\overline{\partial}^{\circ\to\bullet}(w,b)=i(e^{i\vartheta}-e^{i\varphi})=-e^{i(\vartheta+\varphi)/2}\nu(b,w)\quad,\quad\text{
where }\nu(b,w)=2\sin\left(\frac{\vartheta-\varphi}{2}\right)>0.$ (2)

$\nu(b,w)$ is a positive weight function on edges. Note
$\overline{\partial}^{\bullet\to\circ}(b,w)$ acts from black-to-white vertices
and
$\overline{\partial}^{\circ\to\bullet}(w,b)=-\overline{\partial}^{\bullet\to\circ}(b,w)$
acts from white-to-black. Then, we define

$\begin{split}\overline{\partial}^{\bullet\to\circ}(v,v^{\prime})=\overline{\partial}^{\circ\to\bullet}(v,v^{\prime})=0\quad&,\quad\text{
for all vertices }v,v^{\prime}\text{ not neighboring}\end{split}$

and define the full matrix

$\overline{\partial}=\overline{\partial}^{\bullet\to\circ}+\overline{\partial}^{\circ\to\bullet}.$
(3)

The matrix $\overline{\partial}$ can be thought of as a discretization of the
antiholomorphic derivative
$\partial_{\overline{z}}=\frac{1}{2}(\partial_{x}+i\partial_{y})$. For
example, the square-lattice $(\sqrt{2}\mathbb{Z})^{2}$ is an isoradial graph,
and given a function $f(v)$ on vertices of $(\sqrt{2}\mathbb{Z})^{2}$, we have

$(\overline{\partial}f)(v)=\sqrt{2}(f(v^{\rightarrow})-f(v^{\leftarrow}))+i\sqrt{2}(f(v^{\uparrow})-f(v^{\downarrow}))$

where $v^{\rightarrow},v^{\leftarrow},v^{\uparrow},v^{\downarrow}$ are the
vertices to the right,left,above,below $v$ on the grid. As such, a function
$f$ on the vertices is said to be discrete holomorphic or discrete analytic if
$\overline{\partial}f=0$. For most of this paper, we will only be concerned
with the half $\overline{\partial}^{\bullet\to\circ}$ of
$\overline{\partial}$.

For any finite isoradial graph, it turns out that
$\overline{\partial}^{\bullet\to\circ}$ is a Kasteleyn matrix, whose
determinant gives a positive-weighted sum over dimer configurations (i.e.
perfect matchings) of the graph. In particular, we’d have

$\det(\overline{\partial}^{\bullet\to\circ})=e^{i\Phi_{0}}\,\,\sum_{\begin{subarray}{c}\text{perfect
matchings}\\\ \mathcal{M}\text{ of
graph}\end{subarray}}\,\,\,\prod_{\begin{subarray}{c}\text{edges }{\bf
e}=(b,w)\\\ \text{part of }\mathcal{M}\end{subarray}}\,\,\nu(b,w)$ (4)

where $e^{i\Phi_{0}}$ is a unit complex number that doesn’t change even if we
modify the definition of the weight function $\nu(b,w)$ in Eq. (2). For
example, instead choosing $\nu(b,w)=1$ would instead equate
$\det(\overline{\partial}^{\bullet\to\circ})$ to the number of dimer
matchings, an example of the celebrated Kasteleyn’s Theorem [19]. For our
purposes, we instead use the so-called critical weights as in Eq. (2).

For most of the remainder of this paper, we consider infinte isoradial graphs.
In this case, the determinant $\det(\overline{\partial}^{\bullet\to\circ})$ is
infinite, although ratios of related determinants used to characterize certain
dimer correlation functions will be well-defined. On an infinte graph, there
is a special class of discrete holomorphic functions that we’ll refer to as
discrete exponentials or lattice exponentials. Given two vertices $v,w$ on the
rhombus graph and a path $v\to w$ on the rhombus graph, the discrete
exponential is a rational function $g_{v\to w}(z)$ defined recursively. First,
we’d have $g_{v\to v}(z)=1$. Then, if $v,v^{\prime}$ share an edge on the
rhombus graph with the edge oriented $v\to v^{\prime}$ with
$v^{\prime}-v=e^{i\varphi}$, we’ll have $g_{\cdots\to
v^{\prime}}(z)=g_{\cdots\to v}(z)\cdot\frac{1}{z-e^{i\varphi}}$. The rhombus
graph structure means these functions don’t depend on the path $v\to w$. See
Fig. 2 for depictions and an example. For $v$ any vertex, $b$ any black vertex
and $z$ any complex number avoiding the poles of $g_{v\to\cdots}$, it is
simple to check that $\sum_{w\text{ white}}\overline{\partial}(b,w)g_{v\to
w}(z)=0$ so that $g_{v\to\boldsymbol{\cdot}}(z)$ are all discrete holomorphic.
Note that the definitions also make sense for $v,w$ on the rhombus graph
corresponding to faces - such functions are important in this paper’s
analysis. The functions $g_{v\to w}(z)$ are referred to as exponentials
because of their asymptotic behavior for $|z|\ll 1$ and $|z|\gg 1$, which are
reviewed in Appendix A.

The inverse Kasteleyn matrix $\frac{1}{\overline{\partial}}$ comprised of
$\frac{1}{\overline{\partial}^{\bullet\to\circ}}$ and
$\frac{1}{\overline{\partial}^{\circ\to\bullet}}$ would satisfy

$\sum_{w}\overline{\partial}^{\bullet\to\circ}(b,w)\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b^{\prime})=\delta_{b,b^{\prime}}\quad,\quad\sum_{b}\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b)\overline{\partial}^{\bullet\to\circ}(b,w^{\prime})=\delta_{w,w^{\prime}}\quad,\quad\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b)=-\frac{1}{\overline{\partial}^{\circ\to\bullet}}(b,w)\xrightarrow{b\to\infty}0.$
(5)

In fact, an exact formula [4] exists

$\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b)=-\frac{1}{\overline{\partial}^{\circ\to\bullet}}(b,w)=\frac{1}{2\pi}\int_{0}^{-e^{i\phi}\infty}dz\,g_{w\to
b}(z)\quad,\quad\text{ where }|\phi-\arg(b-w)|<\pi.$ (6)

In the above, the contour $0\to-e^{i\phi}\infty$ is taken to be a straight-
line ray starting going $0\to\infty$ going out in the $-e^{i\phi}$ direction.
The exact choice of contour doesn’t depend on $\phi$, because $g_{w\to
b}(z)\xrightarrow{b\to\infty}\frac{1}{z^{2}}$ and the general fact [4, 20]
that the poles of $g_{w\to b}(z)$ are in the set of unit complex numbers
$\\{+e^{i\phi}\,\,|\,\,\phi\in\mathbb{R}\,\text{ and
}\,|\phi-\arg(b-w)|<\pi\\}$.

###  II.2 Edge-Probabilities and Correlation Kernels

Given a finite isoradial graph and a collection of edges
$\textbf{coll}=\\{{\bf e}_{1},\cdots,{\bf
e}_{n}\\}=\\{(b_{1},w_{1}),\cdots,(b_{n},w_{n})\\}$, we may want to ask what
the probability that all edges in coll are contained in a given dimer
matching. This probability is given by (see e.g. [21])

$\text{Prob}(\text{all edges in }\textbf{coll}\text{ are in a
matching})=\left(\prod_{i=1}^{n}\overline{\partial}(b_{i},w_{i})\right)\cdot\det\left({\left.\kern-1.2pt\overline{\partial}^{-1}\vphantom{\big{|}}\right|_{\\{w_{1},\cdots,w_{n}\\},\\{b_{1},\cdots,b_{n}\\}}}\right),$
(7)

i.e. is the product of Kasteleyn edge weights times a corresponding minor of
the inverse Kasteleyn matrix.

Above, the only way that the above probability could be nonzero is if all
$\\{b_{i}\\}$ and $\\{w_{i}\\}$ are distinct because all vertices can only
appear in one edge at a time. In this paper, it’s important to consider
collections coll of edges with repeated $\\{b_{i}\\}$ or $\\{w_{i}\\}$. The
restriction of $\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}$ to the
edges in coll is imporant, defined as

$\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}(b,w)=\begin{cases}\overline{\partial}^{\bullet\to\circ}(b,w)\quad&\text{
if }(b,w)\in\textbf{coll},\\\ 0&\text{ otherwise.}\end{cases}$ (8)

Also, matrices of the form
$\frac{\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}}{\overline{\partial}^{\bullet\to\circ}}:=\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}\frac{1}{\overline{\partial}^{\bullet\to\circ}}$
are important in our analysis. Indeed, we can see it as a certain correlation
kernel as follows. Consider the principal minors of such matrix. For a
collection $\tilde{B}=\\{\tilde{b}_{1}\cdots\tilde{b}_{k}\\}$ of black
vertices of the graph, we can express the corresponding principal minor as:

$\begin{split}\det\left({\left.\kern-1.2pt\frac{\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}}{\overline{\partial}^{\bullet\to\circ}}\vphantom{\big{|}}\right|_{\tilde{B},\tilde{B}}}\right)&=\sum_{\begin{subarray}{c}\tilde{W}:=\\{\tilde{w}_{1},\cdots,\tilde{w}_{k}\\}\\\
\subset\text{ white
vertices}\end{subarray}}\det\left({\left.\kern-1.2pt\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}\vphantom{\big{|}}\right|_{\tilde{B},\tilde{W}}}\right)\cdot\det\left({\left.\kern-1.2pt\frac{1}{\overline{\partial}^{\bullet\to\circ}}\vphantom{\big{|}}\right|_{\tilde{W},\tilde{B}}}\right)\\\
&=\sum_{\begin{subarray}{c}\tilde{W}:=\\{\tilde{w}_{1},\cdots,\tilde{w}_{k}\\}\\\
\subset\text{ white
vertices}\end{subarray}}\sum_{\sigma\in\mathrm{Sym}(n)}(-1)^{\sigma}\left(\prod_{i=1}^{n}\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}(\tilde{b}_{i},\tilde{w}_{\sigma(i)})\right)\cdot\det\left({\left.\kern-1.2pt\overline{\partial}^{-1}\vphantom{\big{|}}\right|_{\\{\tilde{w}_{1},\cdots,\tilde{w}_{n}\\},\\{\tilde{b}_{1},\cdots,\tilde{b}_{n}\\}}}\right)\\\
&=\sum_{\begin{subarray}{c}\tilde{W}:=\\{\tilde{w}_{1},\cdots,\tilde{w}_{k}\\}\\\
\subset\text{ white
vertices}\end{subarray}}\sum_{\sigma\in\mathrm{Sym}(n)}\left(\prod_{i=1}^{n}\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}(\tilde{b}_{i},\tilde{w}_{\sigma(i)})\right)\cdot\det\left({\left.\kern-1.2pt\overline{\partial}^{-1}\vphantom{\big{|}}\right|_{\\{\tilde{w}_{\sigma(1)},\cdots,\tilde{w}_{\sigma(n)}\\},\\{\tilde{b}_{1},\cdots,\tilde{b}_{n}\\}}}\right)\\\
&=\sum_{\begin{subarray}{c}\tilde{W}:=\\{\tilde{w}_{1},\cdots,\tilde{w}_{k}\\}\\\
\text{ordered collection}\\\ \text{of white
vertices}\end{subarray}}\text{Prob}(\text{all edges
}(\tilde{b}_{1},\tilde{w}_{1})\cdots(\tilde{b}_{n},\tilde{w}_{n})\text{ are in
}\textbf{coll}\text{ and in a matching})\\\
&=\text{Prob}(\tilde{b}_{1},\cdots,\tilde{b}_{n}\text{ are all matched in
edges in }\textbf{coll}).\end{split}$ (9)

The first line is the Cauchy-Binet identity. The second line is expanding the
first determinant. The third line absorbs $(-1)^{\sigma}$ into rearranging the
rows. The fourth line uses Eq. (7) and absorbing the two summations into a
single sum. So, all principal minors lie between $0$ and $1$ since they have a
probabilistic interpretation. Consider the modified characteristic polynomial,

$\begin{split}P(t)=\det\left(\text{1}+(t-1)\frac{\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}}{\overline{\partial}^{\bullet\to\circ}}\right)=&\frac{\det\left(\overline{\partial}^{\bullet\to\circ}+(t-1)\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}\right)}{\det\left(\overline{\partial}^{\bullet\to\circ}\right)}=\sum_{k=1}^{|\textbf{coll}|}t^{k}P_{k},\\\
\text{where }P_{k}=\text{Prob}\Big{(}\textit{exactly }k\text{ distinct
vertices of }&\\{b_{1},\cdots,b_{|\textbf{coll}|}\\}\text{ are matched in
edges of }\textbf{coll}\Big{)}.\end{split}$ (10)

This is because
$\overline{\partial}^{\bullet\to\circ}+(t-1)\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}$
is exactly the matrix $\overline{\partial}^{\bullet\to\circ}$ except the edge
weights get modified as $\nu(b,w)\to t\cdot\nu(b,w)$ for
$(b,w)\in\textbf{coll}$ and remain unchanged for other edges, so the
coefficient of $t^{k}$ in
${\det\left(\overline{\partial}^{\bullet\to\circ}+(t-1)\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}\right)}$
is proportional to the number of matching containing exactly $k$ edges of
coll.

Since the $P_{k}$ are all non-negative, we have the following lemma.

######  Lemma 1.

All real eigenvalues of
$\frac{\overline{\partial}^{\bullet\to\circ}_{\textbf{coll}}}{\overline{\partial}^{\bullet\to\circ}}$
lie between $[0,1]$. And all complex eigenvalues come in conjugate pairs.

###### Proof.

Nonzero eigenvalues $\lambda_{0}$ correspond to roots $t_{0}$ with
$P(t_{0})=0$ as $\lambda_{0}=\frac{1}{1-t_{0}}$. Since $P(t)$ has all non-
negative and some positive coefficients, real $t_{0}$ must have $t_{0}\leq 0$
and complex $t_{0}$ must have a conjugate pair. These statements imply the
lemma. ∎

We showed above lemma was for finite graphs and finite coll, the same
statement holds for infinite graphs and finite coll. In particular, since the
analogous $P(t)$ for infinite graphs will be a limit of polynomials of finite
graphs, $P(t)$ would have non-negative coefficients that add up to 1 again.

###  II.3 Graph connections and twisting $\overline{\partial}$

For a group $G$, a $G$-connection (or $G$\- gauge field or $G$\- local system)
on a graph is a function

$U:\\{\text{directed edges of graph}\\}\to G\quad,\quad\text{so that }U(b\to
w)=U(w\to b)^{-1}\text{ for all edges }{\bf e}=(b\leftrightarrow w).$ (11)

Two connections $U$, $\tilde{U}$ are said to be gauge equivalent if there
exists a function

${\bf g}:\\{\text{vertices of graph}\\}\to G\quad,\quad\text{so that
}\tilde{U}(b\to w)={\bf g}(b)U(b\to w){\bf g}(w)^{-1}\text{ for all edges
}{\bf e}=(b\leftrightarrow w).$ (12)

Such a function ${\bf g}$ is called a gauge transformation. Any gauge
transformation (say, on a finite graph) can be written as a sequence of
elementary gauge transformations for which ${\bf g}(v)=g$ for a single vertex
$v$ while $g(v^{\prime})=1$ for all other vertices $v^{\prime}$.

In this paper, we will be concerning ourselves with the matrix-groups
$G\subset GL(N,\mathbb{C})$ that act on $\mathbb{C}^{N}$. In this case, a
coordinate-free description of a $G$-connection can be given by assigning
vector spaces isomorphic to $\mathbb{C}^{N}$ to each vertex and viewing the
function $U$ as a set of isomorphisms between adjacent vertices’ vector spaces
with respect to some basis. Then, a gauge transformation corresponds to a
change-of-basis at each vertex.

Note that for any loop of vertices $\ell=v_{1}\to v_{2}\to\cdots v_{k}\to
v_{1}$, the product $M(\ell)=U(v_{1}\to v_{2})\cdot U(v_{2}\to
v_{3})\cdot\cdots U(v_{k}\to v_{1})$ known as the monodromy matrix of the loop
changes by conjugation $M(\ell)\mapsto\tilde{M}(\ell)={\bf
g}(v_{1})M(\ell){\bf g}(v_{1})^{-1}$. In general, two connections
$U,U^{\prime}$ are gauge equivalent iff for a base-point $v$ and any loop
$\ell$ starting at $v$, the monodromy matrices $M(\ell),M^{\prime}(\ell)$
coming from $U,U^{\prime}$ are related as $M^{\prime}(\ell)=gM(\ell)g^{-1}$
for some $g\in G$ independent of $\ell$. The field $U$ is flat around a face
if the monodromy around it is the identity matrix. For the infinite graphs of
interest to us, we will consider the cases where the monodromy is flat except
only nontrivial around a finite number of faces, referred to as punctures.

