# Massive Scaling Limit of the Ising Model: Subcritical Analysis and
Isomonodromy

S. C. Park  Ecole Polytechnique Fédérale de Lausanne, EPFL SB MATHAA CSFT,
CH-1015 Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract.

We study the spin $n$-point functions of the planar Ising model on a simply
connected domain $\Omega$ discretised by the square lattice
$\delta\mathbb{Z}^{2}$ under near-critical scaling limit. While the scaling
limit on the full-plane $\mathbb{C}$ has been analysed in terms of a fermionic
field theory, the limit in general $\Omega$ has not been studied. We will show
that, in a massive scaling limit wherein the inverse temperature is scaled
$\beta\sim\beta_{c}-m_{0}\delta$ for a constant $m_{0}<0$, the renormalised
spin correlations converge to a continuous quantity determined by a boundary
value problem set in $\Omega$. In the case of $\Omega=\mathbb{C}$ and $n=2$,
this result reproduces the celebrated formula of [WMTB76] involving the
Painlevé III transcendent. To this end, we generalise the comprehensive
discrete complex analytic framework used in the critical setting to the
massive setting, which results in a perturbation of the usual notions of
analyticity and harmonicity.

###### Contents

  1. 1 Introduction
    1. 1.1 Main Results
    2. 1.2 Notation
    3. 1.3 Proof of the Main Theorems
    4. 1.4 Structure of the Paper
  2. 2 Massive Fermions
    1. 2.1 Bounded Domain Fermions and Discrete Analysis
    2. 2.2 Full-Plane Fermions
    3. 2.3 Massive Complex Analysis: Continuous Fermions and their Uniqueness
  3. 3 Discrete Analysis: Scaling Limit
    1. 3.1 Bulk Convergence
    2. 3.2 Analysis near the Singularity
  4. 4 Continuum Analysis: Isomonodromy and Painlevé III
    1. 4.1 Continuous Analysis of the Coefficients
    2. 4.2 Derivation of Painlevé III
  5. A Harmonicity Estimates

##  1\. Introduction

The Ising model is a classical model of ferromagnetism first introduced by
Lenz [Len20], whose simplicity and rich emergent structure have allowed for
applications in various areas of science. In two dimensions, tt famously
exhibits a continuous phase transition [Ons44, Yan52], where the
characteristic length of the model diverges and the model becomes scale
invariant. Consequently the model at the critical temperature $\beta_{c}$ is
expected to exhibit conformal symmetry under scaling limit, a prediction which
has been formalised in terms of the Conformal Field Theory [BPZ84].

Given the infinite dimensional array of 2D local conformal symmetry [DMS97],
it is natural to study the scaling limit of the model not only on the full
plane $\mathbb{C}$ but on an arbitrary simply connected domain $\Omega$.
Accordingly, recent research has focused on giving a rigorous description of
the interaction between various physical quantities of the model under
critical scaling limit and the conformal geometry of $\Omega$, which results
in explicit formulae for the limit of spin correlations at the microscopic
scale [HoSm13, GHP18], at the macroscopic scale [CHI15], or in some mixture of
the two [Hon10, CHI18] in terms of quantities such as the conformal radius and
its derivatives. Convergence of correlations also proves to be useful in
proving convergence in more general senses: [CGN15] proves that the discrete
spin field converges to a continuous random distribution using the convergence
of their correlations.

Central to such analyses of the critical regime are discrete fermion
correlations, which manifest themselves as discrete functions capable of
encoding relevant physical quantities. They are discrete counterparts of the
massless free fermion correlations in the continuous CFT, which turn out to be
explicit holomorphic functions thanks to conformal symmetry. Discrete fermions
instead enjoy a strong notion of discrete analyticity [Smi10], which, unlike
some of its weaker variants, readily lends itself to precompactness estimates
that ultimately yield convergence to the continuous fermion.

However, the free fermion is not an object unique to the massless (conformal)
field theory; indeed, the general field theory of the free fermion specifies a
mass parameter $m$, or equivalently a length scale $\xi\propto
1/\left|m\right|$. In general, the corresponding regime in the Ising model is
the _near-critical_ scaling limit, where the deviation from criticality
$\beta_{c}-\beta$ scales proportionally to the lattice spacing $\delta$. Such
scaling keeps the physical correlation length $\xi$ asymptotically constant,
allowing the limit to be physically described by the massive fermion.

The discrete fermion survives in this near-critical setup, and discrete
analyticity persists albeit in a perturbed sense [DGP14, HKZ15, dT18]. While
the continuous massive fermion is usually described by the two-dimensional
massive Dirac equation, our strong discrete analyticity in fact features twice
as many relations, resulting in (the discrete counterpart of) a perturbation
of the ordinary Cauchy-Riemann equations in 1D. Since there are as many
lattice equations in the near-critical limit as in the critical limit, it is
natural to attempt to carry out in the former the analogues of analyses from
the latter. We note here that in addition to such a _thermal_ perturbation,
one may also consider a _magnetic_ perturbation to introduce mass (e.g.
[CGN16]). Another direction of research has recently focused on universality
with respect to general lattice, see [Che18].

In this paper, we undertake the analysis of macroscopic Ising spin
correlations on a simply connected domain $\Omega$ in the near-critical
scaling limit where $\beta_{c}-\beta$ is held equal to $m_{0}\delta$ for a
fixed $m_{0}<0$ with $+$ boundary conditions. We establish the existence of
scaling functions to which renormalised spin correlations converge, and show
that their logarithmic derivatives are determined by an explicit boundary
value problem set in $\Omega$. This extends the results of [CHI15] to the
massive regime (save for the conformal covariance, which should not hold), and
our proof combines the strategies of that paper with a massive perturbation of
analytic function theory, both in the discrete and the continuous settings. In
the former, massive harmonic and holomorphic functions can be studied via
their relation to massive (extinguished) random walk; in the latter, the
perturbed Cauchy-Riemann equation is dubbed Vekua equation and extensively
treated in a theory established by Carleman, Bers, and Vekua, among others
[Ber56, Vek62].

In the full plane, the massive scaling limit of the spin correlations was
revealed to exhibit a surprising integrability property. Wu, McCoy, Tracy, and
Barouch [WMTB76] first demonstrated that the $2$-point function on the plane
can be described in terms of the Painlevé III transcendent. Subsequently,
Sato, Miwa, and Jimbo [SMJ77] recast the continuous analysis in terms of
isomonodromic deformation theory, where Painlevé equations are known to arise,
and obtained a closed set of differential equations for the $n$-point
function. Letting $\Omega=\mathbb{C}$, we reproduce the $2$-point scaling
limit in the case of full-plane (whose classical treatment is given in, e.g.,
[PaTr83, Pal07]), setting up the continuous analysis. We explicitly carry out
the isomonodromic analysis following the formulation of [KaKo80].

###  1.1. Main Results

Let $\Omega$ be a bounded simply connected domain with smooth boundary. We
will treat the unbounded cases $\Omega=\mathbb{C},\mathbb{H}$ as well. Define
the rotated square lattice
$\Omega_{\delta}:=\Omega\cap\delta(1+i)\mathbb{Z}^{2}=\Omega\cap\mathbb{C}_{\delta}$.
We define the _Ising probability measure_
$\mathbb{P}=\mathbb{P}_{\Omega_{\delta},\beta}^{+}$ with $+$ boundary
conditions at _inverse temperature_ $\beta>0$ on the space of _spin
configurations_ $\left\\{\pm 1\right\\}^{\Omega_{\delta}}$ by

$\mathbb{P}_{\Omega_{\delta},\beta}^{+}\left[\sigma:\Omega_{\delta}\to\left\\{\pm
1\right\\}\right]\propto\exp\sum_{i\sim j}\beta\sigma_{i}\sigma_{j},$

where the sum is over pairs
$\left\\{i,j\right\\}\subset\mathbb{C}_{\delta},i\in\Omega_{\delta}$ such that
$\left|i-j\right|=\sqrt{2}\delta$ and we define $\sigma_{j}=1$ for
$j\notin\Omega_{\delta}$ ($j\in\mathcal{F}\left[\Omega_{\delta}\right]$ in
terms of detailed notation in Section 1.2). If $a\in\Omega$, we understand by
$\sigma_{a}$ the spin at a closest point in $\Omega_{\delta}$ to $a$.

The planar Ising model on the square lattice undergoes a phase transition at
the critical temperature $\beta_{c}=\frac{1}{2}\ln\left(1+\sqrt{2}\right)$.
Henceforth we will fix a negative parameter $m$ and set
$\beta=\beta(\delta)=\beta_{c}-\frac{m\delta}{2}$. This is a _subcritical_
massive limit, where the spins stay in the ordered phase while approaching
criticality.

######  Theorem 1.

Let $\Omega$ be a bounded simply connected domain and suppose
$a_{1},\ldots,a_{n}\in\Omega$. Under $\delta\downarrow
0,\beta=\beta_{c}-\frac{m\delta}{2}$, the spin $n$-point function converges to
a continuous function of $a_{1},\ldots,a_{n}$,

$\delta^{-\frac{n}{8}}\mathbb{E}_{\Omega_{\delta},\beta(\delta)}^{+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]\to\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+},$

and its logarithmic derivative $\partial_{a_{1}}\ln\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+}=\mathcal{A}_{\Omega}^{1}+i\mathcal{A}_{\Omega}^{i}$,
where
$\partial_{a_{1}}=\frac{1}{2}\left(\partial_{x_{1}}-i\partial_{y_{1}}\right)$,
is determined by the solution to the boundary value problem of Proposition 17
set on the domain $\Omega$. The functions $\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+}$ are uniquely determined by
their diagonal and boundary behaviours, see Section 1.3. In particular, as
$a_{1}\to\partial\Omega$,

$\left\langle a_{1}\right\rangle_{\Omega,m}^{+}\sim\left\langle
a_{1}\right\rangle_{\Omega,0}^{+}:=\mathcal{C}\cdot
2^{\frac{1}{4}}\text{rad}^{-\frac{1}{8}}(a_{1},\Omega),$

where $\mathcal{C}:=2^{\frac{1}{6}}e^{-\frac{3}{2}\zeta^{\prime}(-1)}$,
$\text{rad}(a_{1},\Omega)$ is the conformal radius of $\Omega$ as seen from
$a_{1}$, and $\zeta$ is the Riemann zeta function.

If $\phi:\mathbb{D}\to\Omega$ is a conformal map with $\phi(0)=a_{1}$, then
$\text{rad}(a_{1},\Omega)=\left|\phi^{\prime}(a_{1})\right|$. The
normalisation of our continuous functions
$\left\langle\cdot\right\rangle_{\Omega,0}^{+}$ differ to that of [CHI15] by a
factor of $\mathcal{C}^{n}$. The derivation of the following result in our
analytical setup may be of independent interest.

######  Corollary 2 ([WMTB76, SMJ77, KaKo80]).

The $2$-point function in the full-plane is given by

$\displaystyle\left\langle-a,a\right\rangle_{\mathbb{C},m}^{+}$
$\displaystyle=cst\cdot\cosh
h_{0}(am)\cdot\exp\left[\int_{-\infty}^{am}r\left[\left(h_{0}^{\prime}(r)\right)^{2}-4\sinh^{2}2h_{0}(r)\right]dr\right],$
$\displaystyle\left\langle-a,a\right\rangle_{\mathbb{C},-m}^{\text{free}}$
$\displaystyle=cst\cdot\sinh
h_{0}(am)\cdot\exp\left[\int_{-\infty}^{am}r\left[\left(h_{0}^{\prime}(r)\right)^{2}-4\sinh^{2}2h_{0}(r)\right]dr\right],$

where $a>0$ and $\eta_{0}=-\frac{1}{2}\ln h_{0}$ is a solution to the Painlevé
III equation

$r\eta_{0}\eta^{\prime\prime}_{0}=r\left(\eta_{0}^{\prime}\right)^{2}-\eta_{0}\eta_{0}^{\prime}-4r+4r\eta_{0}^{4}.$

The constants are fixed by the condition that as $a\to 0$

$\left\langle-a,a\right\rangle_{\mathbb{C},m}^{+}\sim\left\langle-a,a\right\rangle_{\mathbb{C},0}^{+}:=\frac{\mathcal{C}^{2}}{\left|2a\right|^{1/4}},$

and $\mathcal{C}:=2^{\frac{1}{6}}e^{-\frac{3}{2}\zeta^{\prime}(-1)}$ as above.

###  1.2. Notation

Following signs will be used throughout the paper:

$\lambda:=e^{\frac{i\pi}{4}},\beta_{c}=\frac{1}{2}\ln\left(1+\sqrt{2}\right),\mathbb{H}:=\left\\{z\in\mathbb{C}:\text{Im}z>0\right\\},A\oplus
B:=\left(A\cup B\right)\setminus\left(A\cap B\right).$

We will write partial derivatives in contracted form, i.e.
$\partial_{x}=\frac{\partial}{\partial x}$, etc. And denote by
$\partial_{z},\partial_{\bar{z}}$ the Wirtinger derivatives: where $z=x+iy$,

$\partial_{z}:=\frac{\partial_{x}-i\partial_{y}}{2},\partial_{\bar{z}}:=\frac{\partial_{x}+i\partial_{y}}{2}=\frac{e^{i\theta}\left(\partial_{r}+ir^{-1}\partial_{\theta}\right)}{2}.$

We will also use $\partial$ to denote directional derivatives; i.e.
$\partial_{1}=\partial_{x}$, $\partial_{i}=\partial_{y}$, and so on. If
$z\in\partial\Omega$, denote by $\nu_{\text{out}}(z)\in\mathbb{C}$ as the unit
normal at $x$, i.e. the unit complex number which points to the direction of
outer normal vector at $z$. Then $\partial_{\nu_{\text{out}}}$ is the outer
normal derivative in the direction of $\nu_{\text{out}}$.

We denote by $\partial_{\lambda}^{\delta},\Delta^{\delta}$ the following
discrete operators:

$\displaystyle\partial_{\lambda}^{\delta}f(z)$
$\displaystyle:=f(z+\sqrt{2}\lambda\delta)-f(z)\text{, etc.,}$
$\displaystyle\Delta^{\delta}f(z)$
$\displaystyle:=f(z+\sqrt{2}\lambda\delta)+f(z-\sqrt{2}\bar{\lambda}\delta)+f(z-\sqrt{2}\lambda\delta)+f(z+\sqrt{2}\bar{\lambda}\delta)-4f(z),$

wherever they make sense, if $z\notin\mathcal{V}\left[\Omega_{\delta}\right]$.
On $\mathcal{V}\left[\Omega_{\delta}\right]$, we make a small modification in
the coefficients in $\Delta^{\delta}$; see (2.10). Note that
$(\sqrt{2}\delta)^{-2}\Delta^{\delta}\to\Delta$.

#### Mass Parametrisation.

There are various equivalent ways of parametrising the deviation
$\beta_{c}-\beta$, and we summarise the relation amongst them here at once,
which hold at all times. In this paper, $M,m,\Theta$ will be supposed to be
negative unless otherwise specified. We also assume $\delta$ is small enough
to, e.g., have $\beta\in(0,\infty)$.

  * •

_Discrete mass_ $M:=\beta_{c}-\beta$ is scaled $M=\frac{m\delta}{2}$ with the
continuous mass $m$ being a constant.

  * •

Pure phase factor
$e^{2i\Theta}:=\lambda^{-3}\frac{e^{-2\beta}+i}{e^{-2\beta}-i}$ with
$\Theta\in\left]-\frac{\pi}{8},\frac{\pi}{8}\right[$. Equivalently
$e^{2\beta}=\cot\left(\frac{\pi}{8}+\Theta\right)$. $\Theta$ is scaled
$\Theta\sim\frac{m\delta}{2}$. Also define $M_{H}:=2\sin
2\Theta\sqrt{\frac{2}{\cos 4\Theta}}$ which is the mass coefficient in massive
harmonicity.

#### Graph Notation.

Recall that we work with the rotated square lattice
$\mathbb{C}_{\delta}:=\delta(1+i)\mathbb{Z}^{2}$. Our graph $\Omega_{\delta}$
comprises the following components (see Figure 1.1):

  * •

_faces_
$\mathcal{F}\left[\Omega_{\delta}\right]:=\Omega\cap\delta(1+i)\mathbb{Z}^{2}$,

  * •

_vertices_ $\mathcal{V}\left[\Omega_{\delta}\right]:=\left\\{f\pm\delta,f\pm
i\delta:f\in\mathcal{F}\left[\Omega_{\delta}\right]\right\\}\subset\mathbb{C}_{\delta}^{*}$,

  * •

_edge_
$\mathcal{E}\left[\Omega_{\delta}\right]:=\left\\{(ij)=(ji):i,j\in\mathcal{V}\left[\Omega_{\delta}\right],\left|i-j\right|=\sqrt{2}\delta\right\\}$,
and

  * •

_corners_
$\mathcal{C}\left[\Omega_{\delta}\right]:=\left\\{(vf):v\in\mathcal{V}\left[\Omega_{\delta}\right],f\in\mathcal{F}\left[\Omega_{\delta}\right],\left|v-f\right|=\delta\right\\}$.

For consistency with the low-temperature expansion of the model, we prefer to
visualise the lattice in its dual form. Note that just as the faces are
represented by their midpoints above, an edge $(ij)$ and a corner $(vf)$ will
be identified with their midpoints $\frac{i+j}{2}$ and $\frac{v+f}{2}$,
respectively. Additionally, we draw a _half-edge_ between either an edge
midpoint or a corner to a nearest vertex.

For $\tau=1,i,\lambda,\bar{\lambda}$, a corner
$c\in\mathcal{C}\left[\Omega_{\delta}\right]$ is in
$\mathcal{C}^{\tau}\left[\Omega_{\delta}\right]$ if the nearest vertex is in
the direction $-\tau^{-2}$. The edges in
$\mathcal{C}^{1}\left[\Omega_{\delta}\right],\mathcal{C}^{i}\left[\Omega_{\delta}\right]$
are respectively called _real_ and _imaginary corners_.

We will frequently denote union of various sets by concatenation, e.g.
$\mathcal{EC}\left[\Omega_{\delta}\right]:=\mathcal{E}\left[\Omega_{\delta}\right]\cup\mathcal{C}\left[\Omega_{\delta}\right]$.

