# Piecewise Temperleyan dimers and a multiple SLE8

Nathanaël Berestycki and Mingchang Liu University of Vienna,
<EMAIL_ADDRESS>University,
<EMAIL_ADDRESS>
###### Abstract

We consider the dimer model on piecewise Temperleyan, simply connected
domains, on families of graphs which include the square lattice as well as
superposition graphs. We focus on the spanning tree $\mathcal{T}_{\delta}$
associated to this model via Temperley’s bijection, which turns out to be a
Uniform Spanning Tree with singular alternating boundary conditions.
Generalising the work of the second author with Peltola and Wu [LPW21] we
obtain a scaling limit result for $\mathcal{T}_{\delta}$. For instance, in the
simplest nontrivial case, the limit of $\mathcal{T}_{\delta}$ is described by
a pair of trees whose Peano curves are shown to converge jointly to a multiple
SLE8 pair. The interface between the trees is shown to be given by an
SLE${}_{2}(-1,\ldots,-1)$ curve. More generally we provide an equivalent
description of the scaling limit in terms of imaginary geometry. This allows
us to make use of the results developed by the first author and Laslier and
Ray [BLR20]. We deduce that, universally across these classes of graphs, the
corresponding height function converges to a multiple of the Gaussian free
field with boundary conditions that jump at each non-Temperleyan corner. After
centering, this generalises a result of Russkikh [Rus18] who proved it in the
case of the square lattice. Along the way, we obtain results of independent
interest on chordal hypergeometric SLE8; for instance we show its law is equal
to that of an SLE${}_{8}(\bar{\rho})$ for a certain vector of force points,
conditional on its hitting distribution on a specified boundary arc.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Piecewise Temperleyan domain and Temperley’s bijection
    2. 2.2 Generalisation
    3. 2.3 Coupling of flow lines, counterflow lines and GFF
  3. 3 Coupling of the continuous tree and GFF when $n=2$
    1. 3.1 Convergence of $\gamma_{\delta}$ and discrete Peano curves
    2. 3.2 Convergence of the discrete trees
  4. 4 General case
    1. 4.1 Tightness of the boundary branches
    2. 4.2 Identification of the limit
  5. 5 Convergence of the winding field

##  1 Introduction

The dimer model is one of the simplest but also most intriguing models of
statistical mechanics. Introduced in the pioneering work of Temperley and
Fisher [TF61] and, independently and nearly simultaneously, Kasteleyn [Kas61],
in the 1960s, the model can be defined on any finite weighted graph $G=(V,E)$
admitting a perfect matching, and is the probability measure on perfect
matchings $\mathbf{m}$ of $G$ given by

$\mu(\textbf{m})=\frac{1}{Z}\prod_{e\in\mathbf{m}}w_{e}$ (1.1)

where $(w_{e})_{e\in E}$ denote the edge weights of $G$. The dimer model is
particularly well studied in the case where $G$ is planar and bipartite, in
which case it is in some sense exactly solvable. Nevertheless, despite
remarkable progress over the last sixty years, the dimer model on large but
finite graphs (say, on subdomains of the plane) remains notoriously difficult
to study due to its extreme sensitivity to the microscopic details of the
boundary conditions.

The dimer model is typically studied through its height function,
$h:F\to\mathbb{R}$ introduced by Thurston and defined on the set of faces $F$
of the graph. The height function’s definition depends on the choice of a
reference dimer configuration $\mathbf{m}_{0}$ or more generally reference
flow, and even then is defined only up to constants. However, the centered
height function $h-\mathbb{E}(h)$ is canonically associated with the dimer
model. See [Ken09] for references and relevant definitions. This height
function turns the dimer model into a model of random surfaces and the main
question concern its large scale behaviour. A remarkable conjecture of Kenyon
and Okunkov predicts that the large scale behaviour is in great generality
described by the Gaussian free field, albeit after a possibly nontrivial (and
not conformal) change of coordinates. This conjecture was proved in the case
of Temperleyan boundary conditions in the landmark paper of Kenyon [Ken00],
[Ken01] and in a handful of other cases [Rus18, BLR20, BLR19, Pet15]; see
[Gor21] and references therein for an introduction in the case of lozenge
tilings.

The Temperleyan boundary conditions introduced and studied by Kenyon in
[Ken00] are perhaps the most canonical from the combinatorial point of view;
they appear naturally both in the methods involving the discrete
holomorphicity of the inverse Kasteleyn matrix, and for methods involving the
Temperley bijection as studied say in [BLR20, BLR19, BLR22] and which will be
relevant to this article. It is worth noting that they are in some sense
_trivial_ from the point of view of the phenomenology of the dimer model,
since they exclude the so-called frozen and gaseous phases, and lead to GFF
fluctuations without change of coordinates.

In this paper, we will consider the dimer model in $2n$-black-piecewise
Temperleyan domains on the square lattice (or simply $2n$-piecewise
Temperleyan in the following), which we will introduce precisely in Section
2.1. These boundary conditions, first considered in the work of Russkikh
[Rus18], are perhaps the simplest way to relax the Temperleyan condition on
the boundary. By definition, in such a domain there is a total of $2n$ white
corners for some $n\geq 2$, of which necessarily $n-1$ are concave, and $n+1$
convex (see Figure 2.1 for an illustration). We will make an additional
assumption on these domains: between two consecutive concave white corners of
$\Omega$ (i.e., not separated by intermediate white corners), the boundary
black squares belong to a fixed type of black vertices, which we take to be
$\blacksquare_{0}$.