Now, given a $GL(N,\mathbb{C})$\- gauge field $U$ and our Kasteleyn Matrix
$\overline{\partial}$, we can form a twisted Kasteleyn matrix
$\overline{\partial}(U)$ that is $\mathbb{C}$-valued matrix with indices in
$\\{1,\cdots,N\\}\times\\{\text{vertices}\\}$ or equivalently as a
$GL(N,\mathbb{C})$-valued matrix with indices in $\\{\text{vertices}\\}$
follows. The matrix $\overline{\partial}(U)$ is comprised of the parts
$\overline{\partial}^{\bullet\to\circ}(U)$ and
$\overline{\partial}^{\circ\to\bullet}(U)$ and can be written

$\begin{split}\overline{\partial}^{\bullet\to\circ}(U)(b^{(i)},w^{(j)})=\overline{\partial}^{\bullet\to\circ}(b,w)\cdot
U(b\to
w)_{i,j}\quad&\leftrightarrow\quad\overline{\partial}^{\bullet\to\circ}(U)(b,w)=\overline{\partial}^{\bullet\to\circ}(b,w)\cdot
U(b\to w),\quad\text{and}\\\
\overline{\partial}^{\circ\to\bullet}(U)(w^{(i)},b^{(j)})=\overline{\partial}^{\circ\to\bullet}(w,b)\cdot
U(w\to
b)_{i,j}\quad&\leftrightarrow\quad\overline{\partial}^{\circ\to\bullet}(U)(w,b)=\overline{\partial}^{\circ\to\bullet}(w,b)\cdot
U(w\to b).\end{split}$ (13)

In the above, the $n$ vertices $v^{(1)},\cdots,v^{(N)}$ correspond to the $N$
‘copies’ of the vertex $v$.

Figure 3:  Effects of elementary gauge transformations on the twisted
Kasteleyn matrix.

From the latter perspective, one can phrase a gauge transformation as a
conjugation

$\overline{\partial}(\tilde{U})=\mathrm{diag}({\bf
g})\,\overline{\partial}(U)\,\mathrm{diag}({\bf g})^{-1}\quad,\quad\text{where
}\mathrm{diag}({\bf g})(v,v^{\prime})={\bf g}(v)\delta_{v,v^{\prime}}.$ (14)

As depicted in Fig. 3, an elementary gauge transformation $g$ on a vertex $v$
acts on the pieces $\overline{\partial}^{\bullet\to\circ}(U)$ and
$\overline{\partial}^{\circ\to\bullet}(U)$ as

$\begin{split}&\overline{\partial}^{\bullet\to\circ}(\tilde{U})(b,w)=\overline{\partial}^{\bullet\to\circ}(U)(b,w)\cdot
g^{-1}\quad,\quad\overline{\partial}^{\circ\to\bullet}(\tilde{U})(w,b)=g\cdot\overline{\partial}^{\circ\to\bullet}(U)(w,b)\quad,\quad\text{if
}v=w\\\
&\overline{\partial}^{\bullet\to\circ}(\tilde{U})(b,w)=g\cdot\overline{\partial}^{\bullet\to\circ}(U)(b,w)\quad,\quad\overline{\partial}^{\circ\to\bullet}(\tilde{U})(w,b)=\overline{\partial}^{\circ\to\bullet}(U)(w,b)\cdot
g^{-1}\quad,\quad\text{if }v=b\end{split}$

Note that this implies $\det(\overline{\partial})$ is gauge-invariant, i.e.
doesn’t change upon a gauge transformation. However, the individual components
do indeed change as

$\frac{\det(\overline{\partial}^{\bullet\to\circ}(\tilde{U}))}{\det(\overline{\partial}^{\bullet\to\circ}(U))}=\begin{cases}\det(g)&\text{
if }v\text{ black}\\\ \det(g)^{-1}&\text{ if }v\text{
white}\end{cases}\quad,\quad\frac{\det(\overline{\partial}^{\circ\to\bullet}(\tilde{U}))}{\det(\overline{\partial}^{\circ\to\bullet}(U))}\cdot\begin{cases}\det(g)^{-1}&\text{
if }v\text{ black}\\\ \det(g)&\text{ if }v\text{ white}\end{cases}.$ (15)

So although $\det(\overline{\partial}^{\bullet\to\circ}(U))$ isn’t gauge-
invariant, if we restrict to groups $G\subset SL(N,\mathbb{C})$, then we’ll
have $\det(g)=1$ so that $\det(\overline{\partial}^{\bullet\to\circ}(U))$
would indeed be invariant. Also, note that if we restrict all edge matrices
$U$ and gauge transformations to live in $U(N)$, the absolute-value
$\left|\det(\overline{\partial}^{\bullet\to\circ}(U))\right|$ will be gauge
invariant.

In our setup with infinite graphs, we will only consider connections that are
nontrivial in a finite region so all gauge transformations would indeed be
connected by elementary gauge transformations. In particular, this means that
the monodromy ‘around infinity’ will be trivial.

###  II.4 Height Functions of Dimer Matchings

Now we discuss a prototypical application of graph connections which is the
study of height functions on bipartite planar graphs, for which we only need
$G$ to be an abelian group. Given a reference dimer matching, one can assign
to dimer matching a height function on the faces formed as follows. Orient the
reference matching from white-to-black and orient the dimer matching from
black-to-white. Then the combination of the matchings form a set of closed,
directed loops on the graph. The loops define the contour lines of a height
field $h$, where the value of $h$ at a specific face is fixed, say, to be
zero. In particular, dualizing the directed edges clockwise (and ignoring
doubled edges) defines a divergence-free flow on the dual graph that defines a
function after fixing the value at a face. See Fig. 4 for a depiction.

Figure 4:  (Left) Given a reference matching and a dimer matching, we can
construct a height function $h$ on the dual graph. (Right) Adding a zipper
connection of strength $e^{i\alpha}$ between two faces $F,F^{\prime}$ modifies
Eq. (4) to count matchings with an additional weight of
$e^{i\alpha(h(F)-h(F^{\prime})-H_{0})}$. In this case $H_{0}=6$.

As such, one may ask how to study correlation functions of differences in the
height field $\\{h(F)-h(F^{\prime})|F,F^{\prime}\in\\{\text{faces}\\}\\}$. One
way to do this is to add a $\mathbb{C}^{\times}$-valued connection on the
graph. For example, consider adding a connection $U$ of whose only monodromies
are $e^{-i\alpha}$ counterclockwise around $F$ and $e^{i\alpha}$
counterclockwise around $F^{\prime}$ and localize them on judiciously chosen
edges so that either $U(b\to w)=e^{\pm i\alpha}$ or $U(b\to w)=1$ for each
edge and so that the dual of these edges form a path between $F\to
F^{\prime}$. Such a connection is referred to as a zipper. See Fig. 4 for a
depiction. Then we can compute
$\det(\overline{\partial}^{\bullet\to\circ}(U))$ using the formula Eq. (4),
since one can think of the choice of connection modifying the weight function
$\nu$ as adding a weight $\times\exp{(i\alpha(h(F)-h(F^{\prime})-H_{0}))}$ to
each dimer matching. Here, $H_{0}$ is some constant that depends on the
specific choice of zipper. As such, if one can compute
$\det(\overline{\partial}^{\bullet\to\circ}(U))$, taking moments with respect
to $\alpha$ can give expectation values
$\braket{(h(F)-h(F^{\prime})-H_{0})^{k}}$. Similarly, other height
correlations involving faces $F_{1},F_{2},\cdots$ can be studied by adding
multiple zippers of strengths $e^{i\alpha_{1}},e^{i\alpha_{2}},\cdots$ between
various faces.

Note that the height function depends sensitively on the choice of the
reference matching 222Note that there are more invariant definitions of height
functions (see e.g. [22]). and that different gauge-equivalent connections
will change the value of $H_{0}$. This isn’t surprising, since $e^{i\alpha}$
corresponds to a gauge field that’s not in $SL(1,\mathbb{C})=\\{1\\}$, so
gauge transformations of $U$ should be expected to change
$\det(\overline{\partial}^{\bullet\to\circ})(U)$. As such, studying height
function correlations directly comes with the extra baggage of these
microscopic choices. However, restricting to to $U(1)$ connections and $U(1)$
weights leaves the absolute value
$\left|\det(\overline{\partial}^{\bullet\to\circ}(U))\right|$ invariant, and
the baggage of microscopic choices leads only to a non-universal phase of the
determinant.

Similarly, we will choose to focus on computing determinants of
$U(N)$-connections, which are invariant up to a non-universal phase.

##  III Two Punctures

The first computation we do involves punctures at two far-away faces
$F,F^{\prime}$ on the isoradial graph and a connection $U$ flat away from
these punctures. We compute the ratio of determinants
$\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}$.
Formally, both of these quantities are infinite and must be defined by taking
the limit of larger and larger finite graphs that encompass the gauge field
configuration (which we stipulated were only nontrivial in a a finite region).
Then after that, we’d need to take the limit of $|F-F^{\prime}|$ becoming
large. From our perspective, we won’t need to deal with the technicality of
taking larger and larger graphs, and we only worry about the latter limit.
Furthermore, we will need to restrict our attention to isoradial graphs
satisfying certain regularity conditions as outlined in Appendix A.

Since there are only two punctures and the monodromy at infinity is trivial,
we have that the monodromies around the two punctures will be inverses of each
other $M,M^{-1}$. By diagonalization, we can equate this determinant to a
product of the determinants of $\mathbb{C}^{\times}$-valued connections, where
the eigenvalues are in $\mathbb{C}^{\times}$. Even though these aren’t gauge
invariant individually, their products will be. 333 Although, for eigenvalues
that are unit complex numbers, the changes in partition function Eq. (15) are
only unit complex numbers if we restrict to unitary gauge transformations.
With these restrictions in allowed gauge fields $U$, the absolute values
$\left|\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}\right|$
are well-defined.  Say that we put a weight $e^{i\alpha}$ along a zipper
connecting $F\to F^{\prime}$ to form a twisted matrix
$\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})$. The ratio
$\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}$
is related to expectation values $\braket{e^{i\alpha(h(F)-h(F^{\prime}))}}$
for a suitable choice of height function (see Sec. II.4). The expected
convergence of height fluctuations to a Gaussian process matches the
expectation of convergences to the free fermion determinant of an abelian
connection that’s singular and flat away from the punctures. The discussion
surrounding Eq. (152) (see also Sec. D.2.1) gives the heuristics for such a
determinant. In particular, since there’s a monodromy of $e^{i\alpha}$ around
one puncture and $e^{-i\alpha}$ around the other, we would expect that the
continuum limit satisfies

$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}\xrightarrow{|F^{\prime}-F|\to\infty}-\frac{1}{2\pi^{2}}\alpha^{2}\log(|F^{\prime}-F|)$

However, a subtlety is that the twisted Kasteleyn determinant must necessarily
be periodic in $\alpha$ whereas $\alpha^{2}$ is not periodic in $\alpha$. In
[11], Dubédat establishes that for $-\pi<\alpha<\pi$, this ratio does indeed
converge as

$\log\left|\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}\right|\xrightarrow{|F^{\prime}-F|\to\infty}-\frac{1}{2\pi^{2}}\alpha^{2}\log(|F^{\prime}-F|)+C(\alpha)+o(1)$
(16)

for some undetermined constant $C(\alpha)$ and that this convergence is
uniform on compact subintervals of $(-\pi,\pi)$. This means that for generic
unitary $M\in U(N)$ whose eigenvalues are unit complex numbers
$e^{i\alpha_{1}},\cdots,e^{i\alpha_{N}}$ with $-\pi<\alpha<\pi$, we’ll get

$\begin{split}\log\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}\xrightarrow{|F^{\prime}-F|\to\infty}&-\frac{1}{2\pi^{2}}(\alpha_{1}^{2}+\cdots+\alpha_{N}^{2})\log(|F^{\prime}-F|)+C(\alpha_{1})+\cdots+C(\alpha_{N})+o(1)\\\
&+\text{`imaginary part'}.\end{split}$ (17)

In the above, there will generically be an ‘imaginary part’ that is not
universal in the sense that it will depend on microscopic details of the graph
and zipper (see Secs. III.2.1,III.3.1).

In our analysis, we’ll be able to explicitly identify the constant $C(\alpha)$
(see Sec. III.4) as

$C(\alpha)=-\frac{\alpha^{2}}{2\pi^{2}}(1+\gamma_{\mathrm{Euler}})-4\sum_{n=2}^{\infty}\frac{1}{2n}\left(\frac{\alpha}{2\pi}\right)^{2n}\zeta(2n-1)$

where $\zeta$ is the Riemann-zeta function. In particular, we are rigorously
able to establish a generating function for the coefficients of $C(\alpha)$,
where the coefficients have subleading corrections as
$|F-F^{\prime}|=D\to\infty$. Our analysis by itself does not establish that
the series for $C(\alpha)$ including the subleading corrections actually
converges. But, Dubédat’s convergence result coupled with our calculation of
moments means that $C(\alpha)$ is rigorously determined.

Additionally, even though Dubédat’s convergence results and our analysis focus
on the eigenvalues $e^{i\alpha}$ of $M$ being unit complex numbers, we can
analytically continue the moment generating function to complex $\alpha$ to
get general $\mathbb{C}^{\times}$ eigenvalues, although we don’t know how to
analyze whether that sum converges. We expect that in general, this will lead
to non-universal answers. In particular, the ‘imaginary part’ of the above
answer will end up corresponding to odd powers of $\alpha$, which we expect
not to be universal (see Sec. III.3.1). Analytically continuing $U(1)$ to
$\mathbb{C}^{\times}$ will mix up the real and imaginary parts of the above
expansion and thus mix up the universal and non-universal quantities.

In Sec. III.1, we establish a convenient microscopic choice of the zipper
$F\to F^{\prime}$. In Sec. III.2, we discuss a particular cluster expansion of
the twisted determinant that is useful for this problem. In Sec. III.3, we
discuss exact formulas of the terms in the cluster expansion and intermediate
exact formulas used throughout the paper. In Sec. III.4, we give asymptotic
expressions of the terms in expansion.

###  III.1 Straight-line Zippers

We’ll be given a bipartite isoradial graph. Let $F,F^{\prime}$ be vertices
corresponding to faces of the graph, and overload $F,F^{\prime}$ to also refer
to the complex coordinate of the corresponding vertex on the rhombus graph.
Define an angle $\theta_{0}$ so that $F^{\prime}-F=De^{i\theta_{0}}$ for some
$D>0$. Then, we have in general [20] that there exists a path

$F=:v_{0}\to v_{1}\to v_{2}\to v_{3}\to\cdots\to v_{2P-3}\to v_{2P-2}\to
v_{2P-1}\to v_{2P}:=F^{\prime}$ (18)

of vertices $v_{\cdots}$ in the rhombus graph such that each

$v_{j+1}-v_{j}=e^{i\phi_{j\to j+1}},\quad\text{where
}\theta_{0}-\frac{\pi}{2}<\phi_{j\to j+1}<\theta_{0}+\frac{\pi}{2}.$ (19)

Since the rhombus graph is bipartite (with black or white vertices of the
isoradial graph being neighbors with the faces), each $v_{2j}=:F_{j}$ defines
a face $F_{j}$, so that $F_{0}=F$ and $F_{P}=F^{\prime}$. And, each $v_{2j-1}$
will be a black or white vertex, which we’ll call either $w_{j}$ or $b_{j}$.
See Fig. 6. Furthermore, since the path is finite we can constrain all the
angles to lie in $(\theta_{\text{min}}(F_{0}\to
F_{P}),\theta_{\text{max}}(F_{0}\to F_{P}))$ where
$\theta_{0}-\frac{\pi}{2}<\theta_{\text{min}}<\theta_{\text{max}}<\theta_{0}+\frac{\pi}{2}$.
We refer to these kinds of paths as straight-line paths on the rhombus graph.