#### Graph Boundary.

  * •

$\overline{\mathcal{E}}\left[\Omega_{\delta}\right]:=\left\\{(ij)=(ji):i\in\mathcal{V}\left[\Omega_{\delta}\right],j\in\mathcal{V}\left[\mathbb{C}_{\delta}\right],|i-j|=\sqrt{2}\delta\right\\}$,
$\partial\mathcal{E}\left[\Omega_{\delta}\right]:=\overline{\mathcal{E}}\left[\Omega_{\delta}\right]\setminus\mathcal{E}\left[\Omega_{\delta}\right]$,

  * •

boundary vertices $\partial\mathcal{V}\left[\Omega_{\delta}\right]$ are the
endpoints of edges in $\partial\mathcal{E}\left[\Omega_{\delta}\right]$ not in
$\mathcal{V}\left[\Omega_{\delta}\right]$,

  * •

boundary faces $\partial\mathcal{F}\left[\Omega_{\delta}\right]$ are faces in
$\mathcal{F}\left[\mathbb{C}_{\delta}\right]\setminus\mathcal{F}\left[\Omega_{\delta}\right]$
which are $\delta$ away from a vertex in
$\mathcal{V}\left[\Omega_{\delta}\right]$, and

  * •

boundary corners
$\partial\mathcal{C}\left[\Omega_{\delta}\right]=\left\\{(vf):v\in\mathcal{V}\left[\Omega_{\delta}\right],f\in\partial\mathcal{F}\left[\Omega_{\delta}\right],\left|v-f\right|=\delta\right\\}$

  * •

$\nu_{\text{out}}(z)$ for
$z\in\partial\mathcal{E}\left[\Omega_{\delta}\right]$ is the unit complex
number corresponding to the orientation of $z$ pointing outwards from
$\Omega$.

#### Double Cover.

The fermions we introduce in forthcoming sections are functions defined on the
double cover of the continuous and discrete domains $\Omega,\Omega_{\delta}$.
Define $\left[\Omega,a_{1},\ldots,a_{n}\right]$ as the double cover of
$\Omega$ ramified at distinct interior points $a_{1},\ldots,a_{n}\in\Omega$;
in particular, it is a Riemann surface where
$\sqrt{\left(z-a_{1}\right)\cdots\left(z-a_{n}\right)}$ is well-defined,
smooth and single-valued. In the case where
$\left\\{\bar{a_{1}},\ldots,\bar{a_{n}}\right\\}=\left\\{a_{1},\ldots,a_{n}\right\\}$,
conjugation on the double cover is defined by requiring that
$\sqrt{\left(\bar{z}-a_{1}\right)\cdots\left(\bar{z}-a_{n}\right)}=\overline{\sqrt{\left(z-a_{1}\right)\cdots\left(z-a_{n}\right)}}$.
On $\left[\mathbb{C},a_{1}\right]$, we will refer to the slit domains
$\mathbb{X}^{+}:=\left\\{\text{Re}\sqrt{z}>0\right\\}$ and
$\mathbb{Y}^{+}:=\left\\{\text{Im}\sqrt{z}>0\right\\}$.

When the choice of the lift of $z\in\Omega$ is clear, we will write $z$ to
denote the lift in $\left[\Omega,a_{1},\ldots,a_{n}\right]$. Conversely, if
$z\in\left[\Omega,a_{1},\ldots,a_{n}\right]$, we will write $z$, or, for
clarification, $\pi(z)\in\Omega\setminus\left\\{a_{1},\ldots,a_{n}\right\\}$
for the projection onto the planar domain;
$z^{\cdot}\in\left[\Omega,a_{1},\ldots,a_{n}\right]$ is the lift of $\pi(z)$
which is not $z$. A function which switches sign under switching $z$ and
$z^{\cdot}$ is called a _spinor_.

We say that two points $z,w\in\left[\Omega,a_{1},\ldots,a_{n}\right]$ are _on
the same sheet_ if we can draw a straight line segment between them; i.e. the
straight line segment on $\Omega$ which connects $\pi(z),\pi(w)$ can be lifted
to connect $z,w$ on $\left[\Omega,a_{1},\ldots,a_{n}\right]$.

For the discrete double cover
$\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$, we will take closest faces
in $\Omega_{\delta}$ to $a_{1},\ldots,a_{n}\in\Omega$, and then lift
components of $\Omega_{\delta}$ minus those $n$ faces. Clearly,
$\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$ is a lattice which is
locally isomorphic to the planar lattice $\Omega_{\delta}$. Given the first
monodromy face $a_{1}$, we will fix a lift of $a_{1}+\frac{\delta}{2}$ and
refer to it throughout this paper.

Figure 1.1. The square lattice. Proposition 22:
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}=O(1)$ on boundary edges
and corners in orange, since
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}=O(\delta)$ on
adjacent vertices in
$\mathcal{V}\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$.

###  1.3. Proof of the Main Theorems

###### Proof of Theorem 1..

Given the fermion convergence of Theorem 24 and identification of Ising
quantities in terms of the fermions of Proposition 5, we can integrate the
discrete logarithmic derivative which converges in the scaling limit (see also
[CHI15, Proposition 2.22, Remark 2.23])

$\displaystyle\frac{1}{2\delta}\left[\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}+2\delta}\sigma_{a_{2}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}-1\right]$
$\displaystyle\to\mathcal{A}_{\Omega}^{1}(a_{1},\ldots,a_{n}),$
$\displaystyle\frac{1}{2\delta}\left[\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}+2i\delta}\sigma_{a_{2}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}-1\right]$
$\displaystyle\to-\mathcal{A}_{\Omega}^{i}(a_{1},\ldots,a_{n}),$

to get that
$\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{b_{1}}\cdots\sigma_{b_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}$
scales to a continuous limit $\frac{\left\langle
b_{1},\ldots,b_{n}\right\rangle_{\Omega,m}^{+}}{\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+}}$ for
$a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}\in\Omega$. The convergence for
$\mathcal{A}_{\Omega}^{i}$ follows by considering the result in a
$-90^{\circ}$ rotated domain (see also [CHI15, Proof of Theorems 1.5 and
1.7]).

Now it remains to uniquely relate the massive convergence rate to the massless
convergence rate:
$\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}\to\frac{\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+}}{\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,0}^{+}}$ for some
$a_{1},\ldots,a_{n}\in\Omega$. Given the convergence of
$\delta^{-\frac{n}{8}}\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]$
to a continuous limit $\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,0}^{+}$, we have convergence of the
massive correlation to unique $\left\langle
a_{1},\ldots,a_{n}\right\rangle_{\Omega,m}^{+}$.

The procedure partially relies on the process in the massless case of relating
the bounded domain correlations to full-plane correlations from [CHI15].

Note that $\beta>\beta_{c}$, and denote the dual temperature by
$\beta^{*}<\beta_{c}$. We always assume a scaling
$\beta_{c}-\beta=\frac{m\delta}{2}$ for $m<0$.

  1. (1)

Relating two point functions:
$\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}$
tends to a continuous limit, since

(1.1)
$1\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta^{*},+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta^{*},\text{free}}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}\rightarrow\left|\mathcal{B}_{\Omega}(a_{1},a_{2}|m)\right|^{-1},$

where we successively used the monotonicity of spin correlation in inverse
temperature (e.g. by coupling with FK-Ising) and in boundary condition (FKG
inequality, [FrVe17, Theorem 3.21]). We also used the convergence of Theorem
24 of the Ising ratio to $\left|\mathcal{B}_{\Omega}(a_{1},a_{2}|m)\right|$.
We conclude by noting that $\left|\mathcal{B}_{\Omega}(a_{1},a_{2}|m)\right|$
can be made arbitrarily close to $1$ (Lemma 29) by merging $a_{1},a_{2}$.

  2. (2)

One point functions: note that by above, we may write

$\displaystyle\frac{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,m}^{+}}{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,0}^{+}}$
$\displaystyle=\exp\left[\int_{a_{2}}^{a_{1}}\left[\mathcal{A}_{\Omega}^{1}(z,a_{2}|m)-\mathcal{A}_{\Omega}^{1}(z,a_{2}|0)\right]dx\right.$
(1.2)
$\displaystyle-\left.\left[\mathcal{A}_{\Omega}^{i}(z,a_{2}|m)-\mathcal{A}_{\Omega}^{i}(z,a_{2}|0)\right]dy\right],$

along any line from $a_{2}$ to $a_{1}$ if the integral converges. Choose
$a_{1}$ and $a_{2}$ as in Lemma 33 (see Figure 1.2), then we can bound the
integral: $\frac{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,m}^{+}}{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,0}^{+}}\leq
e^{cst\cdot\left(\epsilon^{-\gamma}\left|a_{1}-a_{2}\right|+\left|a_{1}-a_{2}\right|^{1-\gamma}\right)}$
for some fixed $\gamma\in(0,1)$. Now choose
$\left|a_{1}-a_{2}\right|=\epsilon^{\kappa}$ for some $\gamma<\kappa<1$. Then
$\frac{\left\langle a_{1},a_{2}\right\rangle_{\Omega,m}^{+}}{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,0}^{+}}\to 1$ as $\epsilon\to 0$, while the
hyperbolic distance between $a_{1}$ and $a_{2}$ grows; indeed, the hyperbolic
distance is comparable to
$\ln\frac{\left|a_{1}-a_{2}\right|}{\text{dist}(\left\\{a_{1},a_{2}\right\\},\partial\Omega)}\propto(1-\kappa)\left|\ln\epsilon\right|$.
Therefore, by [CHI15, (1.3)], $\frac{\left\langle
a_{1},a_{2}\right\rangle_{\Omega,0}^{+}}{\left\langle
a_{1}\right\rangle_{\Omega,0}^{+}\left\langle
a_{2}\right\rangle_{\Omega,0}^{+}}\to 1$. Now we may relate the massive one-
point functions to the massless ones, so that the former are uniquely
identified in terms of the latter. By the monotonicity of spin correlation,
$\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\right]\leq\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]$,
and by GKS inequality [FrVe17, Theorem 3.20],
$\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{2}}\right]\leq\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]$,
so

(1.3)
$1\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\right]}\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\right]\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{2}}\right]}=\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\right]\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{2}}\right]}\cdot\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}.$

By the above discussion, both factors on the right is made arbitrarily close
to $1$ by taking small enough $\epsilon$. Moreover, such choice of $\epsilon$
is uniform in the smoothness of the domain (concretely, the bound on the
derivative of a conformal map to the disc).

  3. (3)

More points: we again use the GKS inequality, in that

(1.4)
$1\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]\cdots\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{n}}\right]}\leq\frac{\mathbb{E}_{\Omega_{\delta}^{1}}^{\beta,+}\left[\sigma_{a_{1}}\right]\cdots\mathbb{E}_{\Omega_{\delta}^{n}}^{\beta,+}\left[\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]\cdots\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{n}}\right]},$

where $\Omega^{1}\ni a_{1},\ldots,\Omega^{n}\ni a_{n}$ are choices of disjoint
smooth simply connected subdomains of $\Omega$, each sharing a macroscopic
boundary arc with $\Omega$. By FKG inequality, requiring the spins in
$\Omega\setminus\bigcup_{j=1}^{n}\Omega^{j}$ to be plus raises the
correlation, which gives the second inequality above. By the uniform
identification of the one point functions, there is $\epsilon>0$ such that if
each of $a_{1},\ldots,a_{n}$ are $\epsilon$-close to the boundary arc that
$\Omega^{1},\ldots,\Omega^{n}$ shares with $\Omega$, both
$\mathbb{E}_{\Omega_{\delta}^{j}}^{\beta,+}\left[\sigma_{a_{j}}\right],\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{j}}\right]$
become close to their massless counterparts. Since the massless correlations
are Caratheodory stable ([CHI15, (1.3)]), as $\epsilon$ decreases,
$\mathbb{E}_{\Omega_{\delta}^{j}}^{\beta_{c},+}\left[\sigma_{a_{j}}\right]/\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{j}}\right]$
become close to $1$. Clearly, any ratio between
$\mathbb{E}_{\Omega_{\delta}^{j}}^{\beta,+}\left[\sigma_{a_{j}}\right],\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{j}}\right],\mathbb{E}_{\Omega_{\delta}^{j}}^{\beta_{c},+}\left[\sigma_{a_{j}}\right],\mathbb{E}_{\Omega_{\delta}}^{\beta_{c},+}\left[\sigma_{a_{j}}\right]$
may be made arbitrarily close to $1$ by setting $\epsilon$ small enough, and
thus
$\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\right]\cdots\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{n}}\right]}$
as well; this fixes the normalisation of an arbitrary $n$-point function.

∎

Figure 1.2. Setup for applying Lemma 33 to have decorrelation between
$a_{1},a_{2}$.

###### Proof of Corollary 2..

Note that the argument for the scaling limit of the two-point functions in the
proof of Theorem 1 apply in the full plane as well. It also fixes the
normalisation as $a\to 0$ by [CHI15, Remark 2.26]. From Theorem 24, we need to
integrate $\mathcal{A}_{\mathbb{C}}^{1}(-a,a)$. Recall $r=am$, and define
$\left\langle
r\right\rangle_{\mathbb{C}}^{+}:=\left\langle-a,a\right\rangle_{\mathbb{C},m}^{+}$.

By (4.22),
$-m^{-1}\left.\partial_{a}\ln\left\langle-a,b\right\rangle_{\mathbb{C},m}^{+}\right|_{b=a}=\left.-\frac{1}{2}\left(\ln\cosh
h_{0}(r)\right)^{\prime}-r\left[\frac{1}{2}\left(h_{0}^{\prime}(r)\right)^{2}-2\sinh^{2}2h_{0}(r)\right]\right|_{r=am}$,
which can be rephrased as

$\partial_{r}\ln\left\langle
r\right\rangle_{\mathbb{C}}^{+}=m^{-1}\partial_{a}\left\langle-a,a\right\rangle_{\mathbb{C},m}^{+}=\left(\ln\cosh
h_{0}(r)\right)^{\prime}+r\left[\left(h_{0}^{\prime}(r)\right)^{2}-4\sinh^{2}2h_{0}(r)\right].$

Then

$\left\langle-a,a\right\rangle_{\mathbb{C},m}^{+}=\left\langle
r\right\rangle_{\mathbb{C}}^{+}=cst\cdot\cosh
h_{0}(am)\cdot\exp\left[\int_{-\infty}^{am}r\left[\left(h_{0}^{\prime}(r)\right)^{2}-4\sinh^{2}2h_{0}(r)\right]dr\right].$

Then the definition $\tanh h_{0}:=\mathcal{B}_{0}$ gives the other case. ∎

###  1.4. Structure of the Paper

This paper contains four sections and one appendix to which technical
calculations and estimates are deferred. Section 2 defines our main analytical
tool, the discrete fermions. The combinatorial definition in 2.1 involves
contours on the discrete bounded domain and is seen to naturally encode the
logarithmic derivative of the spin correlation. Its discrete complex analytic
properties are then established, which are exploited in Section 2.2 to give a
definition on the full plane by an infinite volume limit.

Since analysis of the continuous fermions is needed for the scaling limit
process (for a unique characterisation of the continuous limit), we carry out
the continuum analysis first in Section 2.3. We formulate the boundary value
problem on $\Omega$ for our continuous fermions, which will be a massive
perturbation of holomorphic functions treated in [Ber56, Vek62]. We verify
various properties we will use: such as the expansion in terms of formal
’powers’. Analysis in the continuum is continued in Section 4.1.

Convergence of the discrete fermion under scaling limit is done in Section 3.
The analysis is divided into two parts: bulk convergence (Section 3.1), where
the discrete fermion evaluated on compact subsets of
$\left[\Omega,a_{1},\ldots,a_{n}\right]$ is shown to uniformly converge to the
continuous fermion, and analysis near the singularity (Section 3.2), where the
discrete fermion evaluated at a point in
$\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$ microscopically away from a
monodromy face is identified from the coefficients of a massive analytic
version of power series expansion of the continuous fermion. Bulk convergence
is done in a standard manner, by first showing that the set of discrete
fermion correlations are precompact and then uniquely identifying the limit.
Analysis near the singularity mainly uses ideas from [CHI15], where the
continuous power series expansion is modelled in the discrete setting then the
coefficients carefully matched.

In Section 4, we collect the analysis of the massive fermions necessary for
the integration of the logarithmic derivatives and isomonodromic analysis in
Section 4.1, and finally in Section 4.2 we carry out the isomonodromic
analysis and obtain the Painlevé III transcendent, which can be identified in
the logarithmic derivative of spin correlations in $\mathbb{C}$ given the
convergence results.

### Acknowledgements

The author would like to thank Clément Hongler for guidance, Konstantin
Izyurov, Kalle Kytölä, Julien Dubedat, Béatrice de Tilière, and Francesco
Spadaro for helpful discussions, and the ERC under grant SG CONSTAMIS for
financial support.

##  2\. Massive Fermions

In this section, we introduce the main discrete tool of our analysis, the
massive discrete fermion correlations. While the object and the terminology
hearkens back to the physical analysis of the Ising model, we shall give an
explicit definition in Section 2.1 as a complex function which extrapolates
the desired local physical quantity to the entire domain. In the same
subsection we also show that, as a discrete function, it exhibits a notion in
discrete complex analysis called (massive) s-holomorphicity. At first we only
define the fermion in bounded discretised domains, i.e. finite sets; however
we will define them in the complex plane in Section 2.2. Then, we carry out
analysis of the continuous spinors, to which the discrete spinors presented in
the previous section is shown to converge in Section 3. Since the proof of
scaling limit requires unique identification of the continuous limit, we first
give necessary analytic background in Section 2.3.

###  2.1. Bounded Domain Fermions and Discrete Analysis

We introduce here the main object of our analysis, the discrete fermion $F$.
Note that this function is essentially the same object as in [CHI15,
Definition 2.1] albeit at general $\beta$, and we try to keep the same
normalisation and notation where appropriate. The contents of this subsection
are valid for any $\beta>0$.