This can always be assumed without loss of generality if $n=2,3$ but not if
$n\geq 4$ (in that case there exist examples of $2n$-piecewise Temperleyan
domains which do not satisfy this additional condition). In fact our results
will be valid much more generally for graphs obtained by planar superposition
satisfying very mild assumptions, see Section 2.2 for a precise description.
For such graphs we will formulate a natural analogue of the boundary
conditions described above (piecewise Temperleyan boundary together with the
additional assumption). Our main result, valid both for the square lattice and
more generally for superposition graphs defined in Section 2.2, is as follows.

######  Theorem 1.1.

Suppose $\Omega$ is a simply connected domain with $C^{1}$ simple boundary and
suppose that $\\{\Omega_{\delta}\\}_{\delta>0}$ is a sequence of $2n$-black-
piecewise Temperleyan domains satisfying the above assumption, which converges
to $\Omega$. We denote by $h_{\delta}$ the height function of the dimer model
on $\Omega_{\delta}$ and denote by $h$ the Gaussian free field on $\Omega$
with Dirichlet boundary conditions. Define $\chi:=\frac{1}{\sqrt{2}}$. Then,
we have for every $f\in C_{c}^{\infty}(\Omega)$, and $k\geq 1$, we have

$\mathbb{E}\left[\left|(h_{\delta}-\mathbb{E}[h_{\delta}],f)-\frac{1}{\chi}(h,f)\right|^{k}\right]\to
0.$

See (4.1) for the precise notion of convergence of $\Omega_{\delta}$ to
$\Omega$. In the case of the square lattice, this result had already been
proved by Russkikh [Rus18]. Russkikh’s proof follows the method developed in
the groundbreaking work of Kenyon [Ken00, Ken01] which relies on the analysis
of the inverse Kaseteleyn matrix. In the Temperleyan setting, Kenyon had shown
that the inverse Kasteleyn matrix converges in a suitable to the holomorphic
function associated with the derivative of the Green function with Dirichlet
boundary conditions. These eventually lead Dirichlet boundary conditions for
the field. By contrast, in the piecewise Temperleyan case analysed by
Russkikh, the inverse Kasteleyn matrix corresponds to a Green function with
mixed Neumann–Dirichlet boundary conditions. Yet surprisingly, as a result of
fairly striking cancellations, she demonstrated that the limiting centered
height field has purely Dirichlet boundary conditions, as is the case in
Theorem 1.1. We shed some light on this phenomenon, showing in a precise
geometric way how the presence of non-Temperleyan corners influences the
limiting geometry of the model.

This geometric phenomenon can in fact also be formulated purely in terms of
height functions. This relies on the connection to imaginary geometry
initiated in [BLR20], and which we now begin to discuss. As mentioned above,
$h_{\delta}-\mathbb{E}[h_{\delta}]$ does not depend on the choice of a
reference flow. There is however _one_ particular choice of reference flow
which turns out to be quite canonical, and which corresponds to the normalised
winding of the edges in the graph (see e.g. [KS04, BLR16] or [BLR19] for
definitions). For this choice, it will turn out that $\mathbb{E}[h_{\delta}]$
converges to a harmonic function $u$ (which will be defined in Section 2.3)
locally uniformly. This harmonic function will give the boundary values of the
Gaussian free field. In contrast to the purely Temperleyan case, these
boundary values are now nonconstant, and have jumps at each non-Temperleyan
corner. Thus the effect of the non-Temperleyan boundary conditions is to
create jumps at the corner (these jumps can already be seen to occur at the
discrete level). Of course, when we subtract the expectation, the
corresponding field becomes purely Dirichlet, and this cancels the effect of
the non-Temperleyan corners. This explains why the limiting field in
Russkikh’s work [Rus18] is the Dirichlet Gaussian free field, just as in the
purely Temperleyan case.

#### Temperley’s bijection.

To formulate this geometrically, we make use of Temperley’s bijection which
associates to any dimer configuration a spanning tree on a related graph. Let
$\mathcal{T}_{\delta}$ denote the random tree associated to the dimer
configuration $\mathbf{m}$ of (1.1) in a piecewise-Temperleyan domain (to be
precise, $\mathcal{T}_{\delta}$ is the $\blacksquare_{0}$-tree; Temperley’s
bijection also associates to $\mathbf{m}$ a tree living on the
$\blacksquare_{1}$-lattice, but this is simply the planar dual of
$\mathcal{T}_{\delta}$). Let $u$ denote the harmonic function from $\Omega$
which jumps alternatively by $\pm\lambda$ at each non-Temperleyan corner of
$\partial\Omega$ (see Figure 2.3 for an illustration): that is, let us write
the $2n$ white corners as $x_{1},\ldots,x_{2n}$ as we go along the boundary in
counterclockwise order, with both $x_{1}$ and $x_{2n}$ assumed to be convex
(recall there are two more convex corners than concave). Assume first that
$\Omega=\mathbb{H}$. We define $u=u_{\mathbb{H}}$ on the real line to be
$+\lambda$ on $(-\infty,x_{1})$ and then jump by $-\lambda$ on each convex
corner, and $+\lambda$ at each concave corner. Note that $u$ will be equal to
$-\lambda$ on $(x_{2n},\infty)$. Then $u_{\mathbb{H}}$ is simply the harmonic
extension of this boundary data. When $\Omega$ is not the upper-half plane,
$u$ is obtained by applying the conformal change of coordinates of imaginary
geometry associated with $\kappa=2$, that is,

$u_{\Omega}=u_{\mathbb{H}}\circ\varphi^{-1}-\chi\arg(\varphi^{-1})^{\prime};$

where $\varphi:\mathbb{H}\to\Omega$ is a conformal mapping and
$\chi=2/\sqrt{2}-\sqrt{2}/2=1/\sqrt{2}$.