Figure 5: There exists a path between $F=F_{0}=v_{0}\to
F^{\prime}=F_{P}=v_{2P}$ where the angles of the path (red arrows) are
constrained to be in the range
$(\theta_{0}-\frac{\pi}{2},\theta_{0}+\frac{\pi}{2})$ where
$F^{\prime}-F\propto e^{i\theta_{0}}$, as in the right side of the figure. By
finiteness, one can constrain the angles to lie in
$(\theta_{\text{min}}(F_{0}\to F_{P}),\theta_{\text{max}}(F_{0}\to F_{P}))$
where $\theta_{0}-\frac{\pi}{2}<\theta_{\text{min}}(F_{0}\to
F_{P})<\theta_{\text{max}}(F_{0}\to F_{P})<\theta_{0}+\frac{\pi}{2}$.

Figure 6: A connection on the isoradial graph associated to the path chosen in
Fig. 6, whose edges weights to modify near $v_{2j-1}$ depend on whether
$v_{2j-1}$ is black or white, as described in the main text. The angles
$\phi_{j,k}$ that determine the orientation direction of the rhombus edges can
be bounded between $\theta_{\text{min}}(F_{0}\to
F_{P})<\phi_{j,k}<\theta_{max}(F_{0}\to F_{P})+\pi$.

Now we want to construct a zipper between $F$ and $F^{\prime}$ by adding a
connection on some edges of the graph along a straight-line path. This will be
referred to as a straight-line zipper. We will choose a connection with a
transition matrix of $M\in U(N)$ across the zipper. In particular, we’ll
choose a connection so that the weights of some edges for
$\overline{\partial}$ near each $v_{2j-1}=w_{j}\text{ or }b_{j}$ get their
weights multiplied by $M^{\pm 1}$. The particular choice of which edge weights
get multiplied will depend on whether $v_{2j-1}$ is white or black. First if
$v_{2j-1}=w_{j}$ is white, then $w_{j}$ has $n_{j}$ black neighbors
$b_{j,1},\cdots,b_{j,n_{j}}$ that lie on the side of the path
$v_{0}\to\cdots\to v_{2P}$ that points towards
$e^{i(\theta_{0}+\frac{\pi}{2})}$. And if $v_{2j-1}=b_{j}$ is black, then
$b_{j}$ has $n_{j}$ white neighbors $w_{j,1},\cdots,w_{j,n_{j}}$ that lie on
the side of the path pointing towards $e^{i(\theta_{0}-\frac{\pi}{2})}$. In
both cases along the edges going around we multiply the $M^{-1},M$ in the
black$\to$white, white$\to$black direction. See Fig. 6.

The reason we chose these specific edge weights is to constrain the directions
of the oriented edges in the rhombus graph. Let’s define
$e^{i\phi_{j,1}},\cdots,e^{i\phi_{j,n_{j}+1}}$ be the rhombus edge directions
pointing away from $w_{j}$ going clockwise, and
$e^{i\phi_{j,1}},\cdots,e^{i\phi_{j,n_{j}+1}}$ be the rhombus edge directions
pointing towards $b_{j}$ going counterclockwise. In particular, we’ll have
$e^{i\phi_{j,1}}=-e^{i\phi_{2j-2\to 2j-1}}$ for $v_{2j-1}$ white and
$e^{i\phi_{j,n_{j}+1}}=-e^{i\phi_{2j-2\to 2j-1}}$ for $v_{2j-1}$ black, with
$e^{i\phi_{j\to j+1}}$ defined in Eq. (19). Then we’ll have that (see Fig. 6)
the angles can all bounded between $\theta_{\text{min}}(F_{0}\to
F_{P})<\phi_{j,k}<\theta_{max}(F_{0}\to F_{P})+\pi$. In particular, this means
that all of these angles will be bounded away from $\theta_{0}-\frac{\pi}{2}$.

###  III.2 Cluster Expansion of the twisted determinant: single straight-line
zipper

Given a straight-line zipper between $F\to F^{\prime}$ as described in Sec.
III.1, we want to expand the determinant in terms of the matrix $M\in U(N)$.
Since we have only one matrix involved, we can study this problem in terms of
the eigenvalues of $M$, or equivalently if we replace $M$ with a complex
number $e^{i\alpha}$ in the construction. We will restrict our attention to
real $\alpha\in(0,\pi)$, the other cases can be handled by complex conjugation
and periodicity. Throughout the rest of this section, we will refer to
$\overline{\partial}(e^{i\alpha})$ as the twisted Kasteleyn matrix. We have

$\overline{\partial}(e^{i\alpha})=\underbrace{\overline{\partial}^{\bullet\to\circ}+(e^{-i\alpha}-1)\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}_{=:\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}+\underbrace{\overline{\partial}^{\circ\to\bullet}+(e^{i\alpha}-1)\overline{\partial}_{\mathrm{zip}}^{\circ\to\bullet}}_{=:\overline{\partial}^{\circ\to\bullet}(e^{i\alpha})}$
(20)

where $\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}$ and
$\overline{\partial}_{\mathrm{zip}}^{\circ\to\bullet}$ are the restrictions of
the $\overline{\partial}$ matrix to the zipper, for the black-to-white and
white-to-black portions of the matrix respectively.

We restrict our attention to evaluating one half of the twisted matrix, namely
$\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}$.
The logarithm of this ratio is more tractable to study using the identity
$\mathrm{tr}\log=\log\det$. As such, we can expand

$\begin{split}\log\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}&=\mathrm{tr}\log\left(\text{1}-(1-e^{-i\alpha})\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)=-\sum_{n=1}^{\infty}\frac{1}{n}(1-e^{-i\alpha})^{n}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{n}\bigg{]}\end{split}$
(21)

where
$\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}:=\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}\frac{1}{\overline{\partial}^{\bullet\to\circ}}$.

####  III.2.1 Real and Imaginary Parts of Expansions

At this point, we will note that only the real parts of the series expansion
are important in height correlations. By Lemma 1, all traces of powers of
$\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}$
will be real. As such, the terms in the expansion corresponding to even powers
of $\alpha$ will be real and those with odd powers of $\alpha$ will be
imaginary. The imaginary terms do not affect the absolute value of the
determinant.

To get the real part of the determinant, consider the equality,

$\begin{split}&\frac{1}{2}\left(\log\left(1-(1-e^{-i\alpha})y\right)+\log\left(1-(1-e^{i\alpha})y\right)\right)=\frac{1}{2}\log\left(1-(2\sin(\alpha/2))^{2}y(1-y)\right)\end{split}$
(22)

This leads to the expansion

$\begin{split}\mathrm{tr}\log\left(1-(1-e^{-i\alpha})\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)=-\sum_{n=1}^{\infty}\frac{1}{2n}(2\sin(\alpha/2))^{2n}\mathrm{tr}\left[\left(\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\left(\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)\right)^{n}\right]+\text{`imaginary
part'}.\end{split}$ (23)

Note that since each
$\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}$
is a finite-rank matrix, we’ll have that each of
$\mathrm{tr}\log\left(1-(1-e^{\pm
i\alpha})\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)$
both converge absolutely for small enough $\alpha$, so both sides of the above
will converge for small $\alpha$, thus their moments with respect to $\alpha$
will be given by moments with respect to the above.

It will turn out that evaluating these traces
$\mathrm{tr}\left[\left(\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\left(\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)\right)^{n}\right]$
are tractable, and the much of this section will be dedicated to evaluating
these.

###  III.3 Exact expressions for zipper traces

This section finds an exact expression for the traces
$\mathrm{tr}\left[\left(\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\left(\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\right)\right)^{n}\right]$
that are part of Eq. (23), culminating in Proposition 4. However, the
intermediate steps to the proposition and the reasoning behind them are also
at the heart of the multiple zipper calculations in Sec. IV.

The following equality is central.

######  Lemma 2.

Let $n>1$. Let ${\bf e}_{1}=(b_{1},w_{1}),\cdots,{\bf e}_{n}=(b_{n},w_{n})$ be
$n$ edges in the zipper. Define $F^{L}_{{\bf e}_{i}},F^{R}_{{\bf e}_{i}}$ to
be the rhombus graph vertices corresponding to the two dual faces on the dual
edge of each ${\bf e}_{i}$ on the left,right side of $w_{i}\to b_{i}$, as in
Fig. 7. Then for any vertex or dual vertex
$\tilde{v}_{1},\cdots,\tilde{v}_{n}$

$\begin{split}&\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{1},w_{1})\cdot
g_{w_{1}\to
b_{2}}(z_{2})\cdot\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{2},w_{2})\cdot
g_{w_{2}\to
b_{3}}(z_{3})\cdot\cdots\cdot\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{n},w_{n})\cdot
g_{w_{n}\to b_{1}}(z_{1})\\\ =&\left(\frac{g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}{g_{\tilde{v}_{2}\to
F^{L}_{1}}(z_{2})}-\frac{g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}{g_{\tilde{v}_{2}\to
F^{R}_{1}}(z_{2})}\right)\left(\frac{g_{\tilde{v}_{2}\to
F^{L}_{2}}(z_{2})}{g_{\tilde{v}_{3}\to
F^{L}_{2}}(z_{3})}-\frac{g_{\tilde{v}_{2}\to
F^{R}_{2}}(z_{2})}{g_{\tilde{v}_{3}\to
F^{R}_{1}}(z_{3})}\right)\cdots\left(\frac{g_{\tilde{v}_{n}\to
F^{L}_{n}}(z_{n})}{g_{\tilde{v}_{1}\to
F^{L}_{2}}(z_{1})}-\frac{g_{\tilde{v}_{n}\to
F^{R}_{n}}(z_{n})}{g_{\tilde{v}_{1}\to F^{R}_{1}}(z_{1})}\right)\\\
&\times\frac{(-i)^{n}}{(z_{1}-z_{2})(z_{2}-z_{3})\cdots(z_{n}-z_{1})}.\end{split}$
(24)

If $n=1$, then the expression above should be taken to modify
$\frac{1}{z_{1}-z_{1}}$ with an appropriate limit which defines a derivative

$\begin{split}&\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{1},w_{1})\cdot
g_{w_{1}\to
b_{1}}(z_{1})=``\frac{-i}{z_{1}-z_{1}}\left(\frac{g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}-\frac{g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}\right)"=-i\left(\frac{\frac{d}{dz_{1}}g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}-\frac{\frac{d}{dz_{1}}g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to F^{R}_{1}}(z_{1})}\right)\end{split}$
(25)

Figure 7: Given edges ${\bf e}_{1},\cdots,{\bf e}_{n}$ along the zipper, the
dual faces $F^{L}_{{\bf e}_{i}},F^{R}_{{\bf e}_{i}}$ are on the left, right
sides of the edge $w_{i}\to b_{i}$. Also, define $e^{i\varphi_{i}}=F_{{\bf
e}_{i}}^{R}-w_{i}$ and $e^{i\vartheta_{i}}=F_{{\bf e}_{i}}^{L}-w_{i}$ to be
the rhombus vectors along each edge.

###### Proof.

We have that

$\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{i},w_{i})=i(e^{i\vartheta_{i}}-e^{i\varphi_{i}})$

where $e^{i\varphi_{1}}=F_{{\bf e}_{i}}^{L}-w_{i},e^{i\vartheta_{1}}=F_{{\bf
e}_{i}}^{R}-w_{i}$ as in Fig. 7. Also, we have that each

$g_{w_{i}\to b_{i+1}}(z_{i+1})=g_{b_{i}\to
b_{i+1}}(z_{i+1})\frac{1}{(z_{i+1}-e^{i\vartheta_{i}})(z_{i+1}-e^{i\varphi_{i}})}$

where $b_{n+1}:=b_{1}$.

Let’s start with the $n=1$ case. We’ll have

$\begin{split}\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{1},w_{1})\cdot
g_{w_{1}\to
b_{1}}(z_{1})&=i(e^{i\vartheta_{i}}-e^{i\varphi_{i}})\frac{1}{(z_{1}-e^{i\varphi_{1}})(z_{1}-e^{i\vartheta_{1}})}=i\bigg{(}\frac{1}{z_{1}-e^{i\vartheta_{1}}}-\frac{1}{z_{1}-e^{i\varphi_{1}}}\bigg{)}\end{split}$

Now, note that for any $\tilde{v}_{1}$, we can write for some $k$,
$\\{\zeta_{j}\\}$, $\\{\pm_{j}\\}$

$g_{\tilde{v}_{1}\to
b_{1}}(z_{1})=\prod_{j=1}^{k}(z_{1}-e^{i\zeta_{j}})^{\pm_{j}1}.$

Also, we’d have

$g_{\tilde{v}_{1}\to F_{{\bf e}_{1}}^{L}}(z_{1})=g_{\tilde{v}_{1}\to
b_{1}}(z_{1})(z_{1}-e^{i\varphi_{1}})\quad,\quad g_{\tilde{v}_{1}\to F_{{\bf
e}_{1}}^{R}}(z_{1})=g_{\tilde{v}_{1}\to
b_{1}}(z_{1})(z_{1}-e^{i\vartheta_{1}}).$

These facts together imply

$\begin{split}&\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{1},w_{1})\cdot
g_{w_{1}\to
b_{1}}(z)=i\bigg{(}\frac{1}{z_{1}-e^{i\vartheta_{1}}}-\frac{1}{z_{1}-e^{i\varphi_{1}}}\bigg{)}\\\
&=-i\left(\bigg{(}\frac{1}{z_{1}-e^{i\varphi_{1}}}+\sum_{j=1}^{k}\pm_{j}\frac{1}{z_{1}-e^{i\zeta_{j}}}\bigg{)}-\bigg{(}\frac{1}{z_{1}-e^{i\vartheta_{1}}}+\sum_{j=1}^{k}\pm_{j}\frac{1}{z_{1}-e^{i\zeta_{j}}}\bigg{)}\right)=-i\left(\frac{\frac{d}{dz_{1}}g_{\tilde{v}_{1}\to
F^{L}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}-\frac{\frac{d}{dz_{1}}g_{\tilde{v}_{1}\to
F^{R}_{1}}(z_{1})}{g_{\tilde{v}_{1}\to F^{L}_{1}}(z_{1})}\right)\end{split}$

the result for $n=1$.

For $n>1$, we have that

$\begin{split}&\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{1},w_{1})\cdot
g_{w_{1}\to
b_{2}}(z_{2})\cdot\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{2},w_{2})\cdot
g_{w_{2}\to
b_{3}}(z_{3})\cdot\cdots\cdot\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}(b_{n},w_{n})\cdot
g_{w_{n}\to b_{1}}(z_{1})\\\
&=\bigg{[}i(e^{i\vartheta_{1}}-e^{i\varphi_{1}})\frac{1}{(z_{2}-e^{i\vartheta_{1}})(z_{2}-e^{i\varphi_{1}})}g_{b_{1}\to
b_{2}}(z_{2})\bigg{]}\cdots\bigg{[}i(e^{i\vartheta_{n}}-e^{i\varphi_{n}})\frac{1}{(z_{1}-e^{i\vartheta_{n}})(z_{1}-e^{i\varphi_{n}})}g_{b_{n}\to
b_{1}}(z_{1})\bigg{]}\\\ &=(-i)^{n}g_{b_{1}\to b_{2}}(z_{2})\cdots g_{b_{n}\to
b_{1}}(z_{1})\bigg{[}\frac{1}{z_{1}-e^{i\varphi_{n}}}-\frac{1}{z_{1}-e^{i\vartheta_{n}}}\bigg{]}\cdots\bigg{[}\frac{1}{z_{n}-e^{i\varphi_{n-1}}}-\frac{1}{z_{n}-e^{i\vartheta_{n-1}}}\bigg{]}\\\
&=(-i)^{n}\frac{g_{\tilde{v}_{2}\to b_{2}}(z_{2})}{g_{\tilde{v}_{2}\to
b_{1}}(z_{2})}\cdots\frac{g_{\tilde{v}_{1}\to
b_{1}}(z_{1})}{g_{\tilde{v}_{1}\to
b_{n}}(z_{1})}\bigg{[}\frac{1}{z_{n}-z_{1}}\bigg{(}\frac{z_{n}-e^{i\varphi_{n}}}{z_{1}-e^{i\varphi_{n}}}-\frac{z_{n}-e^{i\vartheta_{n}}}{z_{1}-e^{i\vartheta_{n}}}\bigg{)}\bigg{]}\cdots\bigg{[}\frac{1}{z_{n-1}-z_{n}}\bigg{(}\frac{z_{n-1}-e^{i\varphi_{n-1}}}{z_{n}-e^{i\varphi_{n-1}}}-\frac{z_{n-1}-e^{i\vartheta_{n-1}}}{z_{n}-e^{i\vartheta_{n-1}}}\bigg{)}\bigg{]}\\\
&=\frac{(-i)^{n}}{(z_{1}-z_{2})\cdots(z_{n}-z_{1})}\bigg{(}\frac{g_{\tilde{v}_{1}\to
F^{L}_{{\bf e}_{1}}}(z_{1})}{g_{\tilde{v}_{2}\to F^{L}_{{\bf
e}_{1}}}(z_{2})}-\frac{g_{\tilde{v}_{1}\to F^{R}_{{\bf
e}_{1}}}(z_{1})}{g_{\tilde{v}_{2}\to F^{R}_{{\bf
e}_{1}}}(z_{2})}\bigg{)}\cdots\bigg{(}\frac{g_{\tilde{v}_{n}\to F^{L}_{{\bf
e}_{n}}}(z_{n})}{g_{\tilde{v}_{1}\to F^{L}_{{\bf
e}_{n}}}(z_{1})}-\frac{g_{\tilde{v}_{n}\to F^{R}_{{\bf
e}_{n}}}(z_{n})}{g_{\tilde{v}_{1}\to F^{R}_{{\bf
e}_{n}}}(z_{1})}\bigg{)}\end{split}$

∎

A corollary of the above lemma is an explicit integral expression for
$\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{n}\bigg{]}$.