In order to use the low-temperature expansion of the Ising model, we first
define $\Gamma_{\Omega_{\delta}}\subset
2^{\mathcal{E}\left[\Omega_{\delta}\right]}$ as the collection of closed
contours, i.e. set $\omega$ of edges in $\Omega_{\delta}$ such that an even
number of edges in $\omega$ meet at any given vertex. Given $+$ boundary
condition, any $\omega$ is clearly in one-to-one correspondence with a spin
configuration $\sigma$ (where $\omega$ delineates clusters of identical
spins), and we can compute the partition function of the model and the
correlation

$\displaystyle\mathcal{Z}_{\Omega_{\delta}}^{+,\beta}$
$\displaystyle:=\sum_{\omega\in\mathcal{C}_{\Omega_{\delta}}}e^{-2\beta|\omega|}$
$\displaystyle\mathbb{E}_{\Omega_{\delta}}^{+,\beta}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]$
$\displaystyle=\left|\mathcal{Z}_{\Omega_{\delta}}^{+,\beta}\right|^{-1}\sum_{\omega\in\Gamma_{\Omega_{\delta}}}e^{-2\beta|\omega|}(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\omega\right)},$

where $\\#\text{loops}_{a_{1},\ldots,a_{n}}$ denotes the parity of loops in
$\omega$ which separate the boundary spins from an odd number of
$a_{1},\ldots,a_{n}$. The unnormalised correlation
$\mathcal{Z}_{\Omega_{\delta}}^{+,\beta}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]:=\mathcal{Z}_{\Omega_{\delta}}^{+,\beta}\cdot\mathbb{E}_{\Omega_{\delta}}^{+,\beta}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]$
will be used for normalisation below.

######  Definition 3.

For a bounded simply connected domain $\Omega\subset\mathbb{C}$ with $n$
distinct interior points $a_{1},\ldots a_{n}$ and inverse temperature
$\beta>0$, define for
$z\in\mathcal{E}\mathcal{C}\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$
which is not a lift of $a_{1}+\frac{\delta}{2}$ the _discrete massive fermion
correlation_ , or simply the _discrete fermion_

$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)=\frac{1}{\mathcal{Z}_{\Omega_{\delta}}^{+,\beta}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]}\sum_{\gamma\in\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},z)}c_{z}e^{-2\beta\left|\gamma\right|}\cdot\phi_{a_{1},\ldots,a_{n}}\left(\gamma,z\right)$

where

  * •

$\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},z)$ is the collection of
$\gamma=\omega\oplus\gamma_{0}$, where $\omega$ runs over elements over
$\Gamma_{\Omega_{\delta}}$, $\gamma_{0}$ is a fixed simple lattice path from
$a_{1}+\frac{\delta}{2}$ to
$\pi(z)\in\mathcal{E}\mathcal{C}\left[\Omega_{\delta}\right]$, and $\oplus$
refers to the XOR (symmetric difference) operation. $|\gamma|$ is the number
of full edges in $\gamma$.
$c_{z}:=\cos\left(\frac{\pi}{8}+\Theta(\beta)\right)^{-1}$ if $z$ is an edge,
and $1$ if it is a corner. Note that none of these definitions refer to the
double cover.

  * •

$\phi_{a_{1},\ldots,a_{n}}\left(\gamma,z\right)$ is a pure phase factor,
independent of $\beta$, defined by

$\phi_{a_{1},\ldots,a_{n}}\left(\gamma,z\right)=e^{-\frac{i}{2}\text{wind}(\text{p}(\gamma))}(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\setminus\text{p}(\gamma)\right)}\text{sheet}_{a_{1},\ldots,a_{n}}(\text{p}(\gamma)),$

where $\text{p}(\gamma)$ is a simple path (we allow for self-touching, as long
as there is no self-crossing) from $a_{1}+\frac{\delta}{2}$ to $\pi(z)$ chosen
in $\gamma$, $\text{wind}(\text{p}(\gamma))$ is the total turning angle of the
tangent of $\text{p}(\gamma)$, and
$\text{sheet}_{a_{1},\ldots,a_{n}}(\text{p}(\gamma))\in\left\\{\pm 1\right\\}$
is defined to be $+1$ if the lift of $\text{p}(\gamma)$ to the double cover
starting from the fixed lift of $a_{1}+\frac{\delta}{2}$ (fixed once forever
in Section 1.2) ends at $z$ and $-1$ if it ends at $z^{\cdot}$.
$\phi_{a_{1},\ldots,a_{n}}$ is well-defined; see e.g. [ChIz13, CHI15].

Note that $F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is naturally a
spinor, i.e.
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z^{\cdot}|\beta\right)=-F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)$.

The massive fermion satisfies a perturbed notion of discrete holomorphicity,
called _massive s-holomorphicity_. First, define the projection operator
$\text{Proj}_{e^{i\theta}\mathbb{R}}x:=\frac{x+e^{2i\theta}\bar{x}}{2}$ to be
the projection of the complex number $x$ to the line $e^{i\theta}\mathbb{R}$.

######  Proposition 4.

The discrete massive fermion
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(\cdot|\beta\right)$
is _massive s-holomorphic_ , i.e. it satisfies

(2.1) $e^{\mp i\Theta}\text{Proj}_{e^{\pm
i\Theta}\tau(c)\mathbb{R}}F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c\mp\frac{\tau(c)^{-2}i\delta}{2}|\beta\right)=F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c|\beta\right),$

between
$c\mp\frac{\tau(c)^{-2}i\delta}{2}\in\mathcal{E}\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$
and $c\in\mathcal{C}\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$ which is
not a lift of $a_{1}+\frac{\delta}{2}$. At (the fixed lift of)
$a_{1}+\frac{\delta}{2}$, we have instead

(2.2) $e^{\mp i\Theta}\text{Proj}_{e^{\pm
i\Theta}i\mathbb{R}}F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{\delta\pm\delta
i}{2}|\beta\right)=\mp i.$

###### Proof.

The proof of massive s-holomorphicity, essentially identical to the massless
case [CHI15, Subsection 3.1], uses the bijection between
$\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},c\mp\frac{\tau(c)^{-2}i\delta}{2})$
and $\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},c)$ by the $\oplus$
(symmetric difference) operation. Without loss of generality, notice that
$\gamma\mapsto\gamma\oplus t$ with the fixed path
$t:=\left\\{\left(c-\frac{\tau(c)^{-2}i\delta}{2},c-\frac{\tau(c)^{-2}\delta}{2}\right)\left(c-\frac{\tau(c)^{-2}\delta}{2},c\right)\right\\}$
maps
$\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},c-\frac{\tau(c)^{-2}i\delta}{2})$
to $\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},c)$ bijectively (see
Figure 1.1).

We now only need to show that the summand in the definition of
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c\mp\frac{\tau(c)^{2}i\delta}{2}|\beta\right)$
for a given $\gamma$ transforms to the summand in
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c|\beta\right)$ for
$\gamma\oplus t$. Given $\text{p}(\gamma)\subset\gamma$, we may take
$\text{p}(\gamma\oplus t):=\text{p}(\gamma)\oplus t$. Clearly
$\left(-1\right){}^{\\#\text{loops}},\text{sheet}$ remain the same for
$\gamma$ and $\gamma\oplus t$, while

$\text{wind}\left(\text{p}(\gamma\oplus
t)\right)=\begin{cases}\text{wind}\left(\text{p}(\gamma)\right)+\frac{\pi}{4}&\text{if
}\gamma\cap t\neq\emptyset\\\
\text{wind}\left(\text{p}(\gamma)\right)-\frac{3\pi}{4}&\text{if }\gamma\cap
t=\emptyset\end{cases}\in\tau(c)\mathbb{R}.$

In the case where $\gamma\cap t\neq\emptyset$, $\left|\gamma\oplus
t\right|=\left|\gamma\right|$ and

$\displaystyle
e^{-i\Theta}\text{Proj}_{e^{i\Theta}\tau(c)\mathbb{R}}e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\right)\right)}$
$\displaystyle=e^{-i\Theta}\text{Proj}_{e^{i\Theta}\tau(c)\mathbb{R}}\left[e^{i\Theta}e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}\right]e^{-\frac{\pi i}{8}-i\Theta}$
$\displaystyle=e^{-i\Theta}\left[e^{i\Theta}e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}\right]\text{Re}e^{-\frac{\pi i}{8}-i\Theta}$
$\displaystyle=e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}\cos\left(\frac{\pi}{8}+\Theta\right),$

while if $\gamma\cap t=\emptyset$, $\left|\gamma\oplus
t\right|=\left|\gamma\right|+1$ and similarly

$\displaystyle
e^{-i\Theta}\text{Proj}_{e^{i\Theta}\tau(c)\mathbb{R}}e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\right)\right)}$
$\displaystyle=e^{-i\Theta}\left[e^{i\Theta}e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}\right]\text{Re}e^{\frac{3\pi i}{8}-i\Theta}$
$\displaystyle=e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}\sin\left(\frac{\pi}{8}+\Theta\right)$
$\displaystyle=e^{-\frac{i}{2}\text{wind}\left(\text{p}\left(\gamma\oplus
t\right)\right)}e^{2\beta}\cos\left(\frac{\pi}{8}+\Theta\right),$

and the result follows.

At $a_{1}+\frac{\delta}{2}$, (say)
$\gamma\in\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},a_{1}+\frac{\delta}{2}+\frac{i\delta}{2})$
is mapped bijectively to $\gamma\oplus t\in\Gamma_{\Omega_{\delta}}$. It is
easy to see that

$\text{wind}\left(\text{p}(\gamma)\right)=\begin{cases}-\frac{5\pi}{4}\pm
2\pi&\text{if }t\subset\gamma,\\\ -\frac{\pi}{4}\pm 2\pi&\text{if }\gamma\cap
t=\emptyset\end{cases},$

and
$(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\setminus\text{p}(\gamma)\right)}\text{sheet}_{a_{1},\ldots,a_{n}}(\text{p}(\gamma))=\left(-1\right)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\oplus
t\right)}$. Now a simple computation similar to above yields the result. ∎

Figure 2.1. The proof of massive s-holomorphicity by bijection.

We will take (2.1) as the definition of _( $M,\beta,\Theta$)-massive
s-holomorphicity_ (or just _s-holomorphicity_ when the mass is clear) _at_ $c$
(or _between_ $c\mp\frac{\tau(c)^{-2}i\delta}{2}$). We will see later that
massive s-holomorphicity corresponds to the continuous notion of perturbed
analyticity $\partial_{z}f=m\bar{f}$.

The fermion defined above is a deterministic function without explicit
connection to the Ising model–we now record how it encodes probabilistic
information.

######  Proposition 5.

For $v=-1,0,1$, we have

(2.3) $\displaystyle
F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\delta+\frac{e^{iv\frac{\pi}{2}}}{2}\delta|\beta\right)$
$\displaystyle=e^{-iv\frac{\pi}{4}}\frac{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}+\delta+e^{iv\frac{\pi}{2}}\delta}\sigma_{a_{2}}\cdots\sigma_{a_{n}}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\cdots\sigma_{a_{n}}\right]},$

where $a_{1}+\delta+\frac{e^{iv\frac{\pi}{2}}}{2}\delta$ is lifted to the same
sheet as the fixed lift of $a_{1}+\frac{\delta}{2}$, and thus for any
$\beta_{1}<\beta_{2}$ there exist constants
$C_{1}(\beta_{1},\beta_{2}),C_{2}(\beta_{1},\beta_{2})>0$, uniform in
$\beta\in\left[\beta_{1},\beta_{2}\right],\text{$\Omega$},\delta$, such that

$C_{1}<\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\delta+\frac{e^{iv\frac{\pi}{2}}}{2}\delta|\beta\right)\right|<C_{2}.$

In the case $n=2$ we have also

(2.4)
$\left|F_{\left[\Omega_{\delta},a_{1},a_{2}\right]}\left(a_{2}+\frac{\delta}{2}|\beta\right)\right|=\frac{\mathbb{E}_{\Omega_{\delta}^{*}}^{\beta^{*},\text{free}}\left[\sigma_{a_{1}+\delta}\sigma_{a_{2}+\delta}\right]}{\mathbb{E}_{\Omega_{\delta}}^{\beta,+}\left[\sigma_{a_{1}}\sigma_{a_{2}}\right]}.$

###### Proof.

(2.3) and (2.4) were proved for the massless case in [CHI15, Lemma 2.6] and
the proof is easily seen to not depend on a specific value of $\beta$.

We carry out the $v=1$ case here: the path
$t:=\left\\{(a_{1}+\frac{\delta}{2},a_{1}+\delta),(a_{1}+\delta,a_{1}+\delta+\frac{i\delta}{2})\right\\}$
is admissible (i.e. without a crossing) as $\text{p}(\gamma)$ with
$\text{wind}(\text{p($\gamma$)})=\frac{\pi}{2},\text{sheet}_{a_{1},\ldots,a_{n}}(\text{p}(\gamma))=1$
in
$\gamma\in\Gamma_{\Omega_{\delta}}(a_{1}+\frac{\delta}{2},a_{1}+\delta+\frac{i\delta}{2})$
if and only if $\gamma$ does not contain a loop separating $a_{1}$ from
$a_{1}+\delta+i\delta$ and
$(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\setminus
t\right)}=(-1)^{\\#\text{loops}_{a_{1}+\delta+i\delta,\ldots,a_{n}}\left(\gamma\setminus
t\right)}$. If $\gamma$ does contain such a loop $L$, we need to choose
$\text{p}(\gamma)=t\cup L$ with
$\text{wind}\left(\text{p}(\gamma)\right)=\frac{\pi}{2}\pm 2\pi$. In this
case, $\text{sheet}_{a_{1},\ldots,a_{n}}\left(\text{p}(\gamma)\right)=1$ if
and only if $L$ encloses an even number of $a_{i}$. This means that

$\displaystyle(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\setminus\text{p}(\gamma)\right)}\text{sheet}_{a_{1},\ldots,a_{n}}(\text{p}(\gamma))$
$\displaystyle=(-1)^{\\#\text{loops}_{a_{1},\ldots,a_{n}}\left(\gamma\setminus
t\right)}$
$\displaystyle=-(-1)^{\\#\text{loops}_{a_{1}+\delta+i\delta,\ldots,a_{n}}\left(\gamma\setminus
t\right)},$

so in all cases the summand in the definition of
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is
$\lambda^{-1}(-1)^{\\#\text{loops}_{a_{1}+\delta+i\delta,\ldots,a_{n}}\left(\gamma\setminus
t\right)}$ and the result follows. The uniform bound comes from the finite
energy property of the Ising model (such expectations are uniformly bounded
from $0$ and $\infty$ in any finite distribution). ∎

A crucial feature of (discrete) massive s-holomorphicity, shared with its
continuous counterpart, is that the line integral
$\text{Re}\int\left[F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)\right]^{2}dz$
can be defined path-independently.

######  Proposition 6.

There is a single valued function
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(x|\beta\right)$ up
to a global constant on $\mathcal{VF}\left[\Omega_{\delta}\right]$ constructed
by

(2.5) $\displaystyle
H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f|\beta\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v|\beta\right)$
$\displaystyle:=2\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(\frac{v+f}{2}|\beta\right)\right|^{2},$

where
$f\in\mathcal{F}\left[\Omega_{\delta}\right]\cup\partial\mathcal{F}\left[\Omega_{\delta}\right]$
and $v\in\mathcal{V}\left[\Omega_{\delta}\right]$ are $\delta$ away from each
other so that $\frac{v+f}{2}$ is the corner between them. Put
$\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{\delta}{2}|\beta\right)\right|^{2}=1$.

At the boundary faces $H^{\circ}$ is constant, so we may put

(2.6) $\displaystyle
H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f|\beta\right):=0$
$\displaystyle\text{ if }f\in\partial\mathcal{F}\left[\Omega_{\delta}\right],$

and further define
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v|\beta\right):=0\text{
if }v\in\partial\mathcal{V}\left[\Omega_{\delta}\right]$, then across a
boundary edge $e=(a_{\text{int}}a)$ with
$a_{\text{int}}\in\mathcal{V}\left[\Omega_{\delta}\right],a\in\partial\mathcal{V}\left[\Omega_{\delta}\right]$,
we have

$\displaystyle\partial_{\nu_{\text{out}}}^{\delta}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(e|\beta\right)$
$\displaystyle:=-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(a_{\text{int}}|\beta\right)$
(2.7)
$\displaystyle=2\cos^{2}\left(\frac{\pi}{8}+\Theta\right)\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e|\beta\right)\right|^{2}$
$\displaystyle\geq 0.$

###### Proof.

(2.5) gives rise to a single valued function because doing a loop around any
edge will give zero: if $e=(v_{1}v_{2})$ is an edge which is incident to faces
$f_{1},f_{2}$,
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{1}\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{1}\right)+H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{2}\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{2}\right)$
according to (2.5) is simply equal to
$2\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(\frac{f_{1}+v_{1}}{2}\right)\right|^{2}+2\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(\frac{f_{2}+v_{2}}{2}\right)\right|^{2}=2\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e\right)\right|^{2}$,
since the two corner values at $\frac{f_{1,2}+v_{1,2}}{2}$ are simply
projections of
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e\right)$ onto
orthogonal lines by s-holomorphicity.
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{2}\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{1}\right)+H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{1}\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{2}\right)$
is also equal to
$2\delta\left|F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e\right)\right|^{2}$,
and increments along the loop $f_{1}\sim v_{1}\sim f_{2}\sim v_{2}\sim f_{1}$
sum to zero.

Then the boundary behaviour can be easily verified noting that
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e|\beta\right)\in\nu_{\text{out
}}(e)^{-1/2}\mathbb{R}$ if $e$ is a boundary edge. See [CHI15, Proposition
3.6] for a massless counterpart. ∎

######  Remark 7.

Writing corner values in terms of edge projections, we obtain that
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is a discrete version of
the integral
$\text{Re}\int\left[F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)\right]^{2}dz$
in that

(2.8)
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{1}|\beta\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}\left(f_{2}|\beta\right)=\sqrt{2}\sin\left(\frac{\pi}{4}+2\Theta\right)\text{Re}\left[F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)^{2}\left(f_{1}-f_{2}\right)\right],$

for all $f_{1},f_{2}$ of distance $\sqrt{2}\delta$ from each other, and

(2.9)
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{1}|\beta\right)-H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(v_{2}|\beta\right)=\sqrt{2}\sin\left(\frac{\pi}{4}-2\Theta\right)\text{Re}\left[F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(z|\beta\right)^{2}\left(v_{1}-v_{2}\right)\right],$

for $v_{1},v_{2}$ of distance $\sqrt{2}\delta$ from each other.