######  Theorem 1.2.

As the mesh size $\delta\to 0$, the Temperleyan tree $\mathcal{T}_{\delta}$
converges to the tree consisting of the flow lines for $\kappa=2$ of the field
$h_{\Omega}+u_{\Omega}$, where $h_{\Omega}$ is a $\operatorname{GFF}$ with
Dirichlet boundary conditions on $\Omega$.

As announced above, note that the effect of the white corners is thus to
change the boundary data of the limiting field (but, as already mentioned,
this disappears when we take away the expectation). We also obtain a more
direct, geometric description of the scaling limit of $\mathcal{T}_{\delta}$
in the particular case when $n=2$, which is the simplest non trivial case. In
that case there are four white corners which we denote respectively by
$a,b,c,d$. Consistent with our earlier notation, the single white concave
corner is denoted by $b$. It can be shown that, deterministically, the arcs
$(ab),\ldots,(da)$ of the topological quadrilateral $(\Omega,a,b,c,d)$
alternate between Dirichlet and Neumann portions for $\mathcal{T}_{\delta}$ in
a precise sense. Furthermore, $\mathcal{T}_{\delta}$ contains a unique curve,
which we call $\gamma_{\delta}$, starting from $b$ and ending in a random
location $z$ on the arc $(cd)$. This curves splits $\mathcal{T}_{\delta}$ into
two trees, which we call $\mathcal{T}_{\delta}^{L},\mathcal{T}_{\delta}^{R}$
respectively (left and right correspond to the planar orientation inherited
from $\gamma_{\delta}$ viewed as a curve oriented from $b$ to $(cd)$). To this
pair of trees we can associate a pair of Peano curves, which we call
$\eta_{\delta}^{L}$ and $\eta_{\delta}^{R}$ respectively. These curves emanate
from $d$ and $c$ respectively, and terminate in $a$ and $b$ respectively. See
Figure 1.1 for an illustration. We now state a scaling limit result describing
the limiting law of these random curves.

Figure 1.1: Setup of Theorem 1.3. In the limit, we get a multiple SLE8
separated by a chordal SLE${}_{2}(-1,\ldots,-1)$ interface.

######  Theorem 1.3.

As $\delta\to 0$, the random curves
$\gamma_{\delta},\eta^{L}_{\delta},\eta^{R}_{\delta}$ converge jointly to a
triple $(\gamma,\eta^{L},\eta^{R})$. The interface $\gamma$ between the two
trees has the law of a chordal SLE${}_{2}(-1,-1,-1,-1)$ from $b$, stopped when
hitting the arc $(cd)$ (call $z$ this hitting point). Given $\gamma$,
$\eta^{R}$ and $\eta^{L}$ are independent standard chordal SLE8 in their
respective domains. Hence the joint law of $(\eta^{L},\eta^{R})$ is a multiple
SLE8 in the sense that given $\eta^{L}$, the curve $\eta^{R}$ is simply a
standard SLE8 in $\Omega\setminus\eta^{L}$, from $c$ to $b$ (and the same
holds for $\eta^{L}$ given $\eta^{R}$).

#### Marginal laws.

The marginal laws of $\eta^{L}$ and $\eta^{R}$ can also be described:
$\eta^{L}$ has the law of a chordal SLE${}_{8}(2,-2)$ in $\Omega$ from $d$ to
$a$ with force points at $c$ (corresponding to weight $\rho=2$) and $b$
(corresponding to weight $\rho=-2$). Similarly, $\eta^{R}$ has the law of a
chordal SLE${}_{8}(2,2)$ in $\Omega$ from $c$ to $b$, with force points at $d$
and $a$. Note that in this theorem, the target point of $\gamma$ needs not be
specified, as given the value of the force points (four force points of weight
$\rho_{i}=-1$, summing up to a total weight of
$\sum_{i}\rho_{i}=-4=\kappa-6$), the curve has the locality property and is
hence target independent.

#### Multiple SLE.

A joint law on curves $(\eta_{1},\eta_{2})$ is called multiple (say chordal)
SLEκ in a domain $D$, if the conditional law of $\eta_{i}$ given $\eta_{2-i}$
is a chordal SLEκ in the complementary domain. When $\kappa\leq 4$ such laws
are known to be unique by a result of Beffara, Peltola and Wu [BPW21]. This is
however not known when $\kappa>4$. In fact, for $\kappa\geq 8$ there can be no
uniqueness as this is the space-filling regime: any choice of a law for the
interface $\gamma$ separating the two curves gives rise to a pair
$(\eta_{1},\eta_{2})$ of multiple SLEs in the sense of this definition. We do
not know how to canonically characterise the law of the multiple SLE8
$(\eta^{L},\eta^{R})$ appearing in the theorem, except by specifying the
marginals as we have done.