######  Corollary 3.

$\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{n}\bigg{]}=\frac{1}{(2\pi
i)^{n}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{1}\cdots\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{n}\frac{1}{(z_{1}-z_{2})\cdots(z_{n}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{n})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}$ (26)

###### Proof.

By the inverse Kasteleyn formula Eq. (6) and the discussion of rhombus angles
in the zipper in Section III.1 and Fig. 6, we have that $0\to-
ie^{i\theta_{0}}\infty$ is a valid integration contour of
$\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b)$ for all $w,b$ part of
the zipper. So,

$\frac{1}{\overline{\partial}^{\bullet\to\circ}}(w,b)=\frac{1}{2\pi}\int_{0}^{-ie^{i\theta_{0}}\infty}dzg_{w\to
b}(z).$

Also, let us label the edges in the zipper from $F\to F^{\prime}$ as ${\bf
E}_{1}\cdots{\bf E}_{Q}$ in order, so that the dual edges $\\{{\bf
E}_{j}^{\vee}\\}$ in the order ${\bf E}_{1}^{\vee}\to\cdots\to{\bf
E}_{Q}^{\vee}$ give the path $F\to F^{\prime}$ on the dual graph. Note that
$F^{L}_{{\bf E}_{1}}=F$, $F^{R}_{{\bf E}_{Q}}=F^{\prime}$, and $F^{R}_{{\bf
E}_{j}}=F^{L}_{{\bf E}_{j+1}}$ for all $j$. As such, summing over all edges of
the right-hand-side of Eq. (24) is a telescoping sum, that if we choose all
$\tilde{v}_{i}=F$

$\begin{split}\sum_{\begin{subarray}{c}{\bf e}_{1}\cdots{\bf e}_{n}\\\ \in\\\
\\{{\bf E}_{1}\cdots{\bf E}_{Q}\\}\end{subarray}}\left(\frac{g_{F\to
F^{L}_{{\bf e}_{1}}}(z_{1})}{g_{F\to F^{L}_{{\bf
e}_{1}}}(z_{2})}-\frac{g_{F\to F^{R}_{{\bf e}_{1}}}(z_{1})}{g_{F\to
F^{R}_{{\bf e}_{1}}}(z_{2})}\right)\cdots\left(\frac{g_{F\to F^{L}_{{\bf
e}_{n}}}(z_{n})}{g_{F\to F^{L}_{{\bf e}_{n}}}(z_{1})}-\frac{g_{F\to
F^{R}_{{\bf e}_{n}}}(z_{n})}{g_{F\to F^{R}_{{\bf
e}_{n}}}(z_{1})}\right)=\left(1-\frac{g_{F\to F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\right)\cdots\left(1-\frac{g_{F\to
F^{\prime}}(z_{n})}{g_{F\to F^{\prime}}(z_{1})}\right)\end{split}$

because all terms cancel except for the $F^{L}_{{\bf E}_{1}}=F$ and
$F^{R}_{{\bf E}_{Q}}=F^{\prime}$ terms. This implies the Eq. (26). ∎

Note that in Corollary 3, all the integration contours are the same, $0\to-
ie^{i\theta_{0}}$. Even though the $\frac{1}{z_{i}-z_{i+1}}$ terms becomes
singular, the full integrand is never singular because $\left(1-\frac{g_{F\to
F^{\prime}}(z_{i})}{g_{F\to F^{\prime}}(z_{i+1})}\right)$ is a rational
function with a zero at $z_{i}=z_{i+1}$.

There’s a similar statement we can show.

######  Proposition 4.

$\begin{split}\mathrm{tr}&\bigg{[}\Bigg{(}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\bigg{(}\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\Bigg{)}^{n}\bigg{]}=:\mathcal{I}_{2n}\\\
&=(-1)^{n}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{2n}\bigg{]}-(-1)^{n}{n\choose
1}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{2n-1}\bigg{]}+(-1)^{n}{n\choose
2}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{2n-2}\bigg{]}\cdots+{n\choose
n}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{2n-n}\bigg{]}\\\
&=\frac{1}{(2\pi)^{2n}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{1}\cdots\int_{0}^{ie^{i\theta_{0}}\infty}dz_{2n}\underbrace{\frac{1}{(z_{1}-z_{2})\cdots(z_{2n}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{2n})}{g_{F\to
F^{\prime}}(z_{1})}\bigg{)}}_{=:I_{2n}(z_{1},\cdots,z_{2n})}\end{split}$ (27)

where the integration contours for $z_{1},\cdots,z_{2n}$ alternate between the
contours $z_{j}:0\to-ie^{i\theta_{0}}\infty$ for $j$ odd and $z_{j}:0\to
ie^{i\theta_{0}}\infty$ for $j$ even.

In the above, we defined the full integral to equal $\mathcal{I}_{2n}$ and
define the integrand to be $I_{2n}(z_{1},\cdots,z_{2n})$.

###### Proof.

This formula relies on the following formula, valid for any $k>1$:

$\begin{split}&\frac{1}{(2\pi
i)^{k}}\int_{0}^{ie^{i\theta_{0}}\infty}dz_{1}\int_{0}^{\pm_{2}ie^{i\theta_{0}}\infty}dz_{2}\cdots\int_{0}^{\pm_{k}ie^{i\theta_{0}}\infty}dz_{k}\frac{1}{(z_{1}-z_{2})\cdots(z_{k}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}\\\ =&\frac{1}{(2\pi
i)^{k}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{1}\int_{0}^{\pm_{2}ie^{i\theta_{0}}\infty}dz_{2}\cdots\int_{0}^{\pm_{k}ie^{i\theta_{0}}\infty}dz_{k}\frac{1}{(z_{1}-z_{2})\cdots(z_{k}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}\\\ &-\frac{1}{(2\pi
i)^{k-1}}\int_{0}^{\pm_{2}ie^{i\theta_{0}}\infty}dz_{2}\cdots\int_{0}^{\pm_{k}ie^{i\theta_{0}}\infty}dz_{k}\frac{1}{(z_{2}-z_{3})\cdots(z_{k}-z_{2})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{2})}{g_{F\to
F^{\prime}}(z_{3})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{2})}\bigg{)}.\end{split}$ (28)

In the above, the contours of $z_{i}$ for $j=2,\cdots,k$ are either one of
$0\to\pm_{j}i\infty$, fixed on both sides of the equation. Because the
integrand is cyclically symmetric, the above Eq. (28) implies the corollary by
changing the $z_{2\ell}:0\to ie^{i\theta_{0}}\infty$ to $z_{2\ell}:0\to-
ie^{i\theta_{0}}\infty$ one at a time. In particular, the ${n\choose j}$
collections of changing $j$ of these contours correspond to the coefficient
${n\choose j}$ of $(-1)^{j}{n\choose
j}\mathrm{tr}\bigg{[}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}^{2n-j}\bigg{]}$.
Note that it also implies that integral is independent the order of the
contours $z_{j}:0\to\pm_{j}i\infty$ (i.e. only depends on the numbers
$\\#\\{\pm_{j}=+\\}$, $\\#\\{\pm_{j}=-\\}$), since it would mean

$\begin{split}&\frac{1}{(2\pi
i)^{k}}\int_{0}^{+ie^{i\theta_{0}}\infty}dz_{1}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{2}\int_{0}^{\pm_{3}ie^{i\theta_{0}}\infty}dz_{3}\cdots\frac{1}{(z_{1}-z_{2})\cdots(z_{k}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}\\\ =&\frac{1}{(2\pi
i)^{k}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{1}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{2}\int_{0}^{\pm_{3}ie^{i\theta_{0}}\infty}dz_{3}\cdots\frac{1}{(z_{1}-z_{2})\cdots(z_{k}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}\\\ &-\frac{1}{(2\pi
i)^{k-1}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{2}\int_{0}^{\pm_{3}ie^{i\theta_{0}}\infty}dz_{3}\cdots\frac{1}{(z_{2}-z_{3})\cdots(z_{k}-z_{2})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{2})}{g_{F\to
F^{\prime}}(z_{3})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{2})}\bigg{)}\\\ =&\frac{1}{(2\pi
i)^{k}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{1}\int_{0}^{+ie^{i\theta_{0}}\infty}dz_{2}\int_{0}^{\pm_{3}ie^{i\theta_{0}}\infty}dz_{3}\cdots\frac{1}{(z_{1}-z_{2})\cdots(z_{k}-z_{1})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}\bigg{)}.\end{split}$

Now we prove Eq. (27). Writing $g_{F\to
F^{\prime}}(z)=\prod_{j=1}^{P}\frac{z-e^{i\alpha_{j}}}{z-e^{i\beta_{j}}}$, we
have that the poles $\\{e^{i\beta_{j}}\\}$ of $g_{F\to F^{\prime}}(z_{1})$ are
of the form $\theta_{0}-\frac{\pi}{2}<\beta_{j}<\theta_{0}+\frac{\pi}{2}\\}$,
whereas all of the poles $\\{e^{i\alpha_{j}}\\}$ of $\frac{1}{g_{F\to
F^{\prime}}(z_{1})}$ are of the form
$\theta_{0}-3\frac{\pi}{2}<\alpha_{j}<\theta_{0}-\frac{\pi}{2}\\}$. As such,
the strategy is to split the product involving $z_{1}$ into two terms

$\cdots\frac{\bigg{(}1-\frac{g_{F\to F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}\bigg{(}1-\frac{g_{F\to F^{\prime}}(z_{k})}{g_{F\to
F^{\prime}}(z_{1})}\bigg{)}}{(z_{k}-z_{1})(z_{1}-z_{2})}\cdots=\cdots\underbrace{\frac{\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{1})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}}{(z_{k}-z_{1})(z_{1}-z_{2})}}_{\text{term
1}}\cdots-\cdots\underbrace{\frac{\bigg{(}\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{1})}-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to
F^{\prime}}(z_{2})}\bigg{)}}{(z_{k}-z_{1})(z_{1}-z_{2})}}_{\text{term
2}}\cdots.$

Then, for each of each of the terms one can shift the contours in such a way
to avoid the poles $\\{e^{i\alpha}\\}$ or $\\{e^{i\beta}\\}$. In particular
one would shift clockwise for ‘term 1’ and counterclockwise for ‘term 2’ since
‘term 1’ has poles at $\\{z_{1}=e^{i\alpha}\\}$ and ‘term 2’ has poles at
$\\{z_{1}=e^{i\beta}\\}$. However, this procedure has an issue in that each of
the ‘term 1’ and ‘term 2’ have a pole at $z_{1}=z_{k}$. (Note that there’s no
pole at $z_{1}=z_{2}$.) So there’s an issue that the integrals of these
individual terms will be undefined on $z_{1}:0\to\pm_{k}i\infty$ on whichever
part of the equation that is. So to rectify this, one first needs to regulate
the integration contours so that the $z_{1}$ contour is separated from the
others, which must be done before spliting up the term. See Fig. 8 for a
visual aid for the following computation.

Figure 8: (Left) Depiction of the poles $\\{e^{i\alpha}\\}$ and
$\\{e^{i\beta}\\}$ related to the angles $e^{i\theta_{0}}$,
$e^{i\theta_{\text{min,max}}(F\to F^{\prime})}$ of Fig. 6. The poles lie on
unit circle, along the thick parts of the circular wedges. (Right) We slighly
deform the $z_{1}$ contour counterclockwise from the original $0\to-
ie^{i\theta_{0}}\infty$. Then, rotating the countours to slightly clockwise
from $0\to ie^{i\theta_{0}}\infty$ separately for each of ‘term 1’ and ‘term
2’, as described in the main text, will give a at $z_{1}=z_{k}$.

First, let $r=\\#\\{\pm_{j}=+\\}$ and $1\leq a_{1}<\cdots<a_{k-r}\leq k$ be
the coordinates for which $\pm_{a_{j}}=-$, and let
$1=b_{1}<b_{2}<\cdots<b_{r}\leq k$ be the coordinates with $\pm_{b_{j}}=+$ .
We set $b_{1}=1$ because on the left-hand-side of this proposition, $z_{1}$
takes the contour $z_{1}:0\to+i\infty e^{i\theta_{0}}$.

We can start by slightly rotating the $z_{1}$ contour to be $0\to
ie^{i(\theta_{0}-\epsilon)}$ for $0<\epsilon\ll 1$ small enough to avoid the
poles $\\{e^{i\beta_{j}}\\}$ before splitting up the term. Then, we’ll rotate
this shifted $z_{1}$ contour to the new contour $z_{1}:0\to-
ie^{i(\theta_{0}+\epsilon)}\infty$ clockwise for ‘term 1’ and counterclockwise
for ‘term 2’. For ‘term 1’, this rotation will only cross the pole at
$z_{1}=z_{k}$ leaving $(-2\pi i)$ times that residue, which leaves a residue
of

$-\frac{1}{(2\pi
i)^{k-1}}\int_{0}^{-ie^{i\theta_{0}}\infty}dz_{2}\int_{0}^{\pm_{3}ie^{i\theta_{0}}\infty}dz_{3}\cdots\frac{1}{(z_{2}-z_{3})\cdots(z_{k}-z_{2})}\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{2})}{g_{F\to
F^{\prime}}(z_{3})}\bigg{)}\cdots\bigg{(}1-\frac{g_{F\to
F^{\prime}}(z_{k})}{g_{F\to F^{\prime}}(z_{2})}\bigg{)}$

while the rotation for ‘term 2’ will leave no residues. At the end of these
rotations we can then recombine the ‘term 1’ and ‘term 2’ into the original
integrand on their common contour and rotate it back to $0\to-
ie^{i\theta_{0}}\infty$. This new integral plus the residue term are what we
wanted to obtain. ∎

####  III.3.1 Non-Universality of Imaginary Parts of log-determinant

In general, the imaginary parts of the above expansion will generally be non-
universal and depend on microscopic lattice details. To see an example, we
consider the $\alpha^{1}$ term in the expansion, corresponds to
$\mathrm{tr}\bigg{[}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{]}$.
By Corollary 3 and writing $g_{F\to
F^{\prime}}(z)=\prod_{j=1}^{P}\frac{z-e^{i\alpha_{j}}}{z-e^{i\beta_{j}}}$,
we’ll have

$\mathrm{tr}\bigg{[}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{]}=-\frac{1}{2\pi
i}\int_{0}^{ie^{i\theta_{0}}\infty}dz\frac{\frac{d}{dz}g_{F\to
F^{\prime}}(z)}{g_{F\to
F^{\prime}}(z)}=\frac{1}{2\pi}\sum_{j=1}^{P}(\alpha_{j}-\beta_{j}).$ (29)

where the angles $\alpha,\beta$ can be uniquely specified by the integration
contour along $0\to ie^{i\theta_{0}}\infty$. This quantity depends sensitively
on the underlying lattice and can’t be written as a universal function of
$|F-F^{\prime}|$, even in the limit of large $|F-F^{\prime}|$.

In the next Section III.4, we’ll see that each of the
$\mathrm{tr}\bigg{[}\Bigg{(}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\bigg{(}\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\Bigg{)}^{n}\bigg{]}$
will asymptotically equal a universal quantity, a uniquely specified function
in $|F-F^{\prime}|$.