The functions $H^{\circ,\bullet}$ constructed in the previous proposition
satisfy a discrete version of a second-order partial differential equation. We
first recall the standard discrete operators
$\partial_{\bar{z}}^{\delta},\Delta^{\delta}$; as alluded to in the notation
section, we make a small modification to the conventional definition for the
laplacian in $\mathcal{V}\left[\Omega_{\delta}\right]$. We make this boundary
modification specifically for $\mathcal{V}\left[\Omega_{\delta}\right]$, which
lets us define
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}=0$ on boundary
vertices; see [ChSm12] for a motivation.

######  Definition 8.

Suppose $A$ is a function defined on $\mathcal{E}\left[\Omega_{\delta}\right]$
and $B$ on $\mathcal{V}\left[\Omega_{\delta}\right]$ (or
$\mathcal{F}\left[\Omega_{\delta}\right]$, or any locally isomorphic graph).
We define the discrete Wirtinger derivative and laplacian by

$\displaystyle\partial_{\bar{z}}^{\delta}A(x)$
$\displaystyle:=\sum_{m=0}^{3}i^{m}e^{i\pi/4}A\left(x+i^{m}e^{i\pi/4}\frac{\delta}{\sqrt{2}}\right)\text{
if, e.g., }x\in\mathcal{VF}\left[\Omega_{\delta}\right],$ (2.10)
$\displaystyle\Delta^{\delta}B(x)$
$\displaystyle:=\sum_{m=0}^{3}c_{m}\left[B\left(x+i^{m}e^{i\pi/4}\sqrt{2}\delta\right)-B(x)\right]\text{
if, e.g., }x\in\mathcal{V}\left[\Omega_{\delta}\right],$

where $c_{m}=1$ if
$x+i^{m}e^{i\pi/4}\sqrt{2}\delta\in\mathcal{V}\left[\Omega_{\delta}\right]$,
and
$c_{m}=\frac{\sin\left(\frac{\pi}{4}-2\Theta\right)}{\cos^{2}\left(\frac{\pi}{8}+\Theta\right)}$
if
$x+i^{m}e^{i\pi/4}\sqrt{2}\delta\in\partial\mathcal{V}\left[\Omega_{\delta}\right]$.
For any other lattice, $c_{m}\equiv 1$. If $\Delta^{\delta}B(x)=M_{H}^{2}B(x)$
for $M_{H}^{2}=\frac{8\sin^{2}2\Theta}{\cos 4\Theta}$, we call $B$ _(
$\Theta$,$M_{H}^{2}$)-massive harmonic_ at $x$.

The spinor $F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is
$\Theta$-massive harmonic [BeDC12, DGP14, HKZ15], at least away from
monodromy; see Proposition 35, and also the proof of Proposition 13 for the
behaviour near monodromy.

Note that in the scaling limit $\delta\downarrow 0$,
$\frac{1}{2\sqrt{2}\delta}\partial_{\bar{z}}^{\delta}\to\partial_{\bar{z}}:=\frac{1}{2}\left(\partial_{x}+i\partial_{y}\right)$
and $\frac{1}{\left(\sqrt{2}\delta\right)^{2}}\Delta^{\delta}\to\Delta$.
Therefore, in the following result, the second terms become negligible as
$\delta\to 0$ compared to the first, at least if $F$ is in some sense regular;
for an analogue in the scaling limit, see (3.8).

######  Proposition 9.

For $x\neq a_{1}+\delta,a_{2},\ldots a_{n},$ we have

$\displaystyle\Delta^{\delta}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}(x)$
$\displaystyle=2\sin\left(\frac{\pi}{4}+2\Theta\right)\delta\cdot$ (2.11)
$\displaystyle\left[A_{\Theta}\sum_{n=0}^{3}\left|F\left(x+i^{n}e^{i\pi/4}\frac{\delta}{\sqrt{2}}\right)\right|^{2}+B_{\Theta}\left|\partial_{\bar{z}}\bar{F}\right|^{2}(x)\right]$
$\displaystyle\Delta^{\delta}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(x)$
$\displaystyle=2\sin\left(\frac{\pi}{4}-2\Theta\right)\delta\cdot$
$\displaystyle\left[-A_{-\Theta}\sum_{n=0}^{3}\left|F\left(x+i^{n}e^{i\pi/4}\frac{\delta}{\sqrt{2}}\right)\right|^{2}-B_{-\Theta}\left|\partial_{\bar{z}}\bar{F}\right|^{2}(x)\right]$

where $A_{\Theta},B_{\Theta}$ are explicit numbers which depend on $\Theta$.
$A$ is odd and $B$ is even in $\Theta$, $A_{\Theta}\sim 4\sqrt{2}\Theta$ and
$B_{\Theta}\sim\frac{1}{2\sqrt{2}}$ as $\Theta\to 0$ (see (A.3)).

###### Proof.

We calculate
$\partial_{\bar{z}}F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{2}$ in
Proposition 35. The laplacians follow straightforwardly by noting that
$\Delta^{\delta}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ,\bullet}(x)=\text{Re}\left[2\sin\left(\frac{\pi}{4}\pm\Theta\right)\delta\partial_{\bar{z}}F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{2}\right]$,
which is true if $x\in\mathcal{V}\left[\Omega_{\delta}\right]$ is adjacent to
$\partial\mathcal{V}\left[\Omega_{\delta}\right]$ as well (the boundary
conductances are defined precisely to preserve this relation). See also Remark
10. ∎

######  Remark 10.

Note that Proposition 9 applies for $x=a_{1}$ as well, since, thanks to the
singularity (2.2), projections of
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+e^{i\frac{\pi}{4}}\cdot\frac{\delta}{\sqrt{2}}\right),F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+e^{i\frac{3\pi}{2}}\cdot
e^{i\frac{\pi}{4}}\cdot\frac{\delta}{\sqrt{2}}\right)$ (on the sheet which is
cut along $\mathbb{R}_{\geq 0}$ and has the fixed lift of
$a_{1}+\frac{\delta}{2}$ on the upper side of $\mathbb{R}_{\geq 0}$) onto the
line $i\mathbb{R}$ are equal to $-i$; on this sheet,
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ does show
s-holomorphicity between $4$ edges surrounding $a_{1}$. The singularity
effectively transfers the monodromy at face $a_{1}$ to the vertex
$a_{1}+\delta$.

###  2.2. Full-Plane Fermions

We now define the fermion
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ for $\Omega=\mathbb{C}$
by taking increasingly bigger balls $B_{R}=B_{R}(0)$, allowing us to extract
informations about the Ising measure on the corresponding discretised domains.

######  Lemma 11.

Fix $\delta>0$ and $\Theta<0$. There exists a constant $C=C(\delta,\Theta)$
such that
$\left|F_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}(c)\right|\leq
C$ for any
$c\in\mathcal{C}\left(\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]\right)$.

###### Proof.

Discrete Green’s formula implies that

(2.12)
$\sum_{v\in\mathcal{V}\left[\left(B_{R}\right)_{\delta}\right]}\Delta^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(v)=\frac{\sin\left(\frac{\pi}{4}-2\Theta\right)}{\cos^{2}\left(\frac{\pi}{8}+\Theta\right)}\sum_{e\in\partial\mathcal{E}\left[\left(B_{R}\right)_{\delta}\right]}\partial_{\nu_{\text{out}}}^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\left(e\right),$

and we can use (2.11) on vertices other than $a_{1}+\delta$ for the laplacian
and (2.7) for the boundary outer derivatives, so leaving just the laplacian at
$(a_{1}+\delta)$ on the left hand side

$\displaystyle\Delta^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{1}+\delta)=$
(2.15)
$\displaystyle\sum_{\begin{array}[]{c}v\in\mathcal{V}\left[\left(B_{R}\right)_{\delta}\right]\\\
v\neq a_{1}+\delta\end{array}}$ $\displaystyle
2\sin\left(\frac{\pi}{4}-\Theta\right)\delta\cdot\left[A_{-\Theta}\sum_{n=0}^{3}\left|F\left(v+i^{n}e^{i\pi/4}\frac{\delta}{\sqrt{2}}\right)\right|^{2}+B_{-\Theta}\left|\partial_{\bar{z}}\bar{F}\right|^{2}(v)\right]$
$\displaystyle+\sum_{e\in\partial\mathcal{E}\left[\left(B_{R}\right)_{\delta}\right]}2\delta\sin\left(\frac{\pi}{4}-2\Theta\right)\left|F_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}\left(e\right)\right|^{2}$
(2.18) $\displaystyle\geq
2\sin\left(\frac{\pi}{4}-\Theta\right)A_{-\Theta}\delta\sum_{\begin{array}[]{c}c\in\mathcal{C}\left[\left(B_{R}\right)_{\delta}\right]\\\
c\neq
a_{1}+\delta+\frac{i^{m}\delta}{2}\end{array}}\left|F\left(c\right)\right|^{2}$

Note that $A_{-\Theta},B_{-\Theta}>0$, so we have the desired bound if
$\Delta^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{1}+\delta)$
is bounded. But by s-holomorphicity, clearly

(2.19)
$\Delta^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{1}+\delta)\leq
16\sqrt{2}\max_{0\leq m\leq
3}\delta\left|F_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\delta+\frac{e^{iv\frac{\pi}{2}}}{2}\delta\right)\right|^{2},$

but the fermion value on the right hand side is bounded by a constant only
depending on $\beta$ by Proposition 5. ∎

Given uniform boundedness, we can use diagonalisation to get a subsequential
limit $F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$ on the whole
of $\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]$. It suffices to show
that such a limit must be unique.

######  Proposition 12.

Any subsequential limit
$F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$ of
$F_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}$ as
$R\to\infty$

  1. (1)

shows massive s-holomorphicity and the singularity at $a_{1}+\frac{\delta}{2}$
as in Proposition 4,

  2. (2)

at infinity: $F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}\to 0$
uniformly and its square integral
$H_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$ tends to a finite
constant,

  3. (3)

has finite discrete ’$L^{2}$ norm’:

(2.22)
$\displaystyle\sum_{\begin{array}[]{c}v\in\mathcal{V}\left[\mathbb{C}_{\delta}\right]\\\
v\neq a_{1}+\delta\end{array}}2$
$\displaystyle\sin\left(\frac{\pi}{4}-\Theta\right)\delta\cdot\left[A_{-\Theta}\sum_{n=0}^{3}\left|F\left(v+i^{n}e^{i\pi/4}\frac{\delta}{\sqrt{2}}\right)\right|^{2}\right]$
(2.25) $\displaystyle\leq$
$\displaystyle\sum_{\begin{array}[]{c}v\in\mathcal{V}\left[\mathbb{C}_{\delta}\right]\\\
v\neq
a_{1}+\delta\end{array}}\left|\Delta^{\delta}H_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(v)\right|<cst\cdot\delta<\infty,$

with constant independent of $\delta$.

Moreover, there is only one such function
$F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$.

###### Proof.

The first entry is immediate from the corresponding properties of
$F_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}$.

The inequality (2.22) can be deduced from a uniform bound for the analogue for
$\left|\Delta^{\delta}H_{\left[\left(B_{R}\right)_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}\right|$.
It is bounded independently of $R$ by (2.12) and (2.19), so monotone
convergence gives the desired inequality for the limit as $R\to\infty$.

The infinity behaviour is then immediate from the fact that the sum of
$\left|F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}\right|^{2}$
along any line in $\mathbb{C}\setminus B_{R}$ vanishes uniformly as
$R\to\infty$ by (2.22).

If there are two such
$F_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$, their difference
$\hat{F}_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$ is everywhere
s-holomorphic, and since the sum of
$\left|\hat{F}_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}\right|^{2}$
along any line in $\mathbb{C}\setminus B_{R}$ is finite and decays to zero,
the square integral
$\hat{H}_{\left[\mathbb{C}_{\delta},a_{1},\ldots,a_{n}\right]}$ is finite and
constant at infinity.
$\hat{H}_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}$ is
everywhere superharmonic and is finite at infinity, so
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\dagger\bullet}$ is
constant and
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\dagger}\equiv 0$ . ∎

Fix $\Theta<0$. On the full plane, we have an explicit characterisation of the
one point spinor in terms of the _massive harmonic measure_ of the slit plane
$\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}$:
$\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}(z|\Theta)$ for
$z\in(1+i)\mathbb{Z}^{2}$ is the probability of a simple random walk started
at $z$ extinguished at each step with probability
$\left(1+\frac{2\sin^{2}2\Theta}{\cos(4\Theta)}\right)^{-1}\frac{2\sin^{2}2\Theta}{\cos(4\Theta)}$
to successfully reach $0$ before hitting
$(1+i)\mathbb{Z}^{2}\cap\mathbb{R}_{<0}$.
$\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}(\cdot|\Theta)$ is
the unique $\Theta$-massive harmonic function (in the sense of (A.2)) on
$(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{\leq 0}$ which has the boundary
values $1$ at $0$ and $0$ on $(1+i)\mathbb{Z}^{2}\cap\mathbb{R}_{<0}$ and
infinity.

######  Proposition 13.

Denote the slit planes
$\mathbb{X}^{+}:=\left\\{z\in\left[\mathbb{C},0\right]:\text{Re}\sqrt{z}>0\right\\}\cong\mathbb{C}\setminus\mathbb{R}_{>0}$
and
$\mathbb{Y}^{+}:=\left\\{z\in\left[\mathbb{C},0\right]:\text{Im}\sqrt{z}>0\right\\}\cong\mathbb{C}\setminus\mathbb{R}_{>0}$.
Then

$F_{\left[\mathbb{C}_{\delta},0\right]}(c\delta|\Theta)=\begin{cases}\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}\left(c-\frac{3}{2}|\Theta\right)&c\delta\in\mathbb{X}^{+}\cap\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]\\\
-i\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}\left(\frac{1}{2}-c|\Theta\right)&c\delta\in\mathbb{Y}^{+}\cap\mathcal{C}^{i}\left[\mathbb{C}_{\delta},0\right]\end{cases}.$

###### Proof.

By Proposition 35, $F_{\left[\mathbb{C}_{\delta},0\right]}(c\delta)$
restricted to $\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ or
$\mathcal{C}^{i}\left[\mathbb{C}_{\delta},0\right]$ is $\Theta$-massive
harmonic, except possibly at the lifts of $\pm\frac{\delta}{2}$ (because there
is no planar neighbourhood $G_{\delta}$ around these points in
$\left[\mathbb{C}_{\delta},0\right]$, see Figure A.1) and $\frac{3\delta}{2}$
(because $F_{\left[\mathbb{C}_{\delta},0\right]}(c\delta)$ is not
s-holomorphic at the lifts of $\frac{\delta}{2}$).

From Definition 3, it is clear that $F_{\left[\mathbb{C}_{\delta}\cap
B_{R},0\right]}(c\delta)=0$ for any
$c\in\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ on the (lift of)
negative real line and $c\in\mathcal{C}^{i}\left[\mathbb{C}_{\delta},0\right]$
on the (lift of) positive real line, since the complex phase
$\phi_{a_{1},\ldots,a_{n}}\left(\gamma,c\right)$ for any
$\gamma\in\Gamma_{\mathbb{C}_{\delta}\cap B_{R}}(\frac{\delta}{2},\pi(c))$
switches sign for the reflection across the real line
$\gamma_{r}:=\left\\{\bar{e}:e\in\gamma\right\\}$, and
$\Gamma_{\mathbb{C}_{\delta}\cap B_{R}}(\frac{\delta}{2},\pi(c))$ is invariant
under the reflection. We conclude that
$F_{\left[\mathbb{C}_{\delta},0\right]}(c\delta)=0$ for
$c\in\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ .

Also note that $F_{\left[\mathbb{C}_{\delta},0\right]}(\frac{3\delta}{2})=1$,
where $\frac{3\delta}{2}$ is evaluated at the lift which is on the same sheet
as the fixed lift of $\frac{\delta}{2}$, since the corresponding Ising
quantity in (2.3) is precisely the ratio of two adjacent magnetisations (spin
$1$-point functions); in the infinite volume limit with $\beta>\beta_{c}$, the
ratio tends to $1$.

All in all, $F_{\left[\mathbb{C}_{\delta},0\right]}(\cdot|\Theta)$ restricted
to $\mathbb{X}^{+}\cap\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ is
the massive harmonic function which takes the boundary values $1$ on the lift
of $\frac{3\delta}{2}$ in
$\mathbb{X}^{+}\cap\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ and $0$
on the lift of the negative real line and infinity, whence the identification
with $\text{hm}_{(1+i)\mathbb{Z}^{2}\setminus\mathbb{R}_{<0}}^{0}$. The same
argument applies for its restriction to
$\mathbb{Y}^{+}\cap\mathcal{C}^{i}\left[\mathbb{C}_{\delta},0\right]$, noting
the boundary value at $\frac{\delta}{2}$ given by (2.2). ∎

###  2.3. Massive Complex Analysis: Continuous Fermions and their Uniqueness

We now present the boundary value problem which the limit of our discrete
fermions solves. Since massive models by definition do not possess conformal
invariance, we cannot expect the limit to be holomorphic as in the critical
case; instead, it satisfies a particular perturbed notion of holomorphicity
$\partial_{\bar{z}}f=m\bar{f}$, which belongs to various similar families of
functions dubbed _generalised analytic_ or _pseudoanalytic_ in writings of,
e.g., L. Bers and I. N. Vekua. Here we use a small excerpt of the established
theory, saying that $f$ is holomorphic up to a continuous factor. Although any
nonzero smooth function is trivially pseudoanalytic (with the non-constant
complex coefficient $\partial_{\bar{z}}/\bar{f}$), fixing the governing
equation $\partial_{\bar{z}}f=m\bar{f}$ allows us to use known functions which
satisfy the same equation to successively cancel out singularities; in other
words, we have a generalised version of Laurent series. Imposing a real
constant $m$ in addition yields the possibility to define the line integral
$\text{Re}\int f^{2}dz$.

In analogy with the discrete terminology, we refer to our particular
continuous condition _( $m$)-massive holomorphicity._ We will assume $m<0$ is
fixed throughout this section.