#### Connection to hypergeometric SLE8.

It is useful at this point to make a comparison with the paper [HLW20], in
which Uniform Spanning Trees with related alternating boundary conditions were
also studied. To explain the model they considered, let $(\Omega,a,b,c,d)$
denote a topological rectangle. The boundary conditions divide the boundary
into four arcs: $(ab),(bc),(cd),(da)$. They impose Neumann (also known as
reflecting or free) boundary conditions on the arcs $(bc)$ and $(da)$, while
the arcs $(ab)$ and $(cd)$ are each wired – but _not_ wired to each other.
They then consider a uniform spanning tree $\tilde{\mathcal{T}}_{\delta}$ with
these boundary conditions, if the mesh size is $\delta>0$. Note that since
$(ab)$ and $(cd)$ are not wired with one another, this forces the presence of
a path (let us call it $\tilde{\gamma}_{\delta}$) connecting $(ab)$ to $(cd)$.
The Temperleyan tree $\mathcal{T}_{\delta}$ associated to a piecewise
Temperleyan dimer model in the case $n=2$ can therefore be viewed as an
asymptotically degenerate version of $\tilde{\mathcal{T}}_{\delta}$ in which
the interface $\tilde{\gamma}_{\delta}$ is conditioned to start in
$b_{\delta}$. One of the main results of [HLW20] is that the pair of discrete
Peano curves $(\tilde{\eta}^{L}_{\delta},\tilde{\eta}^{R}_{\delta})$ on either
side of $\tilde{\gamma}_{\delta}$ in $\tilde{\mathcal{T}}_{\delta}$ converges
to a scaling limit given by a multiple SLE8 pair
$(\tilde{\eta}^{L},\tilde{\eta}^{R})$, but whose marginal law is the so-called
_hypergeometric_ SLE8 with parameter $\nu=0$ (whose driving function will be
described later in (3.10))111We warn the reader that two related notions, both
bearing the name of hypergeometric SLE and the notation hSLE, have appeared
almost simultaneously and independently in two different papers by two
different authors, namely Wei Qian in [Qia18] and Hao Wu in [Wu20]. In both
cases the driving function is given by the same formula involving a
hypergeometric function. To describe the curve unambiguously, one must
additionally specify the value of the three parameters $a,b$ and $c$ entering
the hypergeometric function. In [Qia18] this choice is only explicit for the
value $\kappa=8/3$, which is the main focus of that paper.
In addition, the hypergeometric SLEκ with parameter $\nu$ (in the notations of
[Wu20]) coincides with the so-called _intermediate_ SLEκ introduced earlier by
Dapeng Zhan in [Zha10, Section 3], with the minor difference that there the
range of values of $\kappa$ was restricted to $\kappa\in(0,4)$ and
$\nu\geq\kappa/2-2$ (in fact, when $\nu\leq\kappa/2-6$, the choice of
parameters in the definition of hypergeometric SLE${}_{\kappa}(\nu)$
considered in [Wu20] is different from the case where $\nu\geq\kappa/2-6$;
thus the corresponding hypergeometric SLE${}_{\kappa}(\nu)$ is in fact
different from the intermediate SLEs considered in [Zha10] in that case). For
the avoidance of doubt, in this paper we are concerned with the case
$\kappa=8$, and we will use the word “hypergeometric SLE8” and the notation
hSLE8 to refer to the hypergeometric SLE8 with parameter $\nu=0$ in the
notations of [Wu20]. Once again, this corresponds to Zhan’s intermediate SLE
generalised to $\kappa=8$, and with the parameter $\rho$ from [Zha10] taken to
be $\rho=2$. We thank Wei Qian and Hao Wu for discussions regarding this..
Combining our results with those of [HLW20], we are therefore able to obtain
some simple descriptions of this hypergeometric SLE8 once we condition on the
hitting point of the opposite arc. For instance, consider the curve
$\tilde{\eta}^{R}$ from $c$ to $b$ and condition on the hitting position of
the arc $(ab)$. Despite the apparent complexity of the driving function, once
we condition on the hitting point of the arc $(ab)$ by the curve, then hSLE8
reduces to a more standard SLE${}_{8}(\bar{\rho})$ where the weight vector
$\bar{\rho}$ can be explicitly described.

######  Theorem 1.4.

Suppose $\tilde{\eta}^{R}$ has the law of the hypergeometric
$\operatorname{SLE}_{8}$ in $(\Omega;a,b,c,d)$ from $c$ to $b$ with force
points $d$, $a$. We denote by $T$ the hitting time by $\tilde{\eta}^{R}$ of
$(ab)$. Then, given $\eta^{R}(T)=z$, the conditional distribution of
$\tilde{\eta}^{R}([0,T])$ equals the law a chordal
$\operatorname{SLE}_{8}(2,2,-4)$ up to time $T$ in $\Omega$ from $c$ to $b$,
with marked points $d$, $a$ and $z$ respectively. Furthermore, conditionally
on $\tilde{\eta}^{R}(T)=z$ and on $\tilde{\eta}^{R}([0,T])$, $\eta^{R}$
evolves after time $T$ as a chordal SLE8 from $z$ to $b$ in the domain
$\Omega\setminus\tilde{\eta}^{R}([0,T])$.