However, we also have a series expansion
$t=\sum_{n=1}^{\infty}\frac{1}{n}{2n-2\choose n-1}(t(1-t))^{n}$ for $0<t<1/2$.
Thus, we may expect the equality

$\mathrm{tr}\bigg{[}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{]}\stackrel{{\scriptstyle?}}{{=}}\sum_{n=1}^{\infty}\frac{1}{n}{2n-2\choose
n-1}\mathrm{tr}\bigg{[}\Bigg{(}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\bigg{(}\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\Bigg{)}^{n}\bigg{]}.$

As such, we may be concerned that an infinite sum of ‘universal’ terms may sum
up to a non-universal one. There are two issues with this reasoning. First,
the equality $t=\sum_{k=1}^{\infty}\frac{1}{k}{2n-2\choose n-1}(t(1-t))^{n}$
doesn’t hold for all $t$ so there may be convergence issues for eigenvalues
outside of $[0,1/2]$. Second, each
$\mathrm{tr}\bigg{[}\Bigg{(}\bigg{(}\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\bigg{(}\text{1}-\frac{\overline{\partial}_{\mathrm{zip}}^{\bullet\to\circ}}{\overline{\partial}^{\bullet\to\circ}}\bigg{)}\Bigg{)}^{n}\bigg{]}$
will have sub-leading corrections for which we don’t have uniform control, so
resummation of these lower-order terms may lead to non-universal sums.

In general, odd powers of $\alpha$ in the expansion of
$\log(1-(1-e^{-i\alpha})y)$ will be accompanied by such infinite sums in
$y(1-y)$, whereas even powers of $\alpha$ will be accompanied by finite sums,
by Eq. (22). These finite sums of universal quantities will also be universal.
So we’ll see that even powers are universal while odd powers are not. In
particular, the real part will be universal while the imaginary part isn’t.

###  III.4 Asymptotics of Eq. (27) for $\mathcal{I}_{2n}$

Now, we give asymptotic estimates of the integrals Eq. (27). In this section,
we’ll assume WLOG that $e^{i\theta_{0}}=1$. And for notational compactness,
we’ll write $g(z)=g_{F\to F^{\prime}}(z)$ throughout this section. The
computation relies on the asymptotics of the function $g_{F\to
F^{\prime}}(z)=\prod_{j=1}^{P}\frac{z-e^{i\alpha_{j}}}{z-e^{i\beta_{j}}}$ from
[4] and also Appendix A. We’ll also define in this section

$D:=|F^{\prime}-F|$ (30)

to be the distance between the dual faces on the rhombus graph.

The remainder of this section will be devoted towards the following
proposition.

######  Proposition 5.

$\begin{split}&\mathcal{I}_{2n}\cdot(1+O(1/D))+O(1/D)=\mathcal{J}_{2n}\\\
&:=\frac{2}{(2\pi)^{2n}}\int_{0\xrightarrow{-}1}dw_{1}\int_{0\xrightarrow{+}1}dw_{2}\cdots\int_{0\xrightarrow{-}1}dw_{2n-1}\int_{\mathcal{C}_{\text{out}}}dw_{2n}\frac{1}{(w_{1}-w_{2})\cdots(w_{2n}-w_{1})}\bigg{(}1-e^{-D|w_{2n}-w_{1}|}\bigg{)}.\end{split}$
(31)

In the above (for $\pm\in\\{+,-\\}$), the contours ${0\xrightarrow{\pm}1}$
refer to contours $0\to 1$ in the complex plane, except deformed slightly in
the imaginary direction $\pm i$ away from the real axis. And, we define
$\mathcal{C}_{\text{out}}$ as the contour consisting of the union of the lines
$\\{0\to-\infty\\}\cup\\{\infty\to 1\\}$ along the real axis.

Originally, we wanted to evaluate
$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}$.
The Eq. (23,27,31) imply that

$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}=-\sum_{n=1}^{\infty}\frac{1}{2n}(2\sin(\alpha/2))^{2n}\mathcal{I}_{2n}+\text{`imaginary
part'}.$

In Appendix C, we instead consider the sum over the leading-order
contributions
$-\sum_{n=1}^{\infty}\frac{1}{2n}(2\sin(\alpha/2))^{2n}\mathcal{J}_{2n}$,
which is expected to give
$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha})}{\det\overline{\partial}^{\bullet\to\circ}}$
up to negligible corrections. This gives the final result Eq. (123)

$-\sum_{n=1}^{\infty}\frac{1}{2n}(2\sin(\alpha/2))^{2n}\mathcal{J}_{2n}+O(D^{-1/3})=-\frac{\alpha^{2}}{2\pi^{2}}\left(1+\log(D)+\gamma_{\mathrm{Euler}}\right)-4\sum_{n=2}^{\infty}\frac{1}{2n}\left(\frac{\alpha}{2\pi}\right)^{2n}\zeta(2n-1)$
(32)

where $\zeta$ is the Riemann-zeta function.

By using the series expansion

$-\frac{(2\arcsin(\beta/2))^{2}}{2\pi^{2}}=-\sum_{n=1}^{\infty}\frac{1}{n}\frac{2}{\pi^{2}}\frac{(2n-2)!!}{(2n-1)!!}\beta^{2n},$
(33)

we can substitute in $\alpha=2\arcsin(\beta/2)$ to deduce

$\mathcal{J}_{2n}+O(D^{-1/3})=\frac{4}{\pi^{2}}\frac{(2n-2)!!}{(2n-1)!!}\log(D)+c_{2n}$
(34)

where $c_{2n}$ are constants that can be determined by Eq. (32) but for which
we don’t know a closed form.

####  III.4.1 Setting up the calculation

Using the regularity conditions in Appendix A, the asymptotic form of $g(z)$
and $\frac{1}{g(z)}$ functions for $D\gg 1$ will be

$g_{F\to F^{\prime}}(-|z|)\cdot(1+O(1/D))=\begin{cases}\gamma e^{-D|z|}&\text{
if }|z|<D^{-2/3}\\\ \approx 0&\text{ if }D^{-2/3}<|z|<D^{2/3}\\\
e^{-D/|z|}&\text{ if }{\sqrt{D}}<|z|\end{cases}$ (35)

and

$\frac{1}{g_{F\to
F^{\prime}}(|z|)}\cdot(1+O(1/D))=\begin{cases}\frac{1}{\gamma}\gamma
e^{-D|z|}&\text{ if }|z|<D^{-2/3}\\\ \approx 0&\text{ if
}D^{-2/3}<|z|<D^{2/3}\\\ e^{-D/|z|}&\text{ if }D^{2/3}<|z|\end{cases}$ (36)

where $\approx 0$ means exponentially small and can be treated as negligible
in the integrals (see Prop. 10). And above, we define $\gamma:=g_{F\to
F^{\prime}}(0)=\prod_{j=1}^{P}\frac{e^{i\alpha_{j}}}{e^{i\beta_{j}}}$ which is
a unit complex number whose value depends on the microscopic details of the
specific path and won’t make a difference in the asymptotic evaluation of the
integrals.

Note that these asymptotics only give useful information about $z$ on the
negative real axis for $g(z)$ and on the positive real axis for
$\frac{1}{g(z)}$ and don’t directly give information about the integrand on
the contours $z_{j}:0\to\pm i\infty$. As such, to actually use these
asymptotic expansions, we need to find a way to shift the contours from the
original $0\to\pm i\infty$ to $0\to\pm\infty$. We can’t directly do this, for
two reasons. First, the integrand has poles at both $\\{z_{j}=e^{i\alpha}\\}$
and $\\{z_{j}=e^{i\beta}\\}$ since they contain pieces of both $g(z_{j})$ and
$\frac{1}{g(z_{j})}$, so these contour deformations will cross many poles.
Second, the asymptotics for $g(z)$ and $\frac{1}{g(z)}$ aren’t even useful on
the same halves of the real axis. To deal with this, we need to split up the
terms in such a way that each term only contains one of $g(z_{j})$ or
$\frac{1}{g(z_{j})}$.

To do this, first write the integrand of Eq. (27) as

$\begin{split}I_{2n}=\frac{1}{(z_{1}-z_{2})\cdots(z_{2n-1}-z_{2n})(z_{2n}-z_{1})}&\left[\left(1-\frac{g(z_{1})}{g(z_{2})}\right)\left(1-\frac{g(z_{3})}{g(z_{4})}\right)\cdots\left(1-\frac{g(z_{2n-1})}{g(z_{2n})}\right)\right]\\\
\times&\left[\left(1-\frac{g(z_{2})}{g(z_{3})}\right)\cdots\left(1-\frac{g(z_{2n-2})}{g(z_{2n-1})}\right)\left(1-\frac{g(z_{2n})}{g(z_{1})}\right)\right].\end{split}$
(37)

If we expand out the second line of the above into its $2^{n}$ factors and
consider the terms gotten by multiplying the first line by those factors, then
we can see that every term will only have $g(z_{j})$ either in the numerator
or the denominator. For example, the case $n=2$ has an integrand that can be
written as

$\begin{split}I_{4}=\frac{1}{(z_{1}-z_{2})\cdots(z_{4}-z_{1})}\times\Bigg{[}&\underbrace{\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}}-\underbrace{\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\frac{g(z_{2})}{g(z_{3})}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}}\\\
-&\underbrace{\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}\frac{g(z_{4})}{g(z_{1})}}+\underbrace{\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\frac{g(z_{2})}{g(z_{3})}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}\frac{g(z_{4})}{g(z_{1})}}\Bigg{]}.\end{split}$
(38)

For each $z_{i}$, each of the terms in with an underbrace in the above can be
simplified to only contain a factor of $g(z_{i})$ or $\frac{1}{g(z_{i})}$.
Also, note that the configuration of the contours of integration alternating
between $0\to+i\infty$ and $0\to-i\infty$ mean that we can deform the pieces
of the full integral for $n=2$ to equal without crossing singularities and
without residues

$\begin{split}\mathcal{I}_{4}=\frac{1}{(2\pi)^{4}}\Bigg{[}&\underbrace{\int_{0}^{-\infty}dz_{1}\int_{0}^{+\infty}dz_{2}\int_{0}^{-\infty}dz_{3}\int_{0}^{+\infty}dz_{4}\frac{1}{(z_{1}-z_{2})\cdots(z_{4}-z_{1})}\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}}_{\text{`Part
(00)'}}\\\
-&\underbrace{\int_{0}^{-\infty}dz_{1}\int_{0}^{-\infty}dz_{2}\int_{0}^{+\infty}dz_{3}\int_{0}^{+\infty}dz_{4}\frac{1}{(z_{1}-z_{2})\cdots(z_{4}-z_{1})}\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\frac{g(z_{2})}{g(z_{3})}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}}_{\text{`Part
(10)'}}\\\
-&\underbrace{\int_{0}^{+\infty}dz_{1}\int_{0}^{+\infty}dz_{2}\int_{0}^{-\infty}dz_{3}\int_{0}^{-\infty}dz_{4}\frac{1}{(z_{1}-z_{2})\cdots(z_{4}-z_{1})}\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}\frac{g(z_{4})}{g(z_{1})}}_{\text{`Part
(01)'}}\\\
+&\underbrace{\int_{0}^{+\infty}dz_{1}\int_{0}^{-\infty}dz_{2}\int_{0}^{+\infty}dz_{3}\int_{0}^{-\infty}dz_{4}\frac{1}{(z_{1}-z_{2})\cdots(z_{4}-z_{1})}\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}\frac{g(z_{2})}{g(z_{3})}\bigg{(}1-\frac{g(z_{3})}{g(z_{4})}\bigg{)}\frac{g(z_{4})}{g(z_{1})}\Bigg{]}}_{\text{`Part
(11)'}}.\end{split}$ (39)

In particular, we label the terms ‘Part($x_{1}x_{2}$)’ for
$x_{1},x_{2}\in\\{0,1\\}$. If $x_{j}=1$, then the integrand of the term will
have a factor $-\frac{g(z_{2j})}{g(z_{2j+1})}$ being multiplied in and the
integration contours will be $z_{2j}:0\to-\infty$ and $z_{2j+1}:0\to+\infty$,
whereas if $x_{j}=\text{0}$, the integrand of the term will have a factor $1$
being multiplied in and the integration contours will be $z_{2j}:0\to+\infty$
and $z_{2j+1}:0\to-\infty$.

For general $n$, in the same exact manner as above, we can split up the
integral into $2^{n}$ parts, labeled ‘Part($x_{1}\cdots x_{n}$)’ for
$x_{1},\cdots,x_{n}\in\\{0,1\\}$. Note that since each of the pieces has a
pole at $z_{2j}=z_{2j+1}$ (where again, $z_{2n+1}\equiv z_{1}$), one may be
concerned that moving the contours may produce residues. However, note that in
this procedure that we are shifting the contours $z_{2j}:0\to i\infty$ and
$z_{2j+1}:0\to-i\infty$ to $z_{2j}:0\to\pm_{2j}\infty$ and
$z_{2j+1}:0\to-\pm_{2j}\infty$ for all $2^{n}$ choices of $\\{\pm_{2j}\\}$. So
the fact that these contours always lay opposite to each other means that this
procedure will never have the contours crossing any poles and this procedure
doesn’t produce any residues.

It will be instructive to analyze the asymptotics of the $n=1$ case,
$\mathcal{I}_{2}$, before moving on to the general case.

####  III.4.2 The $n=1$ case, $\mathcal{I}_{2}$

Somewhat simpler, the $\mathcal{I}_{2}$ integral can be written

$\begin{split}\mathcal{I}_{2}&=\frac{1}{(2\pi)^{2}}\Bigg{[}\underbrace{\int_{0}^{-\infty}dz_{1}\int_{0}^{+\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\bigg{(}1-\frac{g(z_{1})}{g(z_{2})}\bigg{)}}_{\text{`Part
(0)'}}-\underbrace{\int_{0}^{+\infty}dz_{1}\int_{0}^{-\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\bigg{(}\frac{g(z_{2})}{g(z_{1})}-1\bigg{)}}_{\text{`Part
(1)'}}\Bigg{]}\end{split}$ (40)

which can be analyzed using the asymptotics in Eqs. (35,36). Let’s first
analyze ‘Part (0)’, as the story for ‘Part (1)’ will be very similar.

Note that we can use the asymptotic form Eq. (35) of $g(z)$ to write

$\begin{split}\text{`Part
(0)'}\cdot(1+O(1/D))&=\frac{1}{(2\pi)^{2}}\int_{0}^{-1}dz_{1}\int_{0}^{\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\bigg{(}1-e^{D(z_{1}-z_{2})}\bigg{)}+O(1/D)\\\
&\quad+\frac{1}{(2\pi)^{2}}\int_{0}^{-1}dz_{1}\int_{0}^{\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\bigg{(}1-e^{D(\frac{1}{z_{1}}-\frac{1}{z_{2}})}\bigg{)}+O(1/D)\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{-1}dz_{1}\int_{0}^{\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\bigg{(}1-e^{D(z_{1}-z_{2})}\bigg{)}+O(1/D)\\\
&=\frac{1}{2\pi^{2}}(\log(D)+\gamma_{\text{Euler}}+1)+O(1/D)\end{split}$ (41)

In the above, the second equality follows from changing variables
$z_{1}\mapsto\frac{1}{z_{1}}$ and $z_{2}\mapsto\frac{1}{z_{2}}$ in the second
term. In the third equality, $\gamma_{\text{Euler}}$ the Euler gamma constant,
and the result can be found from explicitly evaluating the integral. However,
the first line requires some justification, as follows.

First, we note that in the ‘corner region’ of $z_{1}\in(-D^{2/3},-\infty)$ and
$z_{2}\in(0,D^{-2/3})$, the term will be $O(1/D)$ since it will be of the form

$\begin{split}&-\frac{1}{(2\pi)^{2}}\int_{-D^{2/3}}^{-\infty}dz_{1}\int_{0}^{\frac{1}{D^{2/3}}}dz_{1}\frac{1}{(z_{1}-z_{2})^{2}}(1-\gamma
e^{D(\frac{1}{z_{1}}-z_{2})})+O(1/D)\\\
=&-\frac{1}{(2\pi)^{2}}\int_{0}^{-D^{-2/3}}dz_{1}\int_{0}^{D^{-2/3}}dz_{1}\frac{1}{(1-z_{1}z_{2})^{2}}(1-\gamma
e^{D(z_{1}-z_{2})})+O(1/D)=O(1/D)\end{split}$ (42)

The first line uses the asymptotic form of $g(z)$ to write the integral up to
a lower order $O(1/D)$ term. The second line changes variables
$z_{1}\to\frac{1}{z_{1}}$, and the third line follows because $|\frac{1-\gamma
e^{D(z_{1}-z_{2})}}{(1-z_{1}z_{2})^{2}}|<2$ on the region, and the region
itself has area $1/D^{4/3}$. As such, we can asymptotically ignore that part.