######  Lemma 14 (Similarity Principle).

Let $D$ be a bounded domain in $\mathbb{C}$. If a continuously differentiable
function $f:D\to\mathbb{C}$ is $m$-massive meromorphic, i.e. satisfies
$\partial_{\bar{z}}f:=\frac{1}{2}\frac{\partial}{\partial
x}f+\frac{i}{2}\frac{\partial}{\partial y}f=m\bar{f}$ except on a finite set,
there exists a Hölder continuous function $s$ on $\bar{U}$ such that $e^{-s}f$
is holomorphic in $U$.

In particular, $f$ only vanishes at isolated points.

###### Proof.

See [Ber56, Section 10] or [Vek62, Section III.4]; we also explain the
underlying idea and explicitly carry out specific cases in Section 4.1, e.g.
the proof of Lemma 28. ∎

######  Corollary 15.

There exists a family of $m$-massive holomorphic functions
$Z_{n}^{1},Z_{n}^{i}$ for each $n\in\mathbb{Z}$ such that as $z\to 0$

$Z_{n}^{1}(z)\sim z^{n},Z_{n}^{i}(z)\sim iz^{n},$

and any function $f$ that is $m$-massive holomorphic in a punctured
neighbourhood of $a$ can be expressed near $a$ as formal power series in
$Z_{n}$, i.e.

(2.26)
$f(z)=\sum_{n}\left[A_{n}^{1}Z_{n}^{1}(z-a)+A_{n}^{i}Z_{n}^{i}(z-a)\right]$

for real numbers $A_{n}^{1,i}$. If $f(z)$ is a spinor defined on a double
cover ramified at $a$, $f$ admits formal power series in analogous functions
indexed by half-integers $Z_{n+\frac{1}{2}}$. These expansions are uniformly
convergent in a small disc or annulus around $a$, respectively if $f$ has a
singularity at $a$ or not.

###### Proof.

This corollary depends on the fact that $s$ in Lemma 14 is continuous, so that
if a massive holomorphic function is $o((z-a)^{n})$ at $a$ then it is
automatically $O((z-a)^{n+1})$; see [Ber56, Section 5]. ∎

######  Remark 16.

There is no canonical choice of the ‘local formal powers’ $Z_{n}^{1,i}$. We
will need explicit functions to expand continuous fermions around their
singularities and analyse them further to derive Painlevé III in Section 4; we
hereby fix the following radially symmetric functions for half-integers $\nu$:

(2.27) $\displaystyle Z_{\nu}^{1}(re^{i\theta})$
$\displaystyle:=\frac{\Gamma(\nu+1)}{|m|^{\nu}}\left[e^{i\nu\theta}I_{\nu}(2|m|r)+(\text{sgn}m)\cdot
e^{-i(\nu+1)\phi}I_{\nu+1}(2|m|r)\right],$ $\displaystyle
Z_{\nu}^{i}(re^{i\theta})$
$\displaystyle:=\frac{\Gamma(\nu+1)}{|m|^{\nu}}\left[ie^{i\nu\theta}I_{\nu}(2|m|r)-i(\text{sgn}m)\cdot
e^{-i(\nu+1)\phi}I_{\nu+1}(2|m|r)\right],$

where $I_{n}$ is the modified Bessel function of the first kind. One can
easily verify the desired asymptotics and massive holomorphicity from the
corresponding facts about $I_{\nu}$, namely that
$I_{\nu}(r)\stackrel{{\scriptstyle r\to
0}}{{\sim}}\Gamma(\nu+1)\left(\frac{r}{2}\right)^{\nu}$ and
$I_{\nu}^{\prime}(r)=I_{\nu\pm 1}(r)\pm\frac{\nu}{r}I_{\nu}(r)$ [DLMF, Chapter
10] (See Section 4 for useful formulae, and note also that
$\partial_{\bar{z}}=\frac{1}{2}e^{i\theta}\left(\partial_{r}+ir^{-1}\partial_{\theta}\right)$).

As a special case, we have
$Z_{-\frac{1}{2}}^{1}(z)=\frac{e^{2m\left|z\right|}}{\sqrt{z}}$.

######  Proposition 17.

The following boundary value problem on a bounded smooth simply connected
domain $\Omega$ has at most one solution:

$f:\left[\Omega,a_{1},\ldots,a_{n}\right]\to\mathbb{C}$ satisfies

  1. (1)

$f$ is continuously differentiable, square integrable, and $\partial
f=m\bar{f}$ in $\left[\Omega,a_{1},\ldots,a_{n}\right]$,

  2. (2)

$f(z)\in\sqrt{\nu_{out}^{-1}(z)}\mathbb{R}$ on the lift of $\partial\Omega$,
where $\nu_{out}(z)$ is the outer normal at $\pi(z)\in\partial\Omega$,

  3. (3)

$(z-a_{1})^{1/2}f(z)\to 1$ as $z\to a_{1}$, and

  4. (4)

$\exists\mathcal{B}_{j}\in\mathbb{R}$ such that $(z-a_{j})^{1/2}f(z)\to
i\mathcal{B}_{j}$ as $z\to a_{j}$ for $j=2,\ldots,n$.

###### Proof.

If there are any two such functions $f_{1},f_{2}$, applying Green-Riemann’s
formula to their difference $\hat{f}$ on
$\Omega_{r}=\Omega\setminus\cup_{j}B_{r}(a_{j})$ for small $r>0$ yields

(2.28)
$\oint_{\partial\Omega_{r}}\hat{f}{}^{2}dz=2i\iint_{\Omega_{r}}\partial_{\bar{z}}\hat{f}{}^{{}^{2}}=2i\iint_{\Omega_{r}}2m|\hat{f}|^{2}dz\in
i\mathbb{R}.$

$\hat{f}\sim cst\cdot(z-a_{1})^{1/2}$ near $a_{1}$ in view of Corollary 15 so
$\oint_{\partial B_{r}(a_{1})}\hat{f}^{2}dz$ tends to zero as $r\to 0$, and
$\hat{f}\to i\hat{\mathcal{B}_{j}}(z-a_{j})^{-1/2}$ for some
$\hat{\mathcal{B}_{i}}\in\mathbb{R}$ as $z$ tends to any other $a_{j}$. On the
inner circles, we have

(2.29) $\displaystyle\oint_{\partial
B_{r}(a_{j})}\hat{f}{}^{2}dz\xrightarrow{r\to 0}$ $\displaystyle-2\pi
i\hat{\mathcal{B}_{j}^{2}}\in i\mathbb{R}\text{ for }j=2,\ldots,n,$

whereas the boundary condition on the lift of $\partial\Omega$ readily gives
$\frac{1}{i}\oint_{\partial\Omega}\hat{f}{}^{2}dz=\oint_{\partial\Omega}|\hat{f}|^{2}ds\geq
0$. Therefore

$0\leq\frac{1}{i}\oint_{\partial\Omega}\hat{f}{}^{2}dz=2\iint_{\Omega_{r}}2m|\hat{f}|^{2}dz-\sum_{j}2\pi\hat{\mathcal{B}_{j}}^{2}\leq
0,$

so $\hat{f}\equiv 0$. ∎

######  Remark 18.

As in the proof of the Proposition 17, it is easy to show that
$h:=\text{Re}\int f^{2}dz$ is globally well-defined in
$\Omega\setminus\left\\{a_{1},\ldots,a_{n}\right\\}$. In terms of $h$, the
boundary condition $f(z)\in\sqrt{\nu_{out}^{-1}}\mathbb{R}$ and the
asymptotics around $a_{i}$ is equivalent to

  1. (1)

$h$ is constant on $\partial\Omega$ and there is no $x_{0}\in\partial\Omega$
such that $h>h(\partial\Omega)$ in a neighbourhood of $x_{0}$, and

  2. (2)

$h$ is bounded below near $a_{2},\ldots,a_{n}$ and
$h^{\dagger}(w):=\text{Re}\int^{w}\left(f(z)-Z_{-\frac{1}{2}}^{1}(z-a_{1})\right)^{2}dz$
is single valued and bounded near $a_{1}$.

We note that this boundary problem can easily be extended to the case where
$\Omega=\mathbb{C}$, by requiring that $h$ be finite and constant at infinity
(we will in fact require that $f$ decays exponentially fast, see also Lemma
21).

We now define quantities which reflect the geometry of $\Omega$ exploiting the
boundary value problem above, which will turn out to be directly related to
the Ising correlations through the connection which is precisely our main
convergence result in Section 3.2. Determination of these quantities through
isomonodromic deformation is the main subject of Section 4.

######  Definition 19.

Given a solution $f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ of the boundary
value problem presented in Proposition 17 (the _continuous massive fermion_),
define
$\mathcal{A}_{\Omega}^{1,i}=\mathcal{A}_{\Omega}^{1,i}\left(a_{1},\ldots,a_{n}|m\right)$
as the real coefficients in the expansion

(2.30)
$f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}(z)=Z_{-\frac{1}{2}}^{1}(z-a_{1})+2\mathcal{A}_{\Omega}^{1}Z_{\frac{1}{2}}^{1}(z-a_{1})+2\mathcal{A}_{\Omega}^{i}Z_{\frac{1}{2}}^{i}(z-a_{1})+O\left((z-a)^{3/2}\right).$

In addition, in the case where $n=2$, define
$\mathcal{B}_{\Omega}=\mathcal{B}_{\Omega}\left(a_{1},a_{2}|m\right)$ as the
coefficient

(2.31)
$f_{\left[\Omega,a_{1},a_{2}\right]}(z)=\mathcal{B}_{\Omega}Z_{-\frac{1}{2}}^{i}(z-a_{1})+O\left((z-a)^{1/2}\right).$

For notational convenience, we do not assume $\mathcal{B}_{\Omega}>0$ as in
[CHI15]; instead, its sign depends on the sheet choice of
$Z_{-\frac{1}{2}}^{i}(z-a_{1})$, which we will explicitly fix whenever needed.

##  3\. Discrete Analysis: Scaling Limit

In this section, we show the convergence of the discrete fermions introduced
in Section 2 to their continuous counterparts. Then we show that the family of
discrete fermions as $\delta\downarrow 0$ is precompact in Section 3.1, whose
limit satisfies a unique characterisation as laid out in Proposition 17. These
suffice to show convergence to the desired limit. Throughout this section, we
assume a continuous mass $m<0$ is fixed.

###  3.1. Bulk Convergence

We finally undertake convergence of the discrete fermion
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ to its continuous
counterpart $f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ in scaling limit.
First, we need to interpolate the discrete function defined on
$\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]$ on the continuous domain
$\left[\Omega,a_{1},\ldots,a_{n}\right]$. While any reasonable interpolation
(e.g. linear interpolation used in many papers dealing with massless case)
should converge to the unique continuous limit, we will assume an
interpolation scheme with a continuously differentiable
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ as an easy way to show
that the limit itself is continuous differentiable. While we do not explicitly
carry it out, we could show the limit is smooth by using arbitrarily more
regular interpolation scheme.

######  Proposition 20.

Suppose $\Omega$ is bounded, simply connected with smooth boundary or
$\mathbb{C}$. For any compact subset $K\subset\Omega$, any infinite collection
$\left(\frac{2}{\pi}\delta_{k}\right)^{-1/2}F_{\left[\Omega_{\delta_{k}},a_{1},\ldots,a_{n}\right]}=:f_{\left[\Omega_{\delta_{k}},a_{1},\ldots,a_{n}\right]}$
with $\delta_{k}\downarrow 0$ has a subsequence that (suitably interpolated as
above) converges in $C^{1}(K)$-topology to a continuously differentiable
limit.

###### Proof.

By Arzelà-Ascoli, it suffices to show that the discrete derivatives

$\displaystyle\delta^{-1}\partial_{x}^{\delta}f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)$
$\displaystyle:=\delta^{-1}\left[f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z+2\delta)-f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)\right];$
$\displaystyle\delta^{-1}\partial_{y}^{\delta}f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)$
$\displaystyle:=\delta^{-1}\left[f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z+2\delta
i)-f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)\right],$

are equicontinuous on $K$. In view of Proposition 38, it suffices to show that
the ’discrete $L^{2}$ norm’
$\sum_{c\in\mathcal{C}\left(K_{\delta}\right)}\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(c)\right|^{2}\delta^{2}$
is bounded by a universal number.

Apply the discrete Green’s formula as in (2.12) to
$H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$, then we get the
following analogue to (2.22) (dividing both sides by $\delta$),

(3.1) $\displaystyle cst$
$\displaystyle\geq\delta^{-1}\Delta^{\delta}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{1}+\delta)$
(3.4) $\displaystyle\geq
2\sin\left(\frac{\pi}{4}-\Theta\right)\sum_{\begin{array}[]{c}c\in\mathcal{C}\left[\Omega_{\delta}\right]\\\
c\neq a_{1}+\delta+\frac{i^{m}\delta}{2}\end{array}}\delta
A_{-\Theta}\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c\right)\right|^{2}$
$\displaystyle+\sum_{e\in\partial\mathcal{E}\left[\Omega_{\delta}\right]}2\delta\sin\left(\frac{\pi}{4}-2\Theta\right)\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(e\right)\right|^{2},$
(3.7) $\displaystyle\geq
2\sin\left(\frac{\pi}{4}-\Theta\right)\sum_{\begin{array}[]{c}c\in\mathcal{C}\left[\Omega_{\delta}\right]\\\
c\neq a_{1}+\delta+\frac{i^{m}\delta}{2}\end{array}}\delta
A_{-\Theta}\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c\right)\right|^{2}$

and since $A_{-\Theta}\sim-2\sqrt{2}m\delta$, we have the desired $L^{2}$
bound from the sum of $\delta
A_{-\Theta}\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(c\right)\right|^{2}$.
∎

With a sequence $K_{m}$ of increasing compact subsets such that
$\bigcup_{m}K_{m}=\Omega\setminus\left\\{a_{1},\ldots,a_{n}\right\\}$ and
using diagonalisation, we can find a global subsequential limit
$f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ with uniform convergence in
compact subsets of $\Omega$. We finish the proof of convergence by showing
that a limit must satisfy the boundary value problem of Proposition 17, and
thus is unique. We note that continuous differentiability, square
integrability and the condition
$\partial_{\bar{z}}f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}=m\overline{f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}}$
follows straightforwardly from Proposition 35 and Remark 36, so we are left to
verify the boundary conditions, as laid out in Remark 18.

We first treat the following explicit case:

######  Lemma 21.

The one point spinor $f_{\left[\mathbb{C}_{\delta},0\right]}$ converges to
$Z_{-\frac{1}{2}}^{1}(z=re^{i\theta})=\frac{e^{2mr}}{\sqrt{z}}$ uniformly in
compact subsets of $\left[\mathbb{C}_{\delta},0\right]$.

###### Proof.

We show that a subsequential limit $f_{\left[\mathbb{C},0\right]}$ satisfies
the unique characterisation of Remark 18, and thus has to be equal to
$Z_{-\frac{1}{2}}^{1}$, an explicit solution. We first show that
$f_{\left[\mathbb{C},0\right]}$ vanishes at infinity sufficiently fast to
yield that $h_{\left[\mathbb{C},0\right]}:=\text{Re}\int
f_{\left[\mathbb{C},0\right]}^{2}dz$ is constant at infinity. But this follows
from the identification of the discrete spinor with the massive harmonic
measure (the hitting probability of an extinguished random walk) of the tip of
a slit in Proposition 13 and the exponential estimates of Proposition 38.

It now remains to identify the singularity at $0$ as
$f_{\left[\mathbb{C},0\right]}\sim z{}^{-1/2}$. Note that the massless
harmonic measure [Kes87, LaLi04] and thus the spinor [CHI15, Lemma 2.14] (for
the identification $\vartheta(\delta)=\left(\frac{2}{\pi}\delta\right)^{1/2}$,
see [GHP18, Lemma 5.14]) show that exact behaviour. Since the hitting
probability of a massive random walk is dominated by the hitting probability
of a simple random walk, $f_{\left[\mathbb{C},0\right]}\cdot
z^{1/2}\xrightarrow{z\to 0}\alpha$ for some $\alpha\in\left[0,1\right]$. But
the probability that the massive random walk is extinguished can be made
arbitrarily close to $0$ as length scale becomes negligible compared to
$\frac{1}{\left|m\right|}$, so the two harmonic measures become asymptotically
equal as $z\to 0$ and we conclude $\alpha=1$. ∎

By translation invariance of its definition,
$f_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z)$ converges to
$Z_{-\frac{1}{2}}^{1}(z-a_{1})$ uniformly in compact subsets of
$\left[\mathbb{C},a_{1}\right]$.

######  Proposition 22.

A subsequential limit $f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ of
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ satisfies the conditions
of Remark 18, and thus is unique.

###### Proof.

We first suppose $\Omega\neq\mathbb{C}$. Consider the renormalised discrete
square integral
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}:=\left(\frac{2}{\pi}\delta\right)^{-1}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$.
By Propositions 9 and 12,
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}=\delta^{-1}H_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}$
is discrete superharmonic on
$\mathcal{V}\left[\Omega_{\delta}\right]\setminus\left\\{a_{1}+\delta\right\\}$
and takes the boundary value $0$ on
$\partial\mathcal{V}\left[\Omega_{\delta}\right]$. Moreover, by Proposition
20,
$\Delta^{\delta}h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{1}+\delta)\leq
cst=:c_{0}$. Thus, by superharmonicity,
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}$ is lower
bounded by $c_{0}\mathcal{G}_{\delta}$, where $\mathcal{G}_{\delta}$ is the
domain Green’s function
$\mathcal{G}_{\delta}:=\mathcal{G}_{\mathcal{V}\left[\Omega_{\delta}\right]}(\cdot,a_{1}+\delta)$
with Dirichlet boundary condition on
$\partial\mathcal{V}\left[\Omega_{\delta}\right]$.