Given the above result, it is natural to ask if other hypergeometric SLEs
(still in the sense of [Wu20]) can be described in terms of
SLE${}_{\kappa}(\rho)$ after conditioning. This will be addressed in future
work, see [Liu23]; see also the fourth bullet point of Remark 3.12 for
additional discussion. We conclude by noting that Qian [Qia18] was able to
relate the law of the hypergeometric SLE to tilted SLE${}_{\kappa}(\rho)$
processes via a Girsanov change of measure, see Section 4.3.2 of [Qia18].
Nevertheless Theorem 1.4 is new to the best of our knowledge; for instance the
numbers of force points here and in Section 4.3.2 of [Qia18] are not
identical.

#### Acknowledgements.

Nathanaël Berestycki’s research is supported by FWF grant P33083, “Scaling
limits in random conformal geometry”. This work took place while Mingchang Liu
was visiting the University of Vienna, whose hospitality is gratefully
acknowledged. This visit was made possible thanks to the support of the State
Scholarship Fund No.202106210235 from the Chinese government. N.B. also thanks
Benoît Laslier and Marianna Russkikh for many useful discussions which took
place prior to this work.

##  2 Preliminaries

###  2.1 Piecewise Temperleyan domain and Temperley’s bijection

In this subsection, we will recall the definition of black-piecewise
Temperleyan domain and establish the corresponding Temperleyan’s bijection
between dimer configurations and spanning trees.

First of all, let us recall the definition of black-piecewise Temperleyan
domain, which was introduced in [Rus18]. Suppose $\Omega\subset\mathbb{C}$ is
a simply connected domain, whose boundary $\partial\Omega$ is a simple curve
on $\mathbb{Z}^{2}$. $\Omega$ can be divided into squares of size one which
inherit from $\mathbb{Z}^{2}$ the checkboard black-white colouring. It is
further useful to subdivide the black squares into two distinct sublattices:
more precisely, denote by $\blacksquare_{0}$ the black squares on even rows
and denote by $\blacksquare_{1}$ the black color on odd rows. We call $\Omega$
is a $2n$-black-piecewise Temperleyan domain, if it has $2n$ white corners. It
can then be shown that of those, necessarily $n+1$ are convex and $n-1$ are
concave corners. If $n\geq 3$, we assume that between two consecutive concave
white corners of $\Omega$ (i.e., not separated by intermediate white corners),
the boundary black squares belong to $\blacksquare_{0}$. See the left hand
side of Figure 2.1 for an example of black-piecewise Temperleyan domain in the
case $n=2$. When $n\geq 3$, it is possible for two consecutive white corners
to be concave, but the above assumption imposes restrictions on the
combinatorics, as we will see below.

Figure 2.1: On the left is a $4$-black-piecewise Temperleyan domain. The black
squares belong to type $\blacksquare=\blacksquare_{1}$ and the grey squares
belong to type
${{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>The middle picture shows the graph $\Omega_{0}^{*}$. The right picture shows
$\Omega_{0}^{*}$ and its outer boundary, which result in the graph
$\overline{\Omega}_{0}^{*}$. The boundary
$\overline{\Omega}_{0}^{*}\setminus\Omega_{0}^{*}$ is shown in colour and
consists of two components, coloured respectively red and blue.

Now, we introduce the Temperleyan’s bijection when $\Omega$ is a $2n$-black-
piecewise Temperleyan domain. Define $\Omega^{*}$ to be the dual graph of
$\Omega$. The vertices of $\Omega^{*}$ are the centers of the squares of
$\Omega$ and the edges are straight lines connecting nearest pairs of
vertices. We then say that a vertex of $\Omega^{*}$ is $\blacksquare_{0}$,
$\blacksquare_{1}$ or white, depending on the colour of the corresponding
square in $\Omega$. Define $\Omega_{0}^{*}$ to be the graph whose vertices are
the $\blacksquare_{0}$ vertices of $\Omega^{*}$ and whose edges are straight
lines connecting nearest pairs of vertices. Define $\overline{\Omega}^{*}_{0}$
to be the graph whose vertices are the union of $\Omega_{0}^{*}$ and
$\blacksquare_{0}$ vertices adjacent to $\Omega_{0}^{*}$ and whose edges are
straight lines connecting nearest pairs of vertices. The boundary
$\overline{\Omega}_{0}^{*}\setminus\Omega_{0}^{*}$ consists of $n$ connected
components. We call the endpoints of the components $x_{1},\ldots,x_{2n}$,
labelled in counterclockwise order in such a way that the connected components
are respectively formed by $(x_{2}x_{3}),\ldots,(x_{2n}x_{1})$. We can assume
without loss of generality that $x_{1}$ and $x_{2n}$ are adjacent to two
convex corners. (In general, $x_{i}$ will be adjacent to a white corner when
that corner is convex, and will be within distance 3 of that corner if that
corner is concave). See Figure 2.1 for an illustration. In fact, the arc
$(x_{2n}x_{1})$ is distinguished by the fact that it is the only connected
portion of the boundary $\overline{\Omega}_{0}^{*}\setminus\Omega_{0}^{*}$
connecting two vertices adjacent to a convex corner. In other for each $1\leq
i\leq n-1$, among $x_{2i}$ and $x_{2i+1}$ there is exactly one of these two
vertices which is adjacent to a convex corner (the other will be within
distance three of a concave white corner). This comes from our assumption that
between two consecutive concave corners, the black vertices are all of type
$\blacksquare_{0}$. As we see, this does not prevent concave white corners to
come consecutively, but in particular no three consecutive white corners can
exist.