Second, even though asymptotically $g(z_{1})\approx 0$ and
$\frac{1}{g(z_{2})}\approx 0$ on the middle regions
$z_{1}\in(-D^{2/3},-D^{-2/3})$ and $z_{2}\in(D^{-2/3},D^{2/3})$, we can
substitute $g(z_{1})\to e^{Dz_{1}}$ and $\frac{1}{g(z_{2})}\to e^{-Dz_{2}}$
into the integrals on the full interval at the cost of a lower-order term,
since the exponentially small terms will contribute at most an $O(1/D)$
contribution.

The computation for ‘Part (1)’ goes through exactly the same and gives the
same asymptotic result. As such, we have

$\mathcal{I}_{2}=\text{`Part (0)'}+\text{`Part
(1)'}=\frac{1}{\pi^{2}}(\log(D)+\gamma_{\text{Euler}}+1)\cdot(1+O(1/D)),$ (43)

which is consistent with the Proposition 5.

Now, we show how show the equality with the integral over the $\\{w_{j}\\}$
variables in Proposition 5. Let’s analyze ‘Part (0)’ first. We have

$\begin{split}\text{`Part (0)'}&\cdot(1+O(1/D))+O(1/D)\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{-1}dz_{1}\int_{0}^{\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\big{(}1-e^{D(z_{1}-z_{2})}\big{)}\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{-1}dz_{1}\int_{0}^{\infty}dz_{2}\int_{0}^{D}dw_{1}\int_{0}^{-\infty}dw_{2}\,\,e^{z_{1}(w_{1}-w_{2})}e^{z_{2}(w_{2}-w_{1})}\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{D}dw_{1}\int_{0}^{-\infty}dw_{2}\frac{1}{(w_{1}-w_{2})(w_{2}-w_{1})}\big{(}1-e^{(w_{2}-w_{1})}\big{)}.\end{split}$
(44)

The first line was derived previously. The second line uses the ‘Schwinger
parameterization trick’ substituting
$\frac{1-e^{D(z_{1}-z_{2})}}{z_{1}-z_{2}}=-\int_{0}^{D}dw_{1}\,e^{w_{1}(z_{1}-z_{2})}$
and $\frac{1}{z_{2}-z_{1}}=-\int_{0}^{-\infty}dw_{2}\,e^{w_{2}(z_{2}-z_{1})}$.
Note that the $w_{2}$ integral converges since $z_{2}-z_{1}>0$. The last line
follows from evaluating the integrals over $z_{1},z_{2}$, noting that the
$z_{2}$ integral converges if we choose $w_{1}:0\to D$ and $w_{2}:0\to-\infty$
to be straight lines along the real axis, which would imply $w_{2}-w_{1}<0$.

Similarly, we can write

$\begin{split}\text{`Part (1)'}&\cdot(1+O(1/D))+O(1/D)\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{+1}dz_{1}\int_{0}^{-\infty}dz_{2}\frac{1}{(z_{1}-z_{2})(z_{2}-z_{1})}\big{(}1-e^{D(z_{1}-z_{2})}\big{)}e^{D(z_{2}-z_{1})}\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{+1}dz_{1}\int_{0}^{-\infty}dz_{2}\int_{0}^{D}dw_{1}\int_{\infty}^{D}dw_{2}\,\,e^{w_{1}(z_{1}-z_{2})}e^{w_{2}(z_{2}-z_{1})}e^{D(z_{2}-z_{1})}\big{(}e^{D(z_{1}-z_{2})}-1\big{)}\\\
&=\frac{2}{(2\pi)^{2}}\int_{0}^{D}dw_{1}\int_{\infty}^{D}dw_{2}\frac{1}{(w_{1}-w_{2})(w_{2}-w_{1})}\big{(}1-e^{-(w_{2}-w_{1})}\big{)},\end{split}$
(45)

noting similarly that all of the integrals going out to infinity converge. In
total, we find that up to additive and multiplicative factors of $O(1/D)$, we
have ‘Part (0)’ + ‘Part (1)’ equal the integral specified in Proposition 5,
after changing variables $w_{1,2}\mapsto w_{1,2}/D$.

####  III.4.3 The general case

Now, let us consider the general case inputting the asymptotic form of Eqs.
(35,36) into the decomposition discussed following Eq. (39).

First, let’s again consider the ‘corner regions’ of all of the
‘Part($\cdots$)’, where for some $j\neq k$ we have $|z_{j}|\in(0,D^{-2/3})$
and $|z_{k}|\in(D^{2/3},\infty)$ and for all other $\ell\neq j,k$, either
$|z_{\ell}|\in(0,D^{-2/3})$ or $|z_{\ell}|\in(D^{2/3},\infty)$. It will be a
general fact that the integrals on these corner regions will be lower-order
terms as $D\to\infty$, as follows.

######  Lemma 6.

Define the integrands $I_{2n}(z_{1},\cdots,z_{2n};x_{1}\cdots x_{n})$ of each
of the ‘Part($x_{1}\cdots x_{n}$)’. Then, for all $j\neq k$, we have

$\cdots\int_{0}^{\pm D^{-2/3}}dz_{j}\cdots\int_{\pm
D^{2/3}}^{\pm\infty}dz_{k}\cdots I_{2n}(z_{1},\cdots,z_{2n};x_{1}\cdots
x_{n})=O(1/D)$ (46)

where the integration regions for all $z_{\ell},\ell\neq j,k$ can be either
one of $(0,\pm D^{-2/3})$ or $\pm(D^{2/3},\infty)$.

In the above lemma, the choice of signs $\pm$ for the intervals is specified
by the integration contours corresponding to ‘Part($x_{1}\cdots x_{n}$)’.

###### Proof.

First, we change variables $z_{p}\to|z_{p}|=\pm z_{p}$ for all $p$. At this
point, the integrand can be written as a product of various factors of one of
the following forms

$\begin{split}\frac{1-\frac{g(z_{2m-1})}{g(z_{2m})}}{|z_{2m-1}|+|z_{2m}|}\frac{1}{|z_{2m}|+|z_{2m+1}|}&\text{
for }z_{2m-1}\in(0,-\infty)\text{ and }z_{2m}\in(0,+\infty),\text{ i.e.
}x_{m}=x_{m-1}=0,\\\
\frac{1-\frac{g(z_{2m})}{g(z_{2m-1})}}{|z_{2m-1}|+|z_{2m}|}\frac{1}{|z_{2m}|+|z_{2m+1}|}&\text{
for }z_{2m-1}\in(0,+\infty)\text{ and }z_{2m}\in(0,-\infty),\text{ i.e.
}x_{m}=x_{m-1}=1,\\\
\frac{g(z_{2m-1})-g(z_{2m})}{|z_{2m-1}|-|z_{2m}|}\frac{1}{|z_{2m}|+|z_{2m+1}|}&\text{
for }z_{2m-1}\in(0,-\infty)\text{ and }z_{2m}\in(0,-\infty),\text{ i.e.
}x_{m}=1,x_{m-1}=0,\\\
\frac{\frac{1}{g(z_{2m-1})}-\frac{1}{g(z_{2m})}}{|z_{2m-1}|-|z_{2m}|}\frac{1}{|z_{2m}|+|z_{2m+1}|}&\text{
for }z_{2m-1}\in(0,+\infty)\text{ and }z_{2m}\in(0,+\infty),\text{ i.e.
}x_{m}=0,x_{m-1}=1\end{split}$

depending on the specific ‘Part($x_{1}\cdots x_{n}$)’.

Next, we change variables to all lie in $\text{int}_{0}:=(0,D^{-2/3})$ by
sending $|z_{p}|\to\tilde{z}_{p}:=|z_{p}|^{\pm 1}$, depending on the interval
chosen. At this point, we will use the bounds in Propositions 11,12 and Eq.
(86). In addition, we use the fact that for these ranges of $\\{z_{p}\\}$ that

$\left|1-\frac{g(z_{2m-1})}{g(z_{2m})}\right|\leq
2\quad,\quad\left|1-\frac{g(z_{2m})}{g(z_{2m-1})}\right|\leq
2\quad,\quad\left|\frac{1}{g(z_{2m-1})}-\frac{1}{g(z_{2m})}\right|\leq
2\quad,\quad\left|g(z_{2m-1})-g(z_{2m})\right|\leq 2.$

Then, after having distributed the Jacobians $-\frac{1}{z^{2}}$ throughout the
denominators appropriately, we can bound the full integral as a product of
factors of the form

$\begin{split}\frac{2}{\tilde{z}_{2m-1}+\tilde{z}_{2m}}\quad,\quad\frac{2}{1+\tilde{z}_{2m-1}\tilde{z}_{2m}}\quad,\quad\frac{1}{\tilde{z}_{2m}+\tilde{z}_{2m+1}}\quad,\quad\frac{1}{1+\tilde{z}_{2m}\tilde{z}_{2m+1}},\end{split}$

where all $\tilde{z}_{2m}\in(0,D^{-2/3})$. The fact that the original
integration regions were $|z_{j}|:0\to D^{-2/3}$ and
$|z_{k}|:D^{2/3}\to\infty$ means that there’s at least one factor of
$\frac{1}{1+\tilde{z}_{r}\tilde{z}_{r+1}}$ in the bounding integral. Note that
$\frac{1}{1+\tilde{z}_{r}\tilde{z}_{r+1}}\leq 1$. As such, we can bound the
full integral as a constant power of two times the product of at least two
integrals of the form

$\begin{split}&\int_{0}^{D^{-2/3}}d\tilde{z}_{r}\int_{0}^{D^{-2/3}}d\tilde{z}_{r+1}\cdots\int_{0}^{D^{-2/3}}d\tilde{z}_{r+s}\frac{1}{\tilde{z}_{r}+\tilde{z}_{r+1}}\frac{1}{\tilde{z}_{r+1}+\tilde{z}_{r+2}}\cdots\frac{1}{\tilde{z}_{r+s-1}+\tilde{z}_{r+s}}\\\
=&\frac{1}{D^{2/3}}\int_{0}^{1}d\tilde{z}_{r}\int_{0}^{1}d\tilde{z}_{r+1}\cdots\int_{0}^{1}d\tilde{z}_{r+s}\frac{1}{\tilde{z}_{r}+\tilde{z}_{r+1}}\frac{1}{\tilde{z}_{r+1}+\tilde{z}_{r+2}}\cdots\frac{1}{\tilde{z}_{r+s-1}+\tilde{z}_{r+s}},\end{split}$
(47)

where we can allow possibly $s=0$, in which case the relevant integral would
just be $\int_{0}^{D^{-2/3}}d\tilde{z}_{r}=\frac{1}{D^{2/3}}$.

The fact that the integrals
$\int_{0}^{1}d\tilde{z}_{r}\int_{0}^{1}d\tilde{z}_{r+1}\cdots\int_{0}^{1}d\tilde{z}_{r+s}\frac{1}{\tilde{z}_{r}+\tilde{z}_{r+1}}\cdots\frac{1}{\tilde{z}_{r+s-1}+\tilde{z}_{r+s}}$
are convergent, finite numbers independent of $D$ and that there are at least
two of them means that the full integral can be bounded by a constant times
$\frac{1}{D^{4/3}}$, which implies what we wanted to show. ∎

Now, we can proceed to show the proposition. In particular, we will show the
following expression for ‘Part($x_{1}\cdots x_{n}$)’.

######  Lemma 7.

Fix some $x_{1},\cdots,x_{n}\in\\{0,1\\}$. Then,

$\text{`Part}(x_{1},\cdots,x_{n})\text{'}\cdot(1+O(1/D))+O({1}/{D})=\int_{0}^{D}dw_{1}\int_{\mathcal{C}_{x_{1}}}dw_{2}\cdots\int_{0}^{D}dw_{2n-1}\int_{\mathcal{C}_{x_{n}}}dw_{2n}\frac{1-e^{-|w_{2n}-w_{1}|}}{(w_{1}-w_{2})\cdots(w_{2n}-w_{1})}.$
(48)

In the above, the contours for $w_{2j}$ are $\mathcal{C}_{x_{j}}$, defined as
follows. We define $\mathcal{C}_{0}=0\to-\infty$ and
$\mathcal{C}_{1}=+\infty\to D$.

Note that this lemma implies the Proposition 5. In particular, note that all
of the integrands above have the same integrand. As such, adding up all of the
integration contours over all assignments of $\\{x_{j}\\}$ will give (after
changing all variables of all $w_{j}\mapsto w_{j}/D$)

$\begin{split}\mathcal{I}_{2n}&=\sum_{x_{1},\cdots,x_{n}\in\\{0,1\\}}\text{`Part}(x_{1},\cdots,x_{n})\text{'}\cdot(1+O(1/D))\\\
&=\frac{2}{(2\pi)^{2n}}\int_{0}^{1}dw_{1}\int_{\mathcal{C}_{\text{out}}}dw_{2}\cdots\int_{0}^{1}dw_{2n-1}\int_{\mathcal{C}_{\text{out}}}dw_{2n}\frac{1-e^{-D|w_{2n}-w_{1}|}}{(w_{1}-w_{2})\cdots(w_{2n}-w_{1})}+O({1}/{D}).\end{split}$

Note that for fixed $w_{2n}$, one can set $e^{-|w_{2n}-w_{1}|}$ with one of
$e^{w_{2n}-w_{1}}$ or $e^{w_{1}-w_{2n}}$ depending on if $w_{2n}<0$ or
$w{2n}>0$. So, the integrands are all holomorphic as functions for all other
$w_{\cdots}$ and vanish as $\frac{1}{w^{2}_{\cdots}}$. As such, one can deform
the contours without changing the value of the integrands, as long as a
$w_{j}$ contour doesn’t cross any $w_{j\pm 1}$ contour in the process. In
particular, one can change the contours to be as in the integral Proposition
5.

The proof of the lemma will be as follows, similar to the $n=1$ analysis of
$\mathcal{I}_{2}$ from earlier.

###### Proof of Lemma 7.

First, note that we can use the asymptotics of $g(z)$ and the definition of
each of the ‘Part($x_{1},\cdots,x_{n}$)’ as follows. Justification for each
equality will be provided afterward.