Since $\mathcal{G}_{\delta}$ is $O(\delta)$ on any vertex
$a_{\text{int}}\in\mathcal{V}\left[\Omega_{\delta}\right]$ adjacent to
$a\in\mathcal{V}\left[\Omega_{\delta}\right]$ (Lemma 37), we have that
$O(\delta)=c_{0}\mathcal{G}_{\delta}(a_{\text{int}})\leq
h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{\text{int}})\leq
0$ and
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}(a_{\text{int}})=-\left|f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(\frac{a_{\text{int}}+a}{2}\right)\right|^{2}\delta=O(\delta)$
by (2.7). We see that therefore
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is uniformly bounded on
boundary edges and corners (see Figure 1.1). Since $\left|f\right|^{2}$ is
subharmonic by (A.5), $f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is
uniformly bounded on $\Omega_{r}:=\Omega\setminus\bigcup_{j}B_{r}(a_{j})$ for
small fixed $r>0$. Therefore by equicontinuity
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\to
h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ uniformly on
$\overline{\Omega_{r}}$, and $h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ is
continuous up to the boundary. Note that since
$\partial_{\bar{z}}f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}=m\overline{f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}}$,
$h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}=\text{Re}\int
f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}^{2}dz$ satisfies

(3.8) $\Delta
h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}=4\partial_{\bar{z}}\partial_{z}h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}=2\partial_{\bar{z}}f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}^{2}=4m\left|f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}\right|^{2}.$

We now verify the two remaining conditions of the boundary value problem.

  1. (1)

$h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ in $\Omega_{r}$ is a continuous
solution of a Poisson equation with bounded data
$4m\left|f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}\right|^{2}$ and smooth
boundary data; $h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ is continuously
differentiable up to the boundary thanks to standard Green’s function
estimates (e.g. [GiTr15, Theorem 4.3]). From [ChSm12, Remark 6.3], which only
uses the superharmonicity (in their convention, subharmonicity) of
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\bullet}$ and does not
depend on a particular property of
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\circ}$, we see that
there is no neighbourhood of $x_{0}\in\partial\Omega$ where $h>0$.

  2. (2)

$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is bounded below by
$c\mathcal{G}_{\delta}$, which is bounded from negative infinity away from
$a_{1}$ in the scaling limit. So near $a_{2},\ldots,a_{n}$,
$h_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ is bounded below. For the
asymptotic near $a_{1}$, note that
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\dagger}:=f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}-f_{\left[\mathbb{C}_{\delta},a_{1}\right]}$
is s-holomorphic near $a_{1}$ with
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\dagger}(a_{1}+\frac{\delta}{2})=0$,
so by Proposition 35 it is everywhere massive harmonic (unlike in the proof of
Proposition 13, the zero prevents a singularity near monodromy; see also
[GHP18, Remark 2.6]). But both
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]},f_{\left[\mathbb{C}_{\delta},a_{1}\right]}$
are uniformly bounded, say, in a discrete circle
$S_{r}:=B_{r+5\delta}(a_{1})\setminus B_{r}(a_{1})$ for small $r>0$. As above,
we conclude that
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}^{\dagger}$ is uniformly
bounded in $B_{r}(a_{1})$. Given Lemma 21, this suffices to show boundedness
and well-definedness for the continuous limits $f^{\dagger},h^{\dagger}$.

If $\Omega=\mathbb{C}$, we first show that
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\to 0$ at infinity
uniformly in $\delta$ and
$h_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$ is constant and finite
at infinity using Proposition 38 and the uniform $L^{2}$ bound (3.1). Since
$f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\to 0$ at infinity
uniformly in $\delta$, we may compare with the Green’s function of a suitably
large ball to obtain lower boundedness around $a_{2},\ldots,a_{n}$. The above
proof applies near $a_{1}$. ∎

###  3.2. Analysis near the Singularity

To show convergence of
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{3\delta}{2}\right)$,
we model the expansion of $f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}$ around
$a_{1}$ from Definition 19 in the discrete setting, and carefully analyse the
magnitude of the difference. The candidate for discrete $Z_{-\frac{1}{2}}^{1}$
is clear: $F_{\left[\mathbb{C}_{\delta},a_{1}\right]}$ is able to cancel out
the singularity of $F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}$, and
converges to $Z_{-\frac{1}{2}}^{1}$ by Lemma 21.

Then, we need to build a discrete analogue of $Z_{\frac{1}{2}}^{1}$ (or
rather, some massive s-holomorphic function that has square root behaviour
around $0$). Following [CHI15, (3.12)], we define a discrete function
$G_{\left[\mathbb{C}_{\delta},a_{1}\right]}$ by discrete integrating the
$F_{\left[\mathbb{C}_{\delta},a_{1}\right]}$.

######  Proposition 23.

Construct the spinor
$G_{\left[\mathbb{C}_{\delta},a_{1}\right]}:\mathcal{C}^{1}\left[\Omega_{\delta},a_{1}\right]\to\mathbb{R}$
by

(3.9)
$G_{\left[\mathbb{C}_{\delta},a_{1}\right]}\left(z\right):=\delta\sum_{j=0}^{\infty}\Gamma^{j}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta),$

where $\Gamma(\delta|\Theta):=\tan^{2}\left(\frac{\pi}{4}+2\Theta\right)$, and
$z-2j\delta$ is taken on the same sheet as $z$ if $\pi(z)\notin
a_{1}+\mathbb{R}_{>0}$ or if $\pi(z),\pi\left(z-2j\delta\right)\in
a_{1}+\mathbb{R}_{>0}$, while
$F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)=0$ naturally as soon
as $\pi\left(z-2j\delta\right)\in a_{1}+\mathbb{R}_{<0}$ (cf. Proposition 13).

It is massive harmonic with coefficient $M_{H}^{2}$ on
$\mathbb{X}^{+}\cap\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$, and
$\left(\frac{2}{\pi}\delta\right)^{-1/2}G_{\left[\mathbb{C}_{\delta},a_{1}\right]}$
converges uniformly in compact subsets of $\left[\mathbb{C},a_{1}\right]$ to
$g_{\left[\mathbb{C},a_{1}\right]}$ which has the asymptotic behaviour
$g_{\left[\mathbb{C},a_{1}\right]}(z)\sim\text{Re}\sqrt{z-a_{1}}$ near
$a_{1}$.

###### Proof.

We first show that the discrete integrand
$\Gamma^{j}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)$ is
summable. Without loss of generality, work on the sheet where
$F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)>0$. By Proposition 13,
$F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)$ is the probability of
a massive random walk started at the real corner $z-2j\delta$ and extinguished
at each step with probability
$\left(1+\frac{2\sin^{2}2\Theta}{\cos(4\Theta)}\right)^{-1}\frac{2\sin^{2}2\Theta}{\cos(4\Theta)}$
surviving to hit $a_{1}+\frac{3\delta}{2}$ before $a_{1}+\mathbb{R}_{<0}$;
denote by $h_{j}$ the probability of the same event with $\Theta=0$, i.e. the
massless harmonic measure. Project the two-dimensional walk into two
independent one-dimensional walks respectively in the $x,y$-directions with
step lengths $\delta$. If $p_{2j}$ is the probability of the one-dimensional
massive random walk started at $x-2j\delta$ surviving to hit $x$, we have the
upper bound

$\Gamma^{j}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)\leq\Gamma^{j}p_{2j}h_{j}.$

However, $p_{2j}$ is the massive harmonic measure of $0$ as seen from $-2j$ in
$\mathbb{Z}_{\leq 0}$. From the boundary conditions $p_{0}=1$, $p_{-\infty}=0$
and massive harmonicity in 1D
$p_{j-2}+p_{j}-2p_{j-1}=\frac{4\sin^{2}2\Theta}{\cos 4\Theta}p_{j-1}$, it is
straightforward to verify $p_{j}=\Gamma^{-j/2}$, and we have
$\Gamma^{j}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)\leq h_{j}$.
By [CHI15, Lemma 3.4], $h_{j}=O(j^{-3/2})$ uniformly for $z$ in a compact
subset, and thus the sum is finite.

We know that
$\left(\frac{2}{\pi}\delta\right)^{-1/2}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z^{\prime})$
converges to $\text{Re}\frac{e^{2mr^{\prime}}}{\sqrt{z^{\prime}}}$ on
$\mathcal{C}^{1}\left[\Omega_{\delta},a_{1}\right]$ uniformly in compact
subsets away from $a_{1}$. Also note that in the scaling limit
$x-2j\delta=x^{\prime}$, $\Gamma^{j}$ converges to $e^{-2m(x-x^{\prime})}$. By
above, if $z=x+iy$ is in a compact subset, the discrete integrand
$\Gamma^{\frac{x-x^{\prime}}{2\delta}}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z^{\prime})$
decays to zero as $x^{\prime}\to-\infty$ uniformly in $z$, so we conclude that
$g_{\left[\mathbb{C}_{\delta},a_{1}\right]}:=\left(\frac{2}{\pi}\delta\right)^{-1/2}G_{\left[\mathbb{C}_{\delta},a_{1}\right]}$
converges

(3.10)
$g_{\left[\mathbb{C}_{\delta},a_{1}\right]}\left(z=x+iy\right)\xrightarrow{\delta\to
0}\frac{1}{2}\int_{-\infty}^{x}e^{2m(x^{\prime}-x)}\text{Re}\frac{e^{2mr^{\prime}}}{\sqrt{x^{\prime}+iy^{\prime}}}dx^{\prime},$

uniformly for $z$ in a compact subset. As $z\to 0$, we can uniformly bound
$e^{2m(x^{\prime}-x)}e^{2mr^{\prime}}$ close to $1$, which gives the
asymptotic of $g_{\left[\mathbb{C},a_{1}\right]}$ near $z$.

Massive harmonicity is clear unless $\pi(z)\in\mathbb{R}_{>0}$. If $\pi(z)\in
a_{1}+\mathbb{R}_{>0}$,
$\left(\Delta^{\delta}-M_{H}^{2}\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)=0$
if $\pi(z-2j\delta)\in a_{1}+\frac{3\delta}{2}+\mathbb{R}_{\geq 0}$ while
$F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)=0$ if
$\pi(z-2j\delta)\in a_{1}+\mathbb{R}_{<0}$, thus

$\displaystyle\left(\Delta^{\delta}-M_{H}^{2}\right)G_{\left[\mathbb{C}_{\delta},a_{1}\right]}\left(z\right)$
$\displaystyle=\delta\sum_{j=\left\lfloor(z-a_{1})/2\delta\right\rfloor}^{\infty}\Gamma^{j}\left(\Delta^{\delta}-M_{H}^{2}\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(z-2j\delta)$
$\displaystyle=\delta\Gamma^{\left\lfloor(z-a_{1})/2\delta\right\rfloor}\sum_{j=0}^{\infty}\Gamma^{j}\left(\Delta^{\delta}-M_{H}^{2}\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta),$

where the laplacian is taken on the planar slit domain
$\mathbb{X}^{+}\cap\mathcal{C}^{1}\left[\mathbb{C}_{\delta},0\right]$ with
zero boundary values on the slit.

We need to show that the last sum vanishes.

$\displaystyle\sum_{j=0}^{N}\Gamma^{j}\left(\Delta^{\delta}-M_{H}^{2}\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{N}\Gamma^{j}\sum_{s=\pm
1}\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+s)\delta+i\delta)-\tan\left(\frac{\pi}{4}-s2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)\right]$
$\displaystyle+$ $\displaystyle\sum_{j=0}^{N}\Gamma^{j}\sum_{s=\pm
1}\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+s)\delta-i\delta)-\tan\left(\frac{\pi}{4}+s2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)\right],$

where two sums are done respectively above and below the slit. By massive
discrete holomorphicity, we can convert the differences of real corner values
into differences of imaginary corner values: we need to be careful, since the
points in $\mathbb{X}^{+}$ directly above and below the cut are on opposite
sheets. We can in fact think of $a_{1}+\frac{\delta}{2}$ as lying on the slit,
since the singularity (2.2) ascribes two values at $a_{1}+\frac{\delta}{2}$,
coming from above and below $a_{1}+\mathbb{R}$. Above the slit, (A.1) implies

$\displaystyle\sum_{s=\pm
1}\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+s)\delta+i\delta)-\tan\left(\frac{\pi}{4}-s2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)\right]$
$\displaystyle=$
$\displaystyle-i\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta+i\delta)-\tan\left(\frac{\pi}{4}+2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j-1)\delta)\right]$
$\displaystyle+i\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta+i\delta)-\tan\left(\frac{\pi}{4}-2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+1)\delta)\right]$
$\displaystyle=$ $\displaystyle
i\tan\left(\frac{\pi}{4}+2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j-1)\delta)-i\tan\left(\frac{\pi}{4}-2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+1)\delta).$

Thanks to the factor of $\Gamma^{j}$, the sum telescopes (see Figure A.1)

$\displaystyle\sum_{j=0}^{N}\Gamma^{j}\sum_{s=\pm
1}\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+s)\delta+i\delta)-\tan\left(\frac{\pi}{4}-s2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)\right]$
$\displaystyle=$ $\displaystyle
i\tan\left(\frac{\pi}{4}+2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{5\delta}{2})-i\Gamma^{N}\tan\left(\frac{\pi}{4}-2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2N+1)\delta).$

The sum below the slit analogously gives

$\displaystyle\sum_{j=0}^{N}\Gamma^{j}\sum_{s=\pm
1}\left[F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2j+s)\delta-i\delta)-\tan\left(\frac{\pi}{4}+s2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)\right]$
$\displaystyle=-$ $\displaystyle
i\tan\left(\frac{\pi}{4}+2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{5\delta}{2})+i\Gamma^{N}\tan\left(\frac{\pi}{4}-2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2N+1)\delta),$

where $a_{1}+\frac{3\delta}{2}-(2N+1)\delta$ is approached from below the
sheet. Summing the two and taking $a_{1}+\frac{3\delta}{2}-(2N+1)\delta$ from
above the slit,

$\sum_{j=0}^{N}\Gamma^{j}\left(\Delta^{\delta}-M_{H}^{2}\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-2j\delta)=-2i\Gamma^{N}\tan\left(\frac{\pi}{4}-2\Theta\right)F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2N+1)\delta).$

Now, by Proposition 13, $-i\cdot
F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2N+1)\delta)$
is the massive harmonic measure of the point $a_{1}+\frac{\delta}{2}$ in the
discrete plane
$\mathbb{Y}^{+}\cap\mathcal{C}^{i}\left[\mathbb{C}_{\delta},0\right]$ slit by
$a_{1}+\mathbb{R}_{>0}$.
$-i\cdot\Gamma^{N}F_{\left[\mathbb{C}_{\delta},a_{1}\right]}(a_{1}+\frac{3\delta}{2}-(2N+1)\delta)=O(N^{-1/2})$
since we may bound it by the massless harmonic measure as above, and the sum
decays to zero, as desired. ∎

######  Theorem 24.

As $\delta\to 0$, for $a_{1},\ldots,a_{n}$ at a definite distance from the
boundary and each other, we have uniformly

(3.11) $\displaystyle
F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{3\delta}{2}\right)$
$\displaystyle=1+2\delta\mathcal{A}_{\Omega_{\delta}}^{1}\left(a_{1},\ldots,a_{n}\right)+o(\delta),$
(3.12)
$\displaystyle\left|F_{\left[\Omega_{\delta},a_{1},a_{2}\right]}\left(a_{2}+\frac{\delta}{2}\right)\right|$
$\displaystyle=\left|\mathcal{B}_{\Omega}(a_{1},a_{2})\right|+o(1).$

###### Proof.

We adapt the strategy of [CHI15, Subsection 3.5]; the massive harmonic nature
of our functions are hardly visible since at short length scale massive
hitting probabilities approach simple random walk hitting probabilities. While
we try to use the same notation for corresponding notions where appropriate,
the functions we work with are all massive harmonic.

Note that, due to explicit constructions in Propositions 13 and 23, (3.11) can
be written as

$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{3\delta}{2}\right)=F_{\left[\mathbb{C}_{\delta},a_{1}\right]}\left(a_{1}+\frac{3\delta}{2}\right)+2\mathcal{A}_{\Omega_{\delta}}^{1}\left(a_{1},\ldots,a_{n}\right)G_{\left[\mathbb{C}_{\delta},a_{1}\right]}\left(a_{1}+\frac{3\delta}{2}\right)+o(\delta).$

Define the reflection $\mathcal{R}\left(\Omega\right)$ of $\Omega$ across the
line $a_{1}+\mathbb{R}$. Then there is a ball $B(a_{1})$ around $a_{1}$ which
belongs to $\Omega\cap\mathcal{R}\left(\Omega\right)$. Denote $\Lambda$ to be
the lift of the slit neighbourhood
$B(a_{1})\setminus\left[a_{1}+\mathbb{R}_{<0}\right]$ such that both
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]},F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}$
have their fixed lift of the path origin $a_{1}+\frac{\delta}{2}$ on
$\Lambda$. By symmetry arguments about the path set
$\Gamma(a_{1}+\frac{\delta}{2},z)$ similar to the proof of Proposition 13, we
have
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{3\delta}{2}\right)=F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}\left(a_{1}+\frac{3\delta}{2}\right)$,
whereas the boundary values
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)$,$F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}(z)$
on the slit $\pi(z)\in a_{1}+\mathbb{R}_{<0}$ cancel out to give
$F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(z)+F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}(z)=0$
for $z\in\mathcal{C}^{1}\left[\Omega_{\delta},a_{1}\right]\cap\partial\Lambda$
(see also [CHI15, Subsection 3.5]). Then, restricted to
$\mathcal{C}^{1}\left[\Omega_{\delta},a_{1}\right]\cap\Lambda$, (recall
$g_{\left[\mathbb{C}_{\delta},a_{1}\right]}:=\left(\frac{2}{\pi}\delta\right)^{-1/2}G_{\left[\mathbb{C}_{\delta},a_{1}\right]}$)

$K_{\delta}:=\frac{1}{2}\left[f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}+f_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}\right]-f_{\left[\mathbb{C}_{\delta},a_{1}\right]}-2\mathcal{A}_{\Omega_{\delta}}^{1}\left(a_{1},\ldots,a_{n}\right)g_{\left[\mathbb{C}_{\delta},a_{1}\right]},$

is everywhere massive harmonic with zero boundary values on the slit
$\mathcal{C}^{1}\left[\Omega_{\delta},a_{1}\right]\cap\partial\Lambda$. We
have to show

$K_{\delta}(a_{1}+\frac{3\delta}{2})=f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}(a_{1}+\frac{3\delta}{2})-1-2\mathcal{A}_{\Omega_{\delta}}^{1}\left(a_{1},\ldots,a_{n}\right)\delta=o(\delta^{1/2}).$

Noting the expansion of Definition 19, we see that, on $\Lambda$, away from
$a_{1}$,

$K_{\delta}\to\frac{1}{2}\left[f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}+f_{\left[\mathcal{R}\left(\Omega\right),a_{1},\ldots,a_{n}\right]}\right]-Z_{-\frac{1}{2}}^{1}-2\mathcal{A}_{\Omega_{\delta}}^{1}\left(a_{1},\ldots,a_{n}\right)\text{Re}\sqrt{z-a_{1}}=o\left(\left(z-a_{1}\right)^{1/2}\right),$

so on the discrete circle $S_{r}=B_{r+5\delta}(a_{1})\setminus B_{r}(a_{1})$
for $r>0$, $\max_{S_{r}}\left|K_{\delta}\right|=o(r^{1/2})$ as $r\to 0$. Sharp
discrete Beurling estimate (see [CHI15, (3.4)] for the form used here) for
harmonic functions may be used to dominate the value of the massive harmonic
function $S_{\delta}$ at $a_{1}+\frac{3\delta}{2}$, and we have

$\left|K_{\delta}(a_{1}+\frac{3\delta}{2})\right|\leq
cst\cdot\delta^{1/2}r^{-1/2}\max_{S_{r}}\left|K_{\delta}\right|=cst\cdot\delta^{1/2}o(1),$

where $o(1)$ holds for $r\to 0$. We conclude the right hand side is
$o(\delta^{1/2})$ as $\delta\to 0$.