We denote by $z_{1}^{+},\ldots,z_{n-1}^{+}$ the vertices of $\Omega^{*}_{0}$
adjacent to the concave white corners, also labelled in counterclockwise order
starting immediately after $x_{1}$ (so $z_{i}^{+}$ can be identified with a
black corner, and is on the outer boundary of $\Omega_{0}^{*}$). As already
mentioned, by our assumption, for $1\leq i\leq n-1$, there is exactly one
vertex of $\\{x_{2i},x_{2i+1}\\}$ which is adjacent to $z_{i}^{+}$. Let us
denote it by $z_{i}\in\\{x_{2i},x_{2i+1}\\}$. See Figure 2.1.

A dimer cover of $\Omega^{*}$, denoted by $\omega$, is a subset of the edges
of $\Omega^{*}$ such that each vertex belongs to exactly one edge in $\omega$.
Denote by $\mathcal{D}(\Omega^{*})$ the set of dimer covers of $\Omega^{*}$.
Given a dimer configuration $\omega\in\mathcal{D}(\Omega^{*})$, we now
associate to $\omega$ a tree on $\overline{\Omega}_{0}^{*}$ in the following
manner. Let us consider each edge $e$ in $\omega$ as directed from black to
white vertex, and let us extend it to the next black vertex along this
direction (i.e., “double” the oriented edge $e$). We only consider the edges
starting from $\blacksquare_{0}$ vertices. In this manner, we obtain a subset
of edges of $\overline{\Omega}^{*}_{0}$. We add the edges
$\cup_{i=1}^{n}(x_{2i}x_{2i+1})\cup\cup_{i=1}^{n-1}\\{z_{i}z_{i}^{+}\\}$ to
this subset and also direct the edges
$\cup_{i=1}^{n-1}(x_{2i}x_{2i+1})\cup\cup_{i=1}^{n-1}\\{z_{i}z_{i}^{+}\\}$
from the black vertices to white vertices. Note however that we do _not_
direct the edges along $(x_{2n}x_{1})$ (indeed no consistent orientation on
the last arc). Denote by $\omega^{*}$ the subset of edges we obtain, and note
that except on $(x_{2n}x_{1})$ out of every vertex in $\Omega_{0}^{*}$ there
is a single edge going out of this vertex in $\omega^{*}$. See Figure 2.2.

Figure 2.2: In the left picture, the set of dark green edges is a dimer
configuration. In the right picture, the green edges are from the dimer
configuration and the purple edges are added artificially.

Denote by $\mathcal{U}(\overline{\Omega}^{*}_{0})$ the subset of spanning
trees on $\overline{\Omega}^{*}_{0}$ satisfying the following conditions: the
spanning tree contains $\cup_{i=1}^{n}(x_{2i}x_{2i+1})$ and for every $1\leq
i\leq n-1$, the first step of the branch connecting $(x_{2i}x_{2i+1})$ to
$(x_{2n}x_{1})$ is through the edge $z_{i}z_{i}^{+}$. Equivalently, we could
also wire the boundary components $c_{1}=(x_{2}x_{3}),\ldots
c_{n}=(x_{2n}x_{1})$ and view the spanning tree as an oriented (or rooted)
spanning tree, rooted at $c_{n}=(x_{2n}x_{1})$, and such that the path
connecting $c_{i}$ to $c_{n}$ starts with the oriented edge $z_{i}z_{i}^{+}$.

######  Lemma 2.1 (Temperley’s bijection).

The map $\omega\mapsto\omega^{*}$ is a bijection between
$\mathcal{D}(\Omega^{*})$ and $\mathcal{U}(\overline{\Omega}_{0}^{*})$.

###### Proof.

First, we will show that $\omega^{*}$ is indeed an element in
$\mathcal{U}(\overline{\Omega}_{0}^{*})$. Since $\omega$ is a dimer cover, it
is clear that $\omega^{*}$ contains all the vertices of
$\overline{\Omega}_{0}^{*}$. We need to show that $\omega^{*}$ does not
contain loops. Suppose that $L$ is a loop on $\overline{\Omega}_{0}^{*}$ which
is contained in $\omega^{*}$. We denote by $\tilde{L}$ the union of $L$ and
the squares surrounded by $L$. By induction on the number $\blacksquare_{1}$
squares in $\tilde{L}$, we have that the number of squares in $\tilde{L}$ is
odd. Note that if $L$ belongs to $\omega^{*}$, the subgraph $\tilde{L}$ is
covered by edges in $\omega$. This implies that the number of squares in
$\tilde{L}$ should be even, which is a contradiction. Thus, the loop $L$ can
not belong to $\omega^{*}$. This implies that $\omega^{*}$ is a spanning
forest on $\overline{\Omega}_{0}^{*}$.

To show that $\omega^{*}$ is a tree, recall that starting from every vertex in
$\overline{\Omega}_{0}^{*}$, except on $c_{n}=(x_{2n}x_{1})$, there is a
single oriented edge in $\omega^{*}$ going out of that vertex. Following the
edges in the forward direction, we obtain a path which necessarily ends in
$c_{n}$. Thus every vertex is connected to $c_{n}$ and $\omega^{*}$ is in
$\mathcal{U}(\overline{\Omega}_{0}^{*})$ (i.e., is a spanning tree), as by
definition of $\omega^{*}$, this path connecting $c_{i}$ to $c_{n}$ starts
with the oriented edge $z_{i}z_{i}^{+}$.