$\begin{split}&\text{`Part}(x_{1},\cdots,x_{n})\text{'}\cdot(1+O(1/D))+O(1/D)\\\
&=\frac{1}{(2\pi)^{2n}}\int_{0}^{-(-1)^{x_{n}}}dz_{1}\int_{0}^{(-1)^{x_{1}}\infty}dz_{2}\int_{0}^{-(-1)^{x_{1}}\infty}dz_{3}\int_{0}^{(-1)^{x_{2}}\infty}dz_{4}\cdots\int_{0}^{-(-1)^{x_{n-1}}\infty}dz_{2n-1}\int_{0}^{(-1)^{x_{n}}\infty}dz_{2n}\\\
&\quad\quad\left\\{\frac{(1-e^{D(z_{1}-z_{2})})\bigg{(}-e^{D(z_{2}-z_{3})}\bigg{)}^{x_{1}}(1-e^{D(z_{3}-z_{4})})\bigg{(}-e^{D(z_{4}-z_{5})}\bigg{)}^{x_{2}}\cdots(1-e^{D(z_{2n-1}-z_{2n})})\bigg{(}-e^{D(z_{2n}-z_{1})}\bigg{)}^{x_{n}}}{(z_{1}-z_{2})\cdots(z_{2n}-z_{1})}\right\\}\\\
&\,\,+\frac{1}{(2\pi)^{2n}}\int_{-(-1)^{x_{n}}}^{-(-1)^{x_{n}}\infty}dz_{1}\int_{0}^{(-1)^{x_{1}}\infty}dz_{2}\int_{0}^{-(-1)^{x_{1}}\infty}dz_{3}\int_{0}^{(-1)^{x_{2}}\infty}dz_{4}\cdots\int_{0}^{-(-1)^{x_{n-1}}\infty}dz_{2n-1}\int_{0}^{(-1)^{x_{n}}\infty}dz_{2n}\\\
&\quad\quad\left\\{\frac{(1-e^{D(\frac{1}{z_{1}}-\frac{1}{z_{2}})})\bigg{(}-e^{D(\frac{1}{z_{2}}-\frac{1}{z_{3}})}\bigg{)}^{x_{1}}(1-e^{D(\frac{1}{z_{3}}-\frac{1}{z_{4}})})\bigg{(}-e^{D(\frac{1}{z_{4}}-\frac{1}{z_{5}})}\bigg{)}^{x_{2}}\cdots(1-e^{D(\frac{1}{z_{2n-1}}-\frac{1}{z_{2n}})})\bigg{(}-e^{D(\frac{1}{z_{2n}}-\frac{1}{z_{1}})}\bigg{)}^{x_{n}}}{(z_{1}-z_{2})\cdots(z_{2n}-z_{1})}\right\\}\\\
&=\frac{2}{(2\pi)^{2n}}\int_{0}^{-(-1)^{x_{n}}}dz_{1}\int_{0}^{(-1)^{x_{1}}\infty}dz_{2}\int_{0}^{-(-1)^{x_{1}}\infty}dz_{3}\int_{0}^{(-1)^{x_{2}}\infty}dz_{4}\cdots\int_{0}^{-(-1)^{x_{n-1}}\infty}dz_{2n-1}\int_{0}^{(-1)^{x_{n}}\infty}dz_{2n}\\\
&\quad\quad\left\\{\frac{(1-e^{D(z_{1}-z_{2})})\bigg{(}-e^{D(z_{2}-z_{3})}\bigg{)}^{x_{1}}(1-e^{D(z_{3}-z_{4})})\bigg{(}-e^{D(z_{4}-z_{5})}\bigg{)}^{x_{2}}\cdots(1-e^{D(z_{2n-1}-z_{2n})})\bigg{(}-e^{D(z_{2n}-z_{1})}\bigg{)}^{x_{n}}}{(z_{1}-z_{2})\cdots(z_{2n}-z_{1})}\right\\}\\\
&=\frac{2}{(2\pi)^{2n}}\int_{0}^{-(-1)^{x_{n}}}dz_{1}\int_{0}^{(-1)^{x_{1}}\infty}dz_{2}\int_{0}^{-(-1)^{x_{1}}\infty}dz_{3}\int_{0}^{(-1)^{x_{2}}\infty}dz_{4}\cdots\int_{0}^{-(-1)^{x_{n-1}}\infty}dz_{2n-1}\int_{0}^{(-1)^{x_{n}}\infty}dz_{2n}\\\
&\quad\quad\int_{0}^{D}dw_{1}\int_{\mathcal{C}_{x_{1}}}dw_{2}\int_{0}^{D}dw_{3}\int_{\mathcal{C}_{x_{2}}}dw_{4}\cdots\int_{0}^{D}dw_{2n-1}\int_{\mathcal{C}_{x_{2n}}}dw_{2n}\,\,\Big{\\{}e^{w_{1}(z_{1}-z_{2})}e^{w_{2}(z_{2}-z_{3})}\cdots
e^{w_{2n}(z_{2n}-z_{1})}\Big{\\}}\\\
&=\frac{2}{(2\pi)^{2n}}\int_{0}^{D}dw_{1}\int_{\mathcal{C}_{x_{1}}}dw_{2}\cdots\int_{0}^{D}dw_{2n-1}\int_{\mathcal{C}_{x_{n}}}dw_{2n}\frac{1-e^{-|w_{2n}-w_{1}|}}{(w_{1}-w_{2})\cdots(w_{2n}-w_{1})}\end{split}$
(49)

The first equality of the above equation follows from substituting the
asymptotics from Eqs. (35,36) into the integrand of
‘Part($x_{1},\cdots,x_{n}$)’. There, we need to add in the $O(1/D)$ correction
to extend the approximate $g(z)\approx e^{\pm Dz_{j}}$ or $g(z)\approx e^{\pm
D\frac{1}{z_{j}}}$ from the regions $|z|<D^{-2/3}$ or $|z|>D^{2/3}$. The
second equality follows from substituting $z_{j}\to\frac{1}{z_{j}}$ for each
$j$ in the second term. The third equality can be justified slightly
differently for the $w_{j}$ integrals depending on if $j$ even versus $j$ odd.
The integrals over $w_{2j}$ come from the ‘Schwinger parameterization’ of
substituting
$-\frac{\big{(}-e^{D(z_{2j}-z_{2j+1})}\big{)}^{x_{1}}}{z_{2j}-z_{2j+1}}=\int_{\mathcal{C}_{x_{j}}}dw_{2j}\,e^{w_{2j}(z_{2j}-z_{2j+1})}$.
There, the contour $w_{2j}:0\to-\infty$ for $x_{j}=0$ or $w_{2j}:\infty\to D$
for $x_{j}=1$ work for each pair $z_{2j},z_{2j+1}$. In particular, since the
$z_{2j}$ and $z_{2j+1}$ contours are always opposite each other, these
contours always lead to convergent integrals. The integrals $w_{2j-1}$ follow
from
$-\frac{1-e^{D(z_{2j-1}-z_{2j})}}{z_{2j-1}-z_{2j}}=\int_{0}^{D}dw_{2j-1}\,e^{w_{2j-1}(z_{2j-1}-z_{2j})}$.
The fourth equality follows from integrating over the $\\{z_{j}\\}$, noting
that all of the integrals converge because of the orientations of the
$\\{w_{j}\\}$. ∎

##  IV Multiple Punctures

Now, we consider the analogous calculation for multiple punctures and a gauge
field that’s flat away from a finite number of well-separated punctures. A key
example considered by Dubédat in [11] is a $U(1)$-valued gauge field when for
$s$ pairs of faces $F_{1},F_{1}^{\prime},\cdots,F_{s},F_{s}^{\prime}$ are all
well-separated and the field’s monodromy around $F_{j}$ is the unit complex
number $e^{i\alpha_{j}}$ and the monodromy around $F^{\prime}_{j}$ is
$e^{-i\alpha_{j}}$. Such connections are related to the height correlators
$\braket{\exp\left(i\sum_{j=1}^{s}\alpha_{j}(h(F_{j})-h(F^{\prime}_{j}))\right)}$,
and the limit $\alpha_{j}\to\pi$ can be applied to the strength of monomer
insertions around the $F_{j},F^{\prime}_{j}$. Note that by the Footnote 3, the
absolute value
$\left|\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha_{1}},\cdots,e^{i\alpha_{s}})}{\det\overline{\partial}^{\bullet\to\circ}}\right|$
is well-defined if we restrict to unitary gauge transformations.

In accordance with the free fermion determinant and Gaussian field
expectations Eq. (152) and the results for two punctures Eq. (17), one would
expect

$\log\left|\frac{\det\overline{\partial}^{\bullet\to\circ}(e^{i\alpha_{1}},\cdots,e^{i\alpha_{s}})}{\det\overline{\partial}^{\bullet\to\circ}}\right|\xrightarrow{|F^{\prime}-F|\to\infty}-\frac{1}{2\pi^{2}}\sum_{j=1}^{s}\left(\alpha_{j}^{2}\log(|F^{\prime}_{j}-F_{j}|)+C(\alpha_{j})\right)+\frac{1}{\pi^{2}}\sum_{1\leq
i<j\leq
s}\alpha_{i}\alpha_{j}\log\left(\frac{|F_{i}-F_{j}||F_{i}^{\prime}-F^{\prime}_{j}|}{|F_{i}-F^{\prime}_{j}||F^{\prime}_{i}-F_{j}|}\right)+o(1).$

In fact, Dubédat verfies that this holds and converges for
$-\pi<\alpha_{j}<\pi$. Furthermore, in [1], Dubédat extends this analysis to
certain classes of $SL(2,\mathbb{C})$ connections that are well-separated as
above and shows that for non-abelian connections, well-separated zippers the
analogous determinant
$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}$
converges to a certain ‘$\tau$’ function, which can be interpreted as a
generalization of the free fermion determinants except for non-abelian
connections.

In our analysis, we are able to demonstrate that (in a sense of moments, up to
leading-order as the lattice-spacing goes to zero, to be described shortly)
$\log\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}$
matches with the free-fermion determinant. In particular, we’ll have that for
well-separated zippers approximating paths ${\bf
P}_{j}:\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$ in the complex-plane and
endowed with a graph connection representing a jump matrix $U_{j}\in U(N)$
with eigenvalues $e^{i\alpha^{(j)}_{1}},\cdots,e^{i\alpha^{(j)}_{N}}$ going
right-to-left across ${\bf P}_{j}$, the determinant is asymptotically

$\begin{split}&\log\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}\\\
&\xrightarrow[R\to\infty]{\textbf{moments}}\\\
&\sum_{j=1}^{s}\left(-\frac{1}{2\pi^{2}}({\alpha^{(j)}_{1}}^{2}+\cdots+{\alpha^{(j)}_{N}}^{2})\log\left(R|\tilde{F}^{\prime}_{j}-\tilde{F}_{j}|\right)+C(\alpha^{(j)}_{1})+\cdots+C(\alpha^{(j)}_{N})\right)+\text{`imaginary
part'}\\\ &-\sum_{n=2}^{\infty}\frac{1}{n}\sum_{\begin{subarray}{c}j_{1}\cdots
j_{n}\in\\{1\cdots s\\}\\\
!(j_{1}=\cdots=j_{n})\end{subarray}}\frac{\mathrm{tr}\left[(1-U_{j_{1}}^{-1})\cdots(1-U_{j_{n}}^{-1})\right]}{(2\pi
i)^{n}}\left\\{\int_{{\bf P}_{j_{1}}^{(c(1))}}dw_{1}\cdots\int_{{\bf
P}_{j_{n}}^{(c(n))}}dw_{\ell_{1}+\cdots+\ell_{n}}\frac{1}{(w_{1}-w_{2})\cdots(w_{n}-w_{1})}+\mathrm{c.c.}\right\\}\end{split}$
(50)

where above, $c(1)\cdots c(n)$ satisfy
$c(k)=1,c(k+1)=2,\cdots,c(k+\ell)=\ell+1$ for $j_{k-1}\neq
j_{k}=\cdots=j_{k+\ell}\neq j_{k+\ell+1}$ and $j_{k+m}=j_{k}$ for all $k$, the
curves ${\bf P}_{j}^{(1)},{\bf P}_{j}^{(2)},{\bf P}_{j}^{(3)},\cdots$ are
disjoint curves $\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$ near ${\bf P}_{j}$
going right-to-left (see Fig. 26), and $R$ is a parameter for which $1/R$ can
be thought of as the lattice-spacing (see Sec. IV.1). The notation
$\xrightarrow[R\to\infty]{\textbf{moments}}$ refers to the fact all moments
with respect to the matrix elements of $U_{1}^{-1},\cdots,U_{s}^{-1}$ 444More
precisely, writing $U_{j}=\exp[\sum\kappa_{a}^{(j)}T^{a}]$ for $T^{a}$ a basis
of the Lie algebra of whatever group we choose to work with, the moments with
respect to the $\kappa_{a}^{(j)}$ are determined by the formula. In
particular, the matrix elements themselves generally won’t be independent
since we work with $U(N)$. have a well-defined limit as $R\to\infty$ and can
be rigorously computed using the above formula. The cluster expansion Eq.
(52), the results from the single-zipper case Eq. (17), and the Proposition 8
taken together give the equation above. One can compare this the free-fermion
calculation Eq. (179) and see that it agrees.

The first line comes from terms in the cluster expansion that are of the same
form as those in Sec. III.1. In particular, the ‘imaginary part’ above will be
a sum of the ‘imaginary parts’ of the corresponding calculations from Sec.
III.1, which we argued in Sec. III.3.1 be non-universal. The other terms in
the cluster expansion correspond to the second line of the expansion. They
will all be universal in the sense that their asymptotics only depend on the
continuum paths that the gauge-field configurations approximate. These terms
will generically have real and imaginary parts which are both universal.

Above, we’ve been focusing on well-separated zippers, for which all of the
above integrals are well-defined and convergent. When the zippers are allowed
to share endpoints, then some of the integrals will diverge logarithmically.
On the lattice, everything will be a finite number for a fixed $R$, and it
turns out that for well-chosen zippers, a simple modification to the integrals
(see Sec. IV.4) will give a finite result with a logarithmic divergence in
$R$.

While we can rigorously compute everything with respect to the moments as
above, we do not know how to analyze the convergence of the cluster expansion,
in particular of the behavior of the subleading terms. In [1], this
convergence was established for certain subclasses of well-separated zipper
configurations with a reflectional symmetry around
$\mathbb{R}\subset\mathbb{C}$ and for unipotent $U_{j}\in SL(2,\mathbb{R})$.
In this case, the first line in the RHS of Eq. (50) diverging logarithmically
with $R$ vanishes. In particular, choosing unipotent $\\{U_{j}\\}$ will
automatically give zero for the two-puncture calculations of Sec. III,
including for the ‘non-universal’ imaginary parts. As such in this case, our
series expansion is universal. Then, our result can be seen as rigorously
giving a series expansion for Dubédat’s version of the tau function.

In Sec. IV.1, we go through the geometric preliminaries needed to compare the
lattice and continuum zipper setups for well-separated zippers. In Sec. IV.2,
we derive the cluster expansion used in our analysis. In Sec. IV.3, we
evaluate the terms in the cluster expansion for well-separated zippers. In
Sec. IV.4, we explain how to modify the above steps for non-separated zippers.

###  IV.1 Setup for well-separated Zippers

The general strategy of constructing well-separated zippers will be similar
for this computation as it was for the previous case of a single pair of faces
$(F,F^{\prime})$. In particular, we will consider a deformation
$\overline{\partial}(U)$ of the Kasteleyn matrix $\overline{\partial}$ by
multiplying the edge weights on the graph by weights that correspond to a
connection on the graph with monodromies that jump as $U_{j}$ left-to-right
across $F_{j}\to F^{\prime}_{j}$.

We will be able to arrange these monodromies by considering a collection of
$s$ zippers, which are each a path on the dual graph connecting $F_{j}\to
F^{\prime}_{j}$. Then we will multiply the edge weights by $U_{j}^{\pm 1}$ on
the edges corresponding to the dual edges of the path, where $U_{j}$ will be
multiplied in the direction right-to-left relative to the path and
$U_{j}^{-1}$ will be multiplied in the direction right-to-left. 555 Note that
these definitions seem ‘opposite’ to the fact that the monodromy in the
continuum is multiplied by $U_{j}$ going left-to-right. A way to see that they
should be inverses is that on the graph (see Sec. II.3), the gauge-invariant
monodromy for a path $v_{1}\to v_{2}\to\cdots\to v_{k}\to v_{1}$ is computed
as $U(v_{1}\to v_{2})U(v_{2}\to v_{3})\cdot\cdots\cdot U(v_{k}\to v_{1})$.
However in the continuum, the monodromy matrices would be multiplied in the
opposite order $\tilde{U}(v_{k}\to v_{1})\cdot\cdots\cdot\tilde{U}(v_{2}\to
v_{3})\tilde{U}(v_{1}\to v_{2})$. As such, identifying the graph monodromies
of $U$ as the inverses of the continuum monodromies $\tilde{U}$ is natural and
consistent.

However, there will be a difference from the previous section in the specific
zippers we use to do our calculation. Previously the weights $U_{j}^{-1}$ were
always pointing in the direction of black-to-white vertices, and we chose
zippers along a dual that were approximated by a straight line going from
$F\to F^{\prime}$. In the setup for this calculation, we will still keep the
$U_{j}^{-1}$ factors pointing in the black-to-white direction. However, we
will abandon specifically choosing the paths to be ‘straight lines’.
Specifically, we want to keep the zippers themselves well-separated from each
other so that the zippers don’t have any crossings. The reason for this is
with some foresight from the previous discussion is that in the continuum
limit, the series expansions for these will require asymptotics of the
functions $g_{v\to w}(z)$ for $|v-w|$ large. With even more foresight, we’ll
have that the integrals will involve an integrand
$\frac{1}{(w_{1}-w_{2})\cdots(w_{n}-w_{1})}$ where the $w_{j}$ are on contours
that the zipper approximate. If the zippers cross, then these integrals will
diverge, or at least be ambiguous.

Figure 9:  Example of multiple zippers that are well-separated. (Left) For
$j=1\cdots 4$, there are four continuous, nonintersecting curves ${\bf P}_{j}$
in $\mathbb{C}$ that go between $\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$. They
are split into $N_{j}$ segments, where $N_{2}=16,N_{3}=13,N_{4}=10$ and
$N_{1}$ is unspecified. On the curve ${\bf P}_{1}$, a depiction (not to scale)
of the corresponding rhombus graph is given. We associate a matrix $U_{j}$ for
going right-to-left across the path ${\bf P}_{j}$, which gives a flat
connection on the punctured plane. (Right) For some $R\gg 1$, we choose faces
$F_{j,k}\approx R\tilde{F}_{j,k}$. Then for each of the segments ${\bf
P}_{j,k}:\tilde{F}_{j,k-1}\to\tilde{F}_{j,k}$, we associate a monotone path on
the rhombus graph $\mathrm{zip}\,j,k:F_{j,k-1}\to F_{j,k}$. On the top, we
depict the straight-line zipper path together with the connection that we put
in the graph, associated with the connection in the continuum. On the bottom,
we show the allowed range of rhombus angles for these straight-line paths.