For (3.12), without loss of generality assume
$\mathcal{B}_{\Omega}(a_{1},a_{2})>0$. Define this time
$\mathcal{R}\left(\Omega\right)$ as the reflection of $\Omega$ across
$a_{2}+\mathbb{R}$. As above, write $\Lambda$ for the lift of the slit disc
$B(a_{2})\setminus\left[a_{2}+\mathbb{R}_{<0}\right]$ such that we
simultaneously define
$F_{\left[\Omega_{\delta},a_{1},a_{2}\right]}(a_{2}+\frac{\delta}{2})=F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},a_{2}\right]}(a_{2}+\frac{\delta}{2})=:i\mathcal{B}_{\delta}$
with $\mathcal{B}_{\delta}>0$. Again, by symmetry,
$f_{\left[\Omega_{\delta},a_{1},a_{2}\right]}+f_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},a_{2}\right]}$
is zero on the boundary
$\partial\Lambda\cap\mathcal{C}^{i}\left[\Omega_{\delta},a_{1},a_{2}\right]$,
i.e. the lift of $a_{2}+\mathbb{R}_{<0}$. Define the massive harmonic measure
of the point $a_{1}+\frac{\delta}{2}$ in the lattice
$\Lambda\cap\mathcal{C}^{i}\left[\Omega_{\delta},a_{1},a_{2}\right]$ as
$W_{\delta}$. Then define

$T_{\delta}:=\frac{1}{2}\left[F_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}+F_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}\right]-i\mathcal{B}_{\delta}W_{\delta},$

which is massive harmonic on
$\Lambda\cap\mathcal{C}^{i}\left[\Omega_{\delta},a_{1},a_{2}\right]\setminus\left\\{a_{2}+\frac{\delta}{2}\right\\}$,
takes the value $0$ on
$\partial\Lambda\cap\mathcal{C}^{i}\left[\Omega_{\delta},a_{1},a_{2}\right]$
and $\left\\{a_{2}+\frac{\delta}{2}\right\\}$. Since
$\frac{1}{2}\left[f_{\left[\Omega_{\delta},a_{1},\ldots,a_{n}\right]}+f_{\left[\mathcal{R}\left(\Omega_{\delta}\right),a_{1},\ldots,a_{n}\right]}\right]$
restricted to the imaginary corners converges to a continuous limit with
asymptotic
$i\cdot\text{Re}\frac{\mathcal{B}_{\Omega_{\delta}}(a_{1},a_{2})}{\sqrt{z-a_{2}}}+o(\left(z-a_{2}\right)^{-1/2})$
on the discrete circle $S_{r}=B_{r+5\delta}(a_{2})\setminus B_{r}(a_{2})$, it
is easy to see that unless
$\mathcal{B}_{\delta}\to\mathcal{B}_{\Omega}(a_{1},a_{2})$ and
$T_{\delta}\left(a_{2}+\frac{\delta}{2}\right)=o(1)$, we can find a point in
the bulk which is greater than any value on the boundary $S_{r}$,
contradicting the maximum principle; see also [CHI15, (3.23)]. ∎

##  4\. Continuum Analysis: Isomonodromy and Painlevé III

###  4.1. Continuous Analysis of the Coefficients

In this section, we carry out analysis of the continuum coefficients such as
$\mathcal{A}_{\Omega},\mathcal{B}_{\Omega}$ needed for the proof of the main
theorem and the derivation of the Painlevé III transcendent in the next
section. We will assume that a continuous mass parameter $m<0$ is fixed
throughout this section.

#### Preliminaries: Massive Cauchy Formula.

In Section 2.3, we have seen that massive holomorphic functions admit a
generalisation of Laurent-type expansions. Here we give explicit formulae
related to the formal powers $Z_{\nu}^{1,i}$ and note that a Cauchy-type
integral formula holds.

Recall
$\partial_{z}=\frac{1}{2}e^{-i\theta}\left(\partial_{r}-ir^{-1}\partial_{\theta}\right),\partial_{\bar{z}}=\frac{1}{2}e^{i\theta}\left(\partial_{r}+ir^{-1}\partial_{\theta}\right)$.
The following holds ([DLMF, Section 10]):

$\displaystyle I_{\nu}^{\prime}(r)$ $\displaystyle=I_{\nu\pm
1}(r)\pm\frac{\nu}{r}I_{\nu}(r),$ (4.1) $\displaystyle-2\frac{\sin\nu\pi}{\pi
r}$ $\displaystyle=I_{\nu}(r)I_{-\nu-1}(r)-I_{\nu+1}(r)I_{-\nu}(r).$

Define
$W_{\nu}(re^{i\theta}):=e^{i\nu\theta}I_{\nu}\left(2\left|m\right|r\right)$.
The formal powers $Z_{\nu}^{1,i}$ defined in can be written as

(4.2) $\displaystyle Z_{\nu}^{1}$
$\displaystyle=\frac{\Gamma(\nu+1)}{\left|m\right|^{\nu}}\left(W_{\nu}+\left(\text{sgn}m\right)\overline{W_{\nu+1}}\right),$
$\displaystyle Z_{\nu}^{i}$
$\displaystyle=\frac{\Gamma(\nu+1)}{\left|m\right|^{\nu}}\left(iW_{\nu}-i\left(\text{sgn}m\right)\overline{W_{\nu+1}}\right).$

######  Proposition 25 ([Ber56, Section 6]).

Let $f$ is an $m$-massive holomorphic function defined on a ramified disk
$\left[B_{R}(a),a\right]$. The coefficients $A_{\nu}^{1,i}$ of $Z_{\nu}^{1,i}$
in the expansion (2.27) may be extracted by the line integrals:

(4.3) $\displaystyle A_{\nu}^{1}$
$\displaystyle=\frac{\pi}{4\left|m\right|\nu^{2}}\text{Re}\oint_{C}f(z)Z_{-1-\nu}^{i}(z-a)dz,$
$\displaystyle A_{\nu}^{i}$
$\displaystyle=-\frac{\pi}{4\left|m\right|(1+\nu)^{2}}\text{Re}\oint_{C}f(z)Z_{-1-\nu}^{1}(z-a)dz,$

where the line integral is taken on any smooth curve $C$ going once around
$a$.

###### Proof.

As in the discrete case, if $f,g$ are $m$-massive holomorphic functions, the
real part $\text{Re}\int f\cdot gdz$ is well defined: indeed, the increment
around a closed curve $\partial D$ inclosing a region $D$ is $\int_{\partial
D}f\cdot gdz=2i\int\partial_{\bar{z}}(fg)d^{2}z=2i\int
m\text{Re}\left(fg\right)d^{2}z\in i\mathbb{R}$ by the Green-Riemann formula.
So the integral (4.3) has a well-defined value regardless of choice of $C$.

We will take $C=\partial B_{r}(a)$ for some small $r>0$. In view of the
definition (4.2), it is straightforward to verify first that any line integral
of the form
$\text{Re}\int_{C}Z_{\nu}^{1}Z_{\nu^{\prime}}^{1}dz,\text{Re}\int_{C}Z_{\nu}^{i}Z_{\nu^{\prime}}^{i}dz$
vanish. Similarly one can verify the mixed integral

$\displaystyle\text{Re}\int_{C}Z_{\nu}^{1}Z_{\nu^{\prime}}^{i}dz$
$\displaystyle=\delta_{\nu+\nu^{\prime},-1}\frac{-4\left|m\right|\nu^{2}}{\pi},$

with the help of (4.1) and the standard Gamma function equality [DLMF, Section
5.5]

$\Gamma(z)\Gamma(-1-z)=\frac{\Gamma(z)\Gamma(1-z)}{z(1+z)}=\frac{1}{z(1+z)}\frac{\pi}{\sin
z\pi}.$

∎

Finally, we note the derivatives of the formal powers.

$\partial_{r}W_{\nu}(re^{i\theta})=e^{i\nu\theta}\cdot
2\left|m\right|\cdot\left(I_{\nu\pm
1}(2\left|m\right|r)\pm\frac{\nu}{2\left|m\right|r}I_{\nu}(2\left|m\right|r)\right)$
and $\partial_{\theta}W_{\nu}(re^{i\theta})=i\nu
e^{i\nu\theta}I_{\nu}\left(2\left|m\right|r\right)$, and we see that

$\partial_{z}W_{\nu}=\left|m\right|W_{\nu-1},\partial_{\bar{z}}W_{\nu}=\left|m\right|W_{\nu+1},$

and the corresponding identities for $Z_{\nu}^{1,i}$ follow. In fact, we will
record, noting
$\partial_{x}=\partial_{z}+\partial_{\bar{z}},\partial_{y}=i\left(\partial_{z}-\partial_{\bar{z}}\right)$,

$\displaystyle\partial_{x}Z_{\nu}^{1}$ $\displaystyle=\nu
Z_{\nu-1}^{1}+(\nu+1)^{-1}m^{2}Z_{\nu+1}^{1},\partial_{x}Z_{\nu}^{i}=\nu
Z_{\nu-1}^{i}+(\nu+1)^{-1}m^{2}Z_{\nu+1}^{i},$
$\displaystyle\partial_{y}Z_{\nu}^{1}$ $\displaystyle=\nu
Z_{\nu-1}^{i}-(\nu+1)^{-1}m^{2}Z_{\nu+1}^{i},\partial_{y}Z_{\nu}^{i}=-\nu
Z_{\nu-1}^{1}+(\nu+1)^{-1}m^{2}Z_{\nu+1}^{1}.$

#### Preliminaries: Cauchy Transform and Harmonic Conjugate.

To work with generalised analytic functions, the _Cauchy transform_
$\mathfrak{C}=\mathfrak{C}_{\Omega}:t(\cdot)\mapsto-\frac{1}{\pi}\int_{\Omega}\frac{t(w)}{w-z}d^{2}w$
will be used as the inverse of the derivative $\partial_{\bar{z}}$. Define the
Hölder seminorm
$\left[t\right]_{C^{\alpha}(\Omega)}(w):=\sup_{z\in\Omega}\frac{\left|t(w)-t(z)\right|}{\left|w-z\right|^{\alpha}},\left[t\right]_{C^{\alpha}(\Omega)}:=\sup_{w\in\Omega}\left[t\right]_{C^{\alpha}(\Omega)}$
and the norm
$\left|t\right|_{C^{\alpha}(\Omega)}:=\left|t\right|_{C(\Omega)}+\left[t\right]_{C^{\alpha}(\Omega)}$.
The following estimates are standard and may be shown by direct analysis of
the kernel $\frac{1}{w-z}$.

######  Proposition 26.

Let $B=B_{r}(z)$ be a ball of radius $r>0$.

  * •

If $g\in L^{p}(B)$ for some $p>2$, $\mathfrak{C}g\in C^{\alpha}(\mathbb{C})$
for $\alpha=\frac{p-2}{p}$, holomorphic outside of $B$, and vanishes at
infinity, with

(4.4) $\left|\mathfrak{C}g\right|\leq cst\cdot
r^{\alpha}\left|g\right|_{L^{p}(B)};\left[\mathfrak{C}g\right]_{C^{\alpha}(\mathbb{C})}\leq
cst\cdot\left|g\right|_{L^{p}(B)},$

where the constants only depend on $p$;

  * •

If $g\in C^{\alpha}(B)$, $\mathfrak{C}g$ is differentiable at $z$, with

(4.5) $\left|\nabla\mathfrak{C}g(z)\right|\leq
cst\left(\left|g(z)\right|+r^{\alpha}\left[g\right]_{C^{\alpha}(B)}(z)\right);$

where the constant only depends on $\alpha\in(0,1]$.

###### Proof.

See [Vek62, Sections 1.4-8]. Specifically, for (4.4) refer to [Vek62, Theorem
1.19]; (4.5) follows from [Vek62, (8.2), (8.7)]. ∎

Another standard analytic fact that we use is the Hölder regularity of
harmonic conjugates.

######  Proposition 27 (Privalov’s Theorem).

Let $D$ be a smooth bounded simply connected domain, and fix a conformal map
$\varphi:D\to\mathbb{D}$ such that $M^{-1}<\left|\varphi^{\prime}\right|<M$
for some $M>0$.

Let $t\in C^{\alpha}(\partial D)$ be a real-valued function on the boundary.
Then there exists a holomorphic function $g\in C^{\alpha}(\bar{D})$ such that
$\text{Im}g=t$ which is unique up to a real constant. Moreover, if
$\text{Re}g=0$ at any point in $\bar{D}$,

$\left|g\right|_{C^{\alpha}(\bar{D})}\leq
cst\cdot(1+M^{2\alpha})\left|t\right|_{C^{\alpha}(\partial D)},$

where the constant only depends on $\alpha$.

###### Proof.

See, e.g., [GaMa08, Theorem 3.2], for the proof in the unit disc. Then it is
straightforward to transfer the result to $D$ using the conformal map
$\varphi^{\prime}$ given that the Hölder seminorm transforms with a factor of
$\left|\varphi^{\prime}\right|^{\alpha}$. ∎

#### Analysis of the Continuous Observables.

We now study the continuous observables
$f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}(\cdot|m)$ with $m<0$ and its
coefficients. Fix a conformal map $\varphi:\Omega\to\mathbb{D}$ and let $M>0$
be such that $M^{-1}<\varphi^{\prime}<M$. Note that $\text{diam}\Omega\leq
2M$.

We first give the following lemma based on the Lemma 14 (similarity
principle). The following proof in fact contains the idea of the proof of the
principle, but strictly speaking we do rely on it, e.g. for the existence of
the observable itself. Write
$f_{m}(z):=f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}(z|m)$.

######  Lemma 28.

There exists a unique function $s$ in $\bar{\Omega}$ such that

  * •

$s$ is continuous in $\bar{\Omega}$;

  * •

$s$ is real on $\partial\Omega$ and $\text{Re}e^{-s(a)}=1$;

  * •

$e^{-s}f_{m}$ is holomorphic in $\left[\Omega,a_{1},\ldots,a_{n}\right]$.

Then, defining $c_{j}:=\text{Re}\lim_{z\to a_{j}}e^{-s(z)}\sqrt{z-a_{j}}f_{m}$
(note $c_{1}=1$)

(4.6)
$e^{-s(z)}f_{m}(z)=\sum_{j=1}^{n}c_{b}f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}(z|0),$

Moreover, for any $\alpha\in(0,1)$,

$\left|s\right|_{C^{\alpha}}\leq cst\cdot\left|m\right|(1+M^{1+2\alpha}),$

where the constant only depends on $\alpha$.

###### Proof.

First assume existence of such $s$. Suppose there are two such functions
$s_{1},s_{2}$. Then $e^{-s_{1}(z)+s_{2}(z)}$ is bounded, real on
$\partial\Omega$, and holomorphic in $\Omega\setminus\left\\{a,b\right\\}$;
thus the value at $a$ fixes it to be constantly $1$. So $s_{1}-s_{2}\equiv 0$
given that $\text{Im}\left(s_{1}-s_{2}\right)=0$ on $\partial\Omega$ and $s$
is unique.

Then $\hat{f}(z):=e^{-s(z)}f_{m}(z)$ is holomorphic in
$\left[\Omega,a_{1},\ldots,a_{n}\right]$ and
$\sqrt{\nu_{out}}\hat{f}\in\mathbb{R}$ on the boundary $\partial\Omega$. Near
$a_{j}$, $\text{Re}\sqrt{z-a_{j}}\hat{f}(z)\sim c_{j}$. Thus we obtain (4.6)
since the difference of both sides is zero by the uniqueness of the boundary
value problem ([CHI15, Lemma 2.9], i.e. the massless version of Proposition
17).

Now we show the existence of $s$. Define
$s_{0}:=\mathfrak{C}\left[\partial_{\bar{z}}f_{m}/f_{m}\right]$. Note that

(4.7) $\displaystyle\frac{\partial_{\bar{z}}f_{m}}{f_{m}}$
$\displaystyle=\frac{\partial_{\bar{z}}f_{\left[\Omega,a,b\right]}(z|m)}{f_{\left[\Omega,a,b\right]}(z|m)}=m\left(\frac{\overline{f_{\left[\Omega,a,b\right]}}(z|m)}{f_{\left[\Omega,a,b\right]}(z|m)}\right)$

is in $L^{p}(\Omega)$ for any $p$ since $f_{m}$ only vanishes at isolated
points, so $s_{0}$ is bounded by Proposition 26.

We claim $s_{0}$ is differentiable almost everywhere in $\Omega$. $s_{0}$ is
differentiable at any point $z_{0}$ near which
$\partial_{\bar{z}}\bar{f}_{m}/f_{m}$ satisfies a Hölder condition, since
$\mathfrak{C}_{\Omega}\left[\partial_{\bar{z}}f_{m}/f_{m}\right]$ is the sum
of $\mathfrak{C}_{\Omega\setminus
B_{r}(z_{0})}\left[\partial_{\bar{z}}f_{m}/f_{m}\right]$, which is holomorphic
at $z_{0}$, and
$\mathfrak{C}_{B_{r}(z_{0})}\left[\partial_{\bar{z}}f_{m}/f_{m}\right]$, which
is differentiable by Proposition 26. Since $f_{m}$ is smooth on
$\left[\Omega,a_{1},\ldots,a_{n}\right]$, $s_{0}$ is differentiable away from
$a_{j}$ and at isolated points where $f_{m}$ vanishes.