Second, we will show that this is a bijection. Define the dual graph of
$\overline{\Omega}_{1}^{*}$ similarly as defining $\overline{\Omega}_{0}^{*}$
by considering the $\blacksquare_{1}$ squares. Note that every spanning tree
on $\overline{\Omega}_{0}^{*}$ determines a dual forest on the dual graph of
$\overline{\Omega}_{1}^{*}$. Since every dimer cover $\omega$ is determined by
$\omega^{*}$ and the corresponding dual forest, we deduce that the map
$\omega\mapsto\omega^{*}$ is injective. For every
$\omega^{*}\in\mathcal{U}(\overline{\Omega}_{0}^{*})$, it is clear that
dividing the edges of $\omega^{*}$ and the edges of the corresponding dual
forest, we will get a dimer cover on $\Omega$. Thus, the map
$\omega\mapsto\omega^{*}$ is surjective. This completes the proof. ∎

###  2.2 Generalisation

We now explain briefly how the setup considered in this paper can be
generalised to a much bigger class of planar graphs. Roughly these graphs are
obtained by superposing a planar graph and their dual (which is a setup in
which Temperley’s bijection is known to apply, see [KPW00]) but where we pay
attention to the choice of boundary conditions. Let
$\\{G_{\delta}\\}_{\delta>0}$ denote a sequence of graphs embedded in a domain
$\Omega$, with an edge outer-boundary forming a simple curve $\lambda$. Let us
also fix $n\geq 2$ and boundary arcs $(x_{2}x_{3}),\ldots,(x_{2n}x_{1})$ on
$\partial\Omega$. We let $x_{1}^{\delta},x_{2n}^{\delta}$ denote
approximations of $x_{1},\ldots,x_{2n}$ along $\lambda$ (and hence on
$V(G_{\delta})$). Let $c_{i}$ denote the (wired) arc $(x_{2i}x_{2i+1})$
(understood cyclically), and for each $1\leq i\leq n$, fix
$z_{i}\in\\{x_{2i},x_{2i+1}\\}$. As in Theorem 1.1 we assume that $\Omega$ is
a simply connected domain with simple $C^{1}$ boundary. In fact it would
suffice to assume that the reflecting arcs correspond to disjoint simple
$C^{1}$ curves.

Let $\mathcal{Z}=\cup_{i=1}^{n}\\{z_{i}\\}$.

Let $G^{*}_{\delta}$ denote the interior planar dual of $G_{\delta}$, and let
$G^{s}$ denote the superposition of $G_{\delta}$ and $G^{*}_{\delta}$. The
graph $G^{s}$ is bipartite, with $V(G^{s})=B^{s}\cup W^{s}$ with
$B^{s}=V(G_{\delta})\cup V(G^{*}_{\delta})$, and $W=E(G_{\delta})$, and there
is an edge between $e$ between $b\in B^{s}$ and $w\in W^{s}$ if and only if
$(bw)$ is part of a primal or dual edge of $G_{\delta}$. Finally, for each
$1\leq i\leq n$, we remove from the superposition graph $G^{s}$ the arcs
$c_{i}=(x_{2i}x_{2i+1})$, understood cyclically, and if $1\leq i\leq n-1$ we
also remove the edge $\\{z_{i}w_{i}\\}$, where $w_{i}$ is the unique white
vertex adjacent to $z_{i}$ on $\lambda$. We let $\mathcal{G}^{D}_{\delta}$ be
the resulting graph and call it the dimer graph.

To make the parallel with the previous section clearer, it will be useful to
call $z_{i}^{+}$ the black vertex adjacent to $w_{i}$ on $\lambda$ which is
not $z_{i}$, so $w_{i}\in\\{z_{i}z_{i}^{+}\\}$. We call a white vertex on the
boundary of $\mathcal{G}^{D}_{\delta}$ a concave corner if it is adjacent to
the arc $z_{i}^{+}$ for some $1\leq i\leq n-1$. We call it a convex corner if
it is adjacent to $c_{i}$, for $1\leq i\leq n$. Thus there are $n+1$ convex
white corners, and $n-1$ white concave corners.

We make the following assumptions about $G^{\delta}$. We may allow
$G^{\delta}$ to have oriented edges with weights (but all the edge weights on
$G^{*}_{\delta}$ will be set to one). A continuous time simple random walk
$\\{X_{t}\\}_{t\geq 0}$ on such a graph $G^{\delta}$ is defined in the usual
way: the walker jumps from $u$ to $v$ at rate $w(u,v)$ where $w(u,v)$ denotes
the weight of the oriented edge $(u,v)$. Furthermore, the walk is killed as
soon as it reaches a vertex from $G_{\delta}\setminus G^{-}_{\delta}$. Given a
vertex $u$ in $G^{\delta}$, let $\mathbb{P}_{u}$ denote the law of continuous
time simple random walk on $G^{\delta}$ started from $u$.

For $A\subset\mathbb{C}$, we denote by $A^{\delta}$ the set of vertices of
$G^{\delta}$ in $A$.

  1. 1.

(Bounded density) There exists $C$ such that for any $x\in\mathbb{C}$, the
number of vertices of $G^{\delta}$ in the square $x+[0,\delta]^{2}$ is smaller
than $C$.

  2. 2.

(Good embedding) The edges of the graph are embedded in such a way that they
are piecewise smooth, do not cross each other and have uniformly bounded
winding. Also, $0$ is a vertex.