Figure 10:  Depicition of necessary conditions for splitting up the curve into
sufficiently small segments, namely the existence of angles
$\theta^{\mathrm{mid}\,,\,j}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}},\theta^{\mathrm{mid}\,,\,j^{\prime}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$
and a midpoint
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$ for which
$\tilde{F}_{j,k-1},\tilde{F}_{j,k}$ and
$\tilde{F}_{j^{\prime},k^{\prime}-1},\tilde{F}_{j^{\prime},k^{\prime}}$ are in
arcs that subtend an angle smaller than $\epsilon/8$ from the midpoint, and
for which the rays from the midpoints with angles
$\theta^{\mathrm{mid}\,,\,j}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}},\theta^{\mathrm{mid}\,,\,j^{\prime}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$
going in both directions intersect both line segments.

As such, we will need to split up our paths into smaller pieces, and if there
are crossing between the zippers, then possibly the asymptotics for $g_{v\to
w}(z)$ for $v,w$ near the crossings will no longer be valid and the continuum
expressions for the integrals wouldn’t make sense. Throughout this section, we
will assume that all of the zippers in our analysis are contained in a convex
666By ‘convex’, we mean that for any two vertices in the region, the
‘straight-line’ paths analogous to the ones defined in Section III.1 and Fig.
6 are contained within the region. region of the rhombus graph that satisfy
the regularity conditions of Appendix A.

We starting by fixing some notation and talk about how exactly we will phrase
the ‘continuum limit’ of our problem. First, note that in the continuum limit,
the relative distances between the $\\{F_{j}\\}$ and $\\{F^{\prime}_{j}\\}$
will be much larger than the scales relevant to the asymptotics of $g(z)$ and
the length 1 of the rhombus edges. And, we are considering the limit as the
relative distances goes to $\infty$. As such, we will instead find it
convenient to define fixed complex numbers $\\{\tilde{F}_{j}\\}$ and
$\\{\tilde{F}^{\prime}_{j}\\}$ and a additional parameter $R$ for which the
coordinate of the faces will be $F_{j}\approx R\tilde{F}_{j}$ and
$F^{\prime}_{j}\approx R\tilde{F}^{\prime}_{j}$, where ‘$\approx$’ means that
the $F_{j},F^{\prime}_{j}$ are the closest dual face to the given complex
number. We will be looking at terms at leading order in $R$ as $R\to\infty$.

Next, we need to consider how to choose our zippers. First, we want to choose
continuum paths ${\bf P}_{j}:\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$ in the
complex plane that are smooth and do not mutually intersect. We will want our
zippers to ‘approximate’ these continuum paths in some sense. Recall that in
the case of a single zipper, we essentially chose the continuum path ${\bf
P}_{j}$ to be a straight line between $F\to F^{\prime}$, and we were able to
approximate the zipper by choosing a monotone rhombus path. For our case, we
may not be able to choose straight paths without them intersecting. So
instead, we will split the continuous curve into sufficiently small but still
finite-sized pieces on which the zippers can be chosen to be the monotone
paths, where ‘sufficiently small’ will be defined soon. In particular, we’ll
choose to split up the paths into small enough regions that the asymptotics of
the functions $g_{F\to F^{\prime}}(z)$ can be applied for $F,F^{\prime}$ being
endpoints of two different segments. Every continuous path ${\bf
P}_{j}:\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$ will be split up and turned
into some number $N_{j}$ of segments. There will be $N_{j}+1$ total points on
each ${\bf P}_{j}$, labeled by complex numbers
$\tilde{F}_{j}=:\tilde{F}_{j,0},\tilde{F}_{j,1},\tilde{F}_{j,2},\cdots,\tilde{F}_{j,N_{j}-1},\tilde{F}_{j,N_{j}}:=\tilde{F}^{\prime}_{j}$.
We will give a label ${\bf P}_{j,k}:\tilde{F}_{j,k-1}\to\tilde{F}_{j,k}$ to
each of the segments, which each correspond to a monotone, straight-line,
segments. Also, we will use $\mathrm{zip}\,j,k:F_{j,k-1}\to F_{j,k}$ to refer
to these straight-line segments on the rhombus graph. Also, we will find it
useful to define the angle
$e^{i\theta_{j,k}}=\frac{F_{j,k}-F_{j,k-1}}{|F_{j,k}-F_{j,k-1}|}$. See Fig. 10
for a depiction.

Now, we discuss what we mean by ‘sufficiently small’ for the decomposition
into segments. This will require the definition of some new auxiliary
quantities. The estimate Prop. 10 from Appendix A uses the regularity
conditions to show that there exists an angle $\epsilon/8$ (related to the
$\epsilon$ used in the appendix) for which $g_{F\to F^{\prime}}(z)$ is
exponentially suppressed for $z=-|z|e^{i\theta}$ in an intermediate range of
$D^{-2/3}<|z|<D^{2/3}$ angles in a range of
$\theta\in(\arg(F^{\prime}-F)-\epsilon/8,\arg(F^{\prime}-F)+\epsilon/8)$. For
the segments to be ‘sufficiently small’ means that for each pair of segments
$(\mathrm{zip}\,j,k\,,\,\mathrm{zip}\,j^{\prime},k^{\prime})$, there exists a
complex number
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$ in the
region subtended by the pair, and two angles
$\theta^{\mathrm{mid}\,,\,j}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}},\theta^{\mathrm{mid}\,,\,j^{\prime}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$
such that both $\tilde{F}_{j,k},\tilde{F}_{j,k+1}$ are contained in the cone
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}\bigsqcup_{\theta-\theta^{\mathrm{mid}\,,\,j}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}\in(-\epsilon/8,\epsilon/8)}e^{i\theta}\mathbb{R}_{>0}$
emanating from
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$ and so that
both $\tilde{F}_{j^{\prime},k^{\prime}},\tilde{F}_{j^{\prime},k^{\prime}+1}$
are in the cone
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}\bigsqcup_{\theta-\theta^{\mathrm{mid}\,,\,j^{\prime}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}\in(-\epsilon/8,\epsilon/8)}e^{i\theta}\mathbb{R}_{>0}$.
Furthermore, we require that the rays from
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$ going out
in both directions along the angles
$\theta^{\mathrm{mid}\,,\,j}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}},\theta^{\mathrm{mid}\,,\,j^{\prime}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$
intersect both segments. See Fig. 10 for a depiction.

Note that this decomposition is always possible. For $j\neq j^{\prime}$, it is
clear sufficiently subdivided curves will satisfy this since the distance
between different segments is bounded from below. For $j=j^{\prime}$, segments
that are far enough apart and small enough can satisfy this. However for
segments that are close, the fact that the curves are smooth means that within
some neighborhood, the curves well approximate straight lines, and any choice
of midpoint $\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$
along the ${\bf P}_{j}$ between the two segments will be a suitable choice.
For non-adjacent segments within such a neighborhood, we want to choose
$\tilde{F}^{\mathrm{mid}}_{\\{j,j^{\prime}\\},\\{k,k^{\prime}\\}}$ to be along
${\bf P}_{j}$ strictly in between the segments. For adjacent segments ${\bf
P}_{j,k},{\bf P}_{j,k+1}$ that share $\tilde{F}_{j,k}$, we will choose
$\tilde{F}^{\mathrm{mid}}_{\\{j,j\\},\\{k,k+1\\}}=\tilde{F}_{j,k}$.

###  IV.2 Cluster Expansion of the twisted determinant: multiple zippers

Now we will descibe the cluster expansion used in this analysis. Consider the
Kasteleyn matrix $\overline{\partial}(U)$ as described before, with a
connection along the previously mentioned zippers connections contributing
$U_{1},\cdots,U_{s}$ to the monodromy going right-to-left across $F_{1}\to
F^{\prime}_{1},\cdots,F_{s}\to F^{\prime}_{s}$. This is defined by

$\overline{\partial}(U)=\underbrace{\overline{\partial}^{\bullet\to\circ}+(U^{-1}_{1}-1)\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,1}+\cdots+(U^{-1}_{s}-1)\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,s}}_{=:\overline{\partial}^{\bullet\to\circ}(U)}\
\,\,+\,\,\underbrace{\overline{\partial}^{\bullet\to\circ}+(U_{1}-1)\overline{\partial}^{\circ\to\bullet}_{\mathrm{zip}\,1}+\cdots+(U_{s}-1)\overline{\partial}^{\circ\to\bullet}_{\mathrm{zip}\,s}}_{=:\overline{\partial}^{\circ\to\bullet}(U)}$
(51)

where each $\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,k}$ or
$\overline{\partial}^{\circ\to\bullet}_{\mathrm{zip}\,k}$ is the restriction
of the $\overline{\partial}$ operator to the $k$th zipper in the black-to-
white or white-to-black directions of the lattice. As such, we can expand

$\begin{split}&\log\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}=\mathrm{tr}\log\left(\text{1}-(1-U^{-1}_{1})\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,1}}{\overline{\partial}^{\bullet\to\circ}}-\cdots-(1-U^{-1}_{s})\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,s}}{\overline{\partial}^{\bullet\to\circ}}\right)\\\
&=-\sum_{n=1}^{\infty}\frac{1}{n}\sum_{i_{1}\cdots i_{n}\in\\{1\cdots
s\\}}\mathrm{tr}\left[(1-U^{-1}_{1})\cdots(1-U^{-1}_{n})\right]\cdot\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right].\end{split}$

Note that the terms where $i_{1}=i_{2}=\cdots=i_{n}=j$ for all $n$ with some
fixed $j$ correspond to the series expansion of
$\log\frac{\det\left(\overline{\partial}^{\bullet\to\circ}+(1-U_{j}^{-1})\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,j}\right)}{\det\left(\overline{\partial}^{\bullet\to\circ}\right)}$,
which was already analyzed in Sec. III. We will consider subtracting out these
terms and analyzing the remainder. So, we can similarly expand out

$\begin{split}&\frac{\det\overline{\partial}^{\bullet\to\circ}(U)}{\det\overline{\partial}^{\bullet\to\circ}}-\sum_{j=1}^{s}\log\frac{\det\left(\overline{\partial}^{\bullet\to\circ}+(1-U_{j}^{-1})\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,j}\right)}{\det\overline{\partial}^{\bullet\to\circ}}\\\
&=-\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\begin{subarray}{c}i_{1}\cdots
i_{n}\in\\{1\cdots s\\}\\\ i_{1}\cdots i_{n}\text{ not all
same}\end{subarray}}\mathrm{tr}\left[(1-U^{-1}_{1})\cdots(1-U^{-1}_{n})\right]\cdot\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right].\end{split}$
(52)

Similarly to the series expansion of the single-zipper case, we will show that
the leading order contributions to the traces
$\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]$
can be expressed by some continuous integrals over $n$ variables $w_{1}\cdots
w_{n}$.

###  IV.3 Evaluating
$\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]$
for well-separated zippers

We want to asymptotically evaluate the terms on the left-hand side of Eq. (52)
when the $i_{1}\cdots i_{n}$ are not all the same, in the limit of the
parameter $R\to\infty$ for fixed $\\{\tilde{F}_{j},\tilde{F}^{\prime}_{j}\\}$.
In particular the leading order term will schematically be of the form

$\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]\xrightarrow{R\to\infty}\text{``}\frac{1}{(2\pi
i)^{n}}\int_{{\bf P}_{i_{1}}}dw_{1}\cdots\int_{{\bf
P}_{i_{n}}}dw_{n}\frac{1}{(w_{1}-w_{2})\cdots(w_{n}-w_{1})}+\mathrm{c.c.}\text{''}$

A priori, the integral in quotes on the right-hand-side above is not well-
defined because it will be singular whenever $i_{k}=i_{k+1}$. However, we will
see instead that the integral above will be a certain regularization of the
singular expression above. The regularization will involve slightly deforming
the contours with with the same $i_{k}=\cdots=i_{k+k^{\prime}}=:j$ into
$k^{\prime}+1$ contours that are close to ${\bf P}_{j}$ that don’t mutually
intersect except at the endpoints $\tilde{F}_{j},\tilde{F}^{\prime}_{j}$.

We now describe the particular regularization that pops out of the series
expansion. First, note that since ${\bf P}_{j}$ is a directed curve in the
complex plane, there is a canonical ‘left’ and ‘right’ side of the curve. Now
suppose $i_{k}=\cdots=i_{k+r-1}=:j\in\\{1\cdots s\\}$ are a maximal set of
cyclically adjacent indices that are all equal among the indices, i.e. for
which $i_{k-1}\neq j$, $i_{k+r}\neq j$. Then we can define ${\bf
P}^{(1)}_{j}\cdots{\bf P}^{(r)}_{j}$ to be $r$ non-intersecting curves near
${\bf P}_{j}$ with endpoints at $\tilde{F}_{j}\to\tilde{F}^{\prime}_{j}$ with
‘left-to-right’ order ${\bf P}^{(r)}_{j}\to{\bf P}^{(r-1)}_{j}\to\cdots\to{\bf
P}^{(2)}_{j}\to{\bf P}^{(1)}_{j}$. Then, we’ll have the regulated integrals
that show up in the series expansions will have $w_{k},\cdots,w_{k+r-1}$ vary
over the curves ${\bf P}^{(1)}_{j},\cdots,{\bf P}^{(r)}_{j}$. See the below
proposition for the formal statement and Figs. 11,26 for various examples.

Figure 11:  Various examples of the regulated integration contours of the
integrand $\frac{1}{(2\pi i)^{n}}\frac{1}{(w_{1}-w_{2})\cdots(w_{n}-w_{1})}$
that appear in the evaluation of
$\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]$.
The dotted red lines are the original smooth curves $\\{{\bf P}_{\cdots}\\}$
used to define the zippers, and are labeled. The solid pink lines are the
deformed contours $\\{{\bf P}^{(\cdots)}_{\cdots}\\}$, and these paths
together with the assignments of variables $w_{1}\cdots w_{n}$ to the path
(corresponding to the integral in the Prop. 8) are labeled. (Top-Left) An
$n=3$ example with two zippers. (Top-Right) An $n=6$ example with three
zippers. (Bottom) An $n=9$ example with four zippers.

######  Proposition 8.

Let $i_{1}\cdots i_{n}\in\\{1\cdots s\\}$ not all be the same index. Let us
give adjacent labels, where $i_{1}=\cdots=i_{r_{1}}=:j_{1}$,
$i_{r_{1}+1}=\cdots=i_{r_{1}+r_{2}}=:j_{2}$, $\cdots$,
$i_{r_{1}+\cdots+r_{t-1}+1}=\cdots=i_{r_{1}+\cdots+r_{t-1}+r_{t}}=:j_{t}$,
with $r_{1}+\cdots+r_{t}=m$. Without loss of generality, suppose that
$i_{1}=j_{1}\neq j_{t}=i_{n}$. Then, we have

$\begin{split}&\mathrm{tr}\left[\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{1}}}{\overline{\partial}^{\bullet\to\circ}}\cdots\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,i_{n}}}{\overline{\partial}^{\bullet\to\circ}}\right]\cdot(1+O(1/R))+O(1/R)=\mathrm{tr}\left[\left(\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,j_{1}}}{\overline{\partial}^{\bullet\to\circ}}\right)^{r_{1}}\cdots\left(\frac{\overline{\partial}^{\bullet\to\circ}_{\mathrm{zip}\,j_{t}}}{\overline{\partial}^{\bullet\to\circ}}\right)^{r_{t}}\right]\cdot(1+O(1/R))+O(1/R)\\\
&=\frac{1}{(2\pi i)^{n}}\underbrace{\int_{{\bf
P}^{(1)}_{j_{1}}}dw_{1}\cdots\int_{{\bf
P}^{(r_{1})}_{j_{1}}}dw_{r_{1}}}_{i_{1}=\cdots=i_{r_{1}}=j_{1}}\cdots\underbrace{\int_{{\bf
P}^{(1)}_{j_{t}}}dw_{r_{1}+\cdots+r_{t-1}+1}\cdots\int_{{\bf
P}^{(r_{t})}_{j_{t}}}dw_{n}}_{i_{r_{1}+\cdots+r_{t-1}+1}=\cdots=i_{n}=j_{t}}\frac{1}{(w_{1}-w_{2})\cdots(w_{n}-w_{1})}+\mathrm{c.c.}\end{split}$
(53)

where the contours ${\bf P}^{(\cdots)}_{j_{\cdots}}$ are a choice of regulated