So $e^{-s_{0}}f_{m}$ is holomorphic almost everywhere, and by removable
singularity it is holomorphic on $\left[\Omega,a_{1},\ldots,a_{n}\right]$. Now
let $s_{1}$ be a holomorphic function on $\Omega$ with boundary data
$\text{Im}s_{1}=\text{Im}s_{0}$ on $\partial\Omega$. We fix
$\text{Re}e^{s_{1}-s_{0}}=1$. Then $s:=s_{0}-s_{1}$ satisfies the desired
properties.

For the Hölder estimate, recall that $\text{diam}\Omega\leq 2M$ and thus
$\left|\partial_{\bar{z}}f_{m}/f_{m}\right|_{L^{p}(\Omega)}\leq
cst\cdot\left|m\right|M^{\frac{2}{p}}$. Then from Proposition 26 we have
$\left|s_{0}\right|_{C^{\alpha}}\leq
cst\cdot(1+M^{\alpha})\cdot\left|m\right|M^{1-\alpha}$. Then the corresponding
norm for $s_{1}$ is given by Proposition 27, and the sum gives the desired
estimate. ∎

The above lemma allows us to give the following results in the case $n=2$.

######  Lemma 29.

$\left|\mathcal{B}_{\Omega}\left(a_{1},a_{2}|m\right)\right|\to 1$ as
$\frac{\left|a_{1}-a_{2}\right|}{\text{dist}\left(\left\\{a_{1},a_{2}\right\\},\partial\Omega\right)}\to
0$.

###### Proof.

By [CHI15, Remark 2.24], the result holds for $m=0$.

Now consider the decomposition (4.6). Comparing the imaginary parts of the
coefficient of $\frac{1}{\sqrt{z-a_{2}}}$ in both sides, we see that
$\mathcal{B}_{\Omega}\left(a_{1},a_{2}|m\right)\text{Re}\left[e^{-s(a_{2})}\right]=\mathcal{B}_{\Omega}(a_{1},a_{2}|0)$.
Since $\text{Re}e^{-s(a_{1})}=1$ and $e^{-s}$ is Hölder continuous by the
previous lemma, we have
$\left|\frac{\mathcal{B}_{\Omega}\left(a_{1},a_{2}|m\right)}{\mathcal{B}_{\Omega}(a_{1},a_{2}|0)}\right|\to
1$ as $\left|a_{1}-a_{2}\right|\to 0$. ∎

Before we go on to give a more delicate estimate on the two-point observable,
we note a few facts we use.

######  Remark 30.

As mentioned above, the possible zeroes of $f_{m}$ are problematic for the
regularity of $s$. However, in the case where $n=2$, $f_{m}$ cannot vanish in
$\Omega$.

Indeed, $\hat{f}=e^{-s}f_{m}$ is a holomorphic function on
$\left[\Omega,a_{1},a_{2}\right]$ with the boundary condition
$\sqrt{\nu_{out}}\hat{f}\in\mathbb{R}$. By the argument principle for
solutions of a Hilbert boundary problem ([Wen92, Theorem 2.2]), one sees that
$1=\frac{1}{2}N_{\partial\Omega}+N_{\Omega}$, where $1$ is the index of
$\nu_{out}\cdot(z-a_{1})(z-a_{2})$ on $\partial\Omega$,
$N_{\partial\Omega},N_{\Omega}$ are number of zeroes of
$\hat{f}^{2}(z-a_{1})(z-a_{2})$ respectively on $\partial\Omega,\Omega$
counted with multiplicity. But since it is a square, any zero of
$\hat{f}^{2}(z-a_{1})(z-a_{2})$ is second order; the only possible scenario
then is that $N_{\partial\Omega}=2,N_{\Omega}=0$. So $\hat{f}$ does not vanish
in $\Omega$, and since $s$ is bounded, $f_{m}$ does not.

######  Remark 31.

Since $\hat{f}$ is a linear combination of the two observables with $m=0$, we
may estimate $f_{m}$ using properties of them. The massless observables are
conformally covariant: if $\phi:\Omega\to\Omega^{\prime}$ is a conformal map
[CHI15, Lemmas 2.9, 2.21],

$\displaystyle f_{\left[\Omega,a_{1},\ldots,a_{n}\right]}(z|0)$
$\displaystyle=f_{\left[\Omega^{\prime},\phi(a_{1}),\ldots,\phi(a_{n})\right]}(\phi(z)|0)\phi^{\prime}(z)^{1/2};$
(4.8) $\displaystyle f_{\left[\mathbb{H},a,b\right]}(z|0)$
$\displaystyle=\frac{\left(2i\text{Im}a\right)^{1/2}}{\left|b-\bar{a}\right|+\left|b-a\right|}\frac{\left[(\bar{b}-\bar{a})(\bar{b}-a)\right]^{1/2}(z-b)+\left[(b-a)(b-\bar{a})\right]^{1/2}(z-\bar{b})}{\sqrt{(z-a)(z-\bar{a})(z-b)(z-\bar{b})}}.$

Now we list some properties of the half-plane observable which need simple
verifications.

######  Lemma 32.

Suppose $a,b\in\mathbb{H}$ with $\text{Im}a=\text{Im}b=\epsilon$,
$\text{Re}a<\text{Re}b$ (i.e. $\left|b-a\right|=\text{Re}b-\text{Re}a$)

  1. (1)

Let $r\leq\frac{1}{2}\min(\epsilon,d)$. Then

$\displaystyle\left|\sqrt{z-a}f_{\left[\mathbb{H},a,b\right]}(z)\right|$
$\displaystyle\leq cst\text{ for }z\in B_{r}(a),$
$\displaystyle\left|\sqrt{z-b}f_{\left[\mathbb{H},a,b\right]}(z)\right|$
$\displaystyle\leq cst\text{ for }z\in B_{r}(b),$

with constants independent of the positions of $a,b,r$.

  2. (2)

Let $n_{a,b}\in\mathbb{R}$ be the zero of $f_{\left[\mathbb{H},a,b\right]}$.

$\displaystyle 0$ $\displaystyle\leq
n_{a,b}-n_{b,a}\leq\left|b-a\right|+2\epsilon;$
$\displaystyle\left|n_{a,b}-a\right|\geq\left|b-a\right|+\epsilon.$

  3. (3)

$\frac{f_{\left[\Omega,b,a\right]}(z|0)}{f_{\left[\Omega,a,b\right]}(z|0)}$
can be extended to a holomorphic function in $\Omega$, and

$\left|\frac{f_{\left[\Omega,b,a\right]}(a|0)}{f_{\left[\Omega,a,b\right]}(a|0)}\right|\leq
3;\text{
$\left|\left(\frac{f_{\left[\mathbb{H},b,a\right]}(z|0)}{f_{\left[\mathbb{H},a,b\right]}(z|0)}\right)^{\prime}_{z=a}\right|$$\leq\frac{cst}{\left|b-a\right|+\epsilon}.$}$

###### Proof.

These results all follow from the explicit formulae above.

  1. (1)

Note that $\left|b-a\right|\leq\left|b-\bar{a}\right|\leq
2\epsilon+\left|b-a\right|$. We have

$\displaystyle\left|\frac{\left[(\bar{b}-\bar{a})(\bar{b}-a)\right]^{1/2}}{\left|b-\bar{a}\right|+\left|b-a\right|}\right|$
$\displaystyle\leq\left|\frac{\left[\left|b-a\right|(\left|b-a\right|+2\epsilon)\right]^{1/2}}{2\left|b-a\right|+2\epsilon}\right|\leq\frac{\sqrt{\left|b-a\right|}}{\sqrt{2\left|b-a\right|+2\epsilon}}<1.$

Then we estimate the two terms in (4.8) separately, noting that
$\text{Im}a=\epsilon\leq|z-\bar{a}|$, and also
$\left|\sqrt{z-b}\right|\leq\left|\sqrt{z-\bar{b}}\right|$,

(4.9)
$\left|\frac{\left(2i\text{Im}a\right)^{1/2}(z-b)}{\sqrt{(z-a)(z-\bar{a})(z-b)(z-\bar{b})}}\right|\leq\left|\frac{\left(2i\text{Im}a\right)^{1/2}\sqrt{(z-b)}}{\sqrt{(z-a)(z-\bar{a})(z-\bar{b})}}\right|\leq\frac{cst}{\left|\sqrt{z-a}\right|}.$

Since $\left|z-\bar{b}\right|\leq\epsilon+\frac{|b-a|}{2}$,

$\left|\frac{\sqrt{\left|b-a\right|}}{\sqrt{2\left|b-a\right|+2\epsilon}}\frac{\left(2i\text{Im}a\right)^{1/2}(z-\bar{b})}{\sqrt{(z-a)(z-\bar{a})(z-b)(z-\bar{b})}}\right|\leq\left|\frac{cst\sqrt{\left|b-a\right|}}{\sqrt{(z-a)(z-b)}}\right|.$

If $z\in B_{r}(a)$, we have the result from
$\left|z-b\right|>\frac{1}{2}\left|a-b\right|$. If $z\in B_{r}(b)$, similarly
note $\left|z-a\right|\geq\frac{1}{2}\left|a-b\right|$, and apply
$\left|\sqrt{z-a}\right|>\left|\sqrt{z-b}\right|$ to (4.9).

  2. (2)

From the formula on $f_{\left[\mathbb{H},a,b\right]}$, we may write

$\displaystyle n_{a,b}$
$\displaystyle=\frac{\text{Re}\sqrt{(\bar{b}-\bar{a})(\bar{b}-a)}b}{\text{Re}\sqrt{(\bar{b}-\bar{a})(\bar{b}-a)}}=\text{Re}b-\frac{\text{Im}\sqrt{\bar{b}-a}\text{Im}b}{\text{Re}\sqrt{\bar{b}-a}},$
$\displaystyle n_{b,a}$
$\displaystyle=\text{Re}a-\frac{\text{Im}\sqrt{b-\bar{a}}\text{Im}a}{\text{Re}\sqrt{b-\bar{a}}}=\text{Re}a+\frac{\text{Im}\sqrt{\bar{b}-a}\text{Im}a}{\text{Re}\sqrt{\bar{b}-a}}.$

So

$\displaystyle 0\leq n_{a,b}-n_{b,a}$
$\displaystyle\leq\left|b-a\right|+2\epsilon,$

and

$\left|n_{a,b}-a\right|\geq(n_{a,b}-\text{Re}a)+\text{Im}a\geq\left|b-a\right|+\epsilon,$

since $\sqrt{\bar{b}-a}$ belongs to the fourth quadrant and thus

$0<-\frac{\text{Im}\sqrt{\bar{b}-a}}{\text{Re}\sqrt{\bar{b}-a}}\leq 1.$

  3. (3)

Again examining the formula, we have, away from $b,a$ (and then everywhere in
$\mathbb{H}$ by removable singularity),

$\frac{f_{\left[\mathbb{H},b,a\right]}(z|0)}{f_{\left[\mathbb{H},a,b\right]}(z|0)}=\frac{\text{Re}\left[(\bar{a}-\bar{b})(\bar{a}-b)\right]^{1/2}(z-n_{b,a})}{\text{Re}\left[(\bar{b}-\bar{a})(\bar{b}-a)\right]^{1/2}(z-n_{a,b})}=:\rho\frac{z-n_{b,a}}{z-n_{a,b}},$

so

$\displaystyle\left|\frac{f_{\left[\mathbb{H},b,a\right]}(a|0)}{f_{\left[\mathbb{H},a,b\right]}(a|0)}\right|$
$\displaystyle=\left|1+\frac{n_{a,b}-n_{b,a}}{a-n_{a,b}}\right|\leq
1+\frac{\text{$\left|b-a\right|$}+2\epsilon}{\left|b-a\right|+\epsilon}\leq
3;$
$\displaystyle\left|\left(\frac{f_{\left[\mathbb{H},b,a\right]}(z|0)}{f_{\left[\mathbb{H},a,b\right]}(z|0)}\right)^{\prime}_{z=a}\right|$
$\displaystyle=\text{$\left|\frac{n_{a,b}-n_{b,a}}{\left(a-n_{a,b}\right)^{2}}\right|\leq\frac{\left|b-a\right|+2\epsilon}{\left(\left|b-a\right|+\epsilon\right)^{2}}\leq\frac{cst}{\left|b-a\right|+\epsilon}.$}$

∎

Recall that we fix a conformal map $\varphi:\Omega\to\mathbb{D}$ such that
$M^{-1}\leq\varphi^{\prime}\leq M$. Now fix the standard conformal map
$\mathbb{D}\to\mathbb{H}$ such that $0\in\mathbb{D}$ is mapped to
$i\in\mathbb{H}$, and fix $\varphi_{\mathbb{H}}:\Omega\to\mathbb{H}$. Similar
estimate $cst(R)\cdot M^{-1}\leq\varphi_{H}^{\prime}\leq Cst(R)\cdot M$ holds
in $\varphi^{-1}\left(B_{R}\cap\mathbb{H}\right)$ for $R>0$. Write
$\mathcal{A}_{\Omega}^{1}(a_{1},a_{2}|m)+i\mathcal{A}_{\Omega}^{i}(a_{1},a_{2}|m)=:\mathcal{A}_{\Omega}(a_{1},a_{2}|m)$.

######  Lemma 33.

Let
$a_{1},a_{2}\in\varphi_{\mathbb{H}}^{-1}(B_{R}\cap\mathbb{H})\subset\Omega$ be
such that
$\text{Im}\varphi_{\mathbb{H}}(a_{1})=\text{Im}\varphi_{\mathbb{H}}(a_{2})=\epsilon$.
Then for any fixed $0<\gamma<1$, we have

$\left|\mathcal{A}_{\Omega}(a_{1},a_{2}|m)-\mathcal{A}_{\Omega}(a_{1},a_{2}|0)\right|\leq
cst\cdot\left(\epsilon^{-\gamma}+\left|a_{1}-a_{2}\right|^{-\gamma}\right),$

where the constant only depends on $M,R,m,\gamma$.

###### Proof.

We will refer to quantities only depending on $M,R,m,\gamma$ as constants in
this proof. To extract the desired difference
$\mathcal{A}^{\Delta}:=\mathcal{A}_{\Omega}(a_{1},a_{2}|m)-\mathcal{A}_{\Omega}(a_{1},a_{2}|0)$,
we will study
$e^{-2m\left|z-a_{1}\right|}f_{m}(z)/f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)$.
Indeed, around $a_{1}$, we have

$\displaystyle\frac{e^{-2m\left|z-a_{1}\right|}f_{m}(z)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)}$
$\displaystyle=\frac{1}{e^{2m\left|z-a_{1}\right|}}\frac{\left(\frac{e^{2m\left|z-a_{1}\right|}}{\sqrt{z-a_{1}}}+2\mathcal{A}_{\Omega}(a_{1},a_{2}|m)\sqrt{z-a_{1}}+o(\left|z-a_{1}\right|^{1/2})\right)}{\left(\frac{1}{\sqrt{z-a_{1}}}+2\mathcal{A}_{\Omega}(a_{1},a_{2}|0)\sqrt{z-a_{1}}+o(\left|z-a_{1}\right|^{1/2})\right)}$
$\displaystyle=1+2\mathcal{A}^{\Delta}\left(z-a_{1}\right)+o(\left|z-a\right|),$

therefore $\mathcal{A}^{\Delta}$ is half the derivative of
$\frac{e^{-2m\left|z-a_{1}\right|}f_{m}(z)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|m)}$
at $a_{1}$.

By (4.6),

$\frac{e^{-2m\left|z-a_{1}\right|}f_{m}(z|m)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)}=e^{s(z)-2m\left|z-a_{1}\right|}\left(1+c_{2}\frac{f_{\left[\Omega,a_{2},a_{1}\right]}(z|0)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)}\right),$

and we will estimate the derivatives of the two factors separately. Fix
$\alpha=1-\gamma$.

For the second factor, note that
$\left|c_{2}\right|=\left|\text{Im}e^{-s(a_{2})}\mathcal{B}_{\Omega}(a_{1},a_{2}|m)\right|\leq\left|\text{Im}e^{-s(a_{2})}\right|$,
but since $s$ is uniform $\alpha$-Hölder continuous and purely real on
$\partial\Omega$, $\left|\text{Im}e^{-s(a_{2})}\right|\leq
cst\cdot\text{dist}(a_{2},\partial\Omega)^{\alpha}\leq
cst\cdot\left(M\epsilon\right)^{\alpha}$. Now by conformal covariance,
$\frac{f_{\left[\Omega,a_{2},a_{1}\right]}(z|0)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)}=\frac{f_{\left[\Omega,\varphi_{\mathbb{H}}(a_{2}),\varphi_{\mathbb{H}}(a_{1})\right]}(\varphi_{\mathbb{H}}(z)|0)}{f_{\left[\Omega,\varphi_{\mathbb{H}}(a_{1}),\varphi_{\mathbb{H}}(a_{2})\right]}(\varphi_{\mathbb{H}}(z)|0)}$,
so we may apply the third estimate in Lemma 32:

$\displaystyle
1+c_{2}\frac{f_{\left[\Omega,a_{2},a_{1}\right]}(a_{1}|0)}{f_{\left[\Omega,a_{1},a_{2}\right]}(a_{1}|0)}$
$\displaystyle\leq 1+cst\cdot(M\epsilon)^{\alpha}\leq cst,$ (4.10)
$\displaystyle\left(1+c_{2}\frac{f_{\left[\Omega,a_{2},a_{1}\right]}(z|0)}{f_{\left[\Omega,a_{1},a_{2}\right]}(z|0)}\right)_{z=a_{1}}^{{}^{\prime}}$
$\displaystyle\leq\frac{cst\cdot
M^{\alpha+1}\epsilon^{\alpha}}{\left|a_{2}-a_{1}\right|+\epsilon}\leq cst\cdot
M^{\alpha+1}\epsilon^{\alpha-1}.$