  3. 3.

(Irreducible) The continuous-time random walk on $G^{\delta}$, is irreducible
in the sense that for any two vertices $u$ and $v$ in $G^{\delta}$,
$\mathbb{P}_{u}(X_{1}=v)>0$.

  4. 4.

(Invariance principle) The continuous time random walk $\\{X_{t}\\}_{t\geq 0}$
on $G^{\delta}$ started from $0$ satisfies:

${(X_{t/\delta^{2}})_{t\geq 0}}\xrightarrow[\delta\to
0]{(d)}(B_{\phi(t)})_{t\geq 0}$

where $(B_{t},t\geq 0)$ is a Brownian motion in $\Omega$ with normal
reflection along $(x_{1}x_{2})\cup\ldots\cup(x_{2n-1}x_{2n})$, started from
$0$, and $\phi$ is a nondecreasing, continuous, possibly random function
satisfying $\phi(0)=0$ and $\phi(\infty)=\infty$. The above convergence holds
in law in Skorokhod topology.

  5. 5.

(Uniform crossing estimate). Let $\mathcal{R}$ be the horizontal rectangle
$[0,3]\times[0,1]$ and $\mathcal{R}^{\prime}$ be the vertical rectangle with
same dimensions, and let $B_{1}:=B((1/2,1/2),1/4)$ be the _starting ball_ and
$B_{2}:=B((5/2,1/2),1/4)$ be the _target ball_. There exist constants
$\delta_{0}>0$ and $\alpha_{0}>0$ such that for all $z\in\mathbb{C},\delta>0$,
$\ell\geq 1/\delta_{0}$, $v\in\ell\delta B_{1}$ such that $v+z\in G^{\delta}$,

$\mathbb{P}_{v+z}(X\text{ hits }(\ell\delta B_{2}+z)\text{ before exiting
}(\ell\delta\mathcal{R}+z))>\alpha_{0}.$ (2.1)

The same statement as above holds for crossing from right to left, i.e., for
any $v\in\ell\delta B_{2}$, (2.1) holds if we replace $B_{2}$ by $B_{1}$.
Also, the corresponding statements hold for the vertical rectangle
$\mathcal{R}^{\prime}$.

These assumptions essentially mirror those of [BLR20]. In addition, in order
to handle potential difficulties near the interface between reflecting and
wired boundary pieces, we make the following additional assumption:

  1. 6.

For every topological quadrilateral $(\Lambda,a,b,c,d)\subset\Omega$, the
discrete and continuous extremal lengths of $\Lambda$ are uniformly
comparable.

For precise definitions of the notions of discrete and continuous extremal
lengths above we refer the reader for instance to [Che16]. In particular, by
Theorem 7.1 in [Che16], this additional assumption holds as soon as the edges
of the graph $G_{\delta}$ are straight, edge lengths are locally comparable,
the edge weights are uniformly elliptic, and the angles between edges are
uniformly bounded away from 0 and $\pi$; see again [Che16] for details.

We will consider the set $\mathcal{U}(G_{\delta})$ of spanning trees on
$G_{\delta}$ with the boundary conditions that the arcs
$c_{1}=(x_{2}x_{3}),\ldots c_{n}=(x_{2n}x_{1})$ are each wired (but not to one
another). Since $G_{\delta}$ is directed, by definition, a spanning tree
$\mathbf{t}\in\mathcal{U}(G_{\delta})$ is such that there is exactly one
forward edge out of every non-wired vertex. There is a natural measure on
$\mathcal{U}(G_{\delta})$ which is simply

$\mathbb{P}(\mathbf{t})\propto\prod_{e\in\mathbf{t}}w_{e};\quad\mathbf{t}\in\mathcal{U}(G_{\delta}).$

We also consider the set $\mathcal{D}(\mathcal{G}^{D}_{\delta})$ of dimer
configurations on $\mathcal{G}^{D}_{\delta}$, which we equip with a
probability measure

$\mathbb{P}(\mathbf{m})\propto\prod_{e\in\mathbf{m}}w_{e};\quad\mathbf{m}\in\mathcal{D}(\mathcal{G}^{D}_{\delta}),$

where an edge $e$ of $\mathcal{G}^{D}_{\delta}$ inherits the weight $w_{e}=1$
if it was part of a dual edge of $G_{\delta}$, and otherwise the weight of the
corresponding primal edge oriented from black to white.

Temperley’s bijection has an obvious extension to this setting: namely, given
$\mathbf{m}\in\mathcal{D}(\mathcal{G}^{D}_{\delta})$ we associate to it the
spanning tree $\mathbf{t}\in\mathcal{U}(G_{\delta})$ by considering all the
edges of $\mathbf{m}$ emanating out of a primal black vertex and doubling them
up: i.e., for each $(bw)\in\mathbf{m}$ with $b\in V(G_{\delta})$, we add to
$\mathbf{t}$ the unique (primal) edge $bb^{\prime}$ such that
$w\in(bb^{\prime})$.

It is not hard to see that Temperley’s bijection can be generalised to this
setup:

######  Lemma 2.2 (Generalised Temperley’s bijection).

The map $\mathbf{m}\mapsto\mathbf{t}$ is a measure-preserving bijection
between $\mathcal{D}(\mathcal{G}^{D}_{\delta})$ and $\mathcal{U}(G_{\delta})$.

As a result of this bijection, all the results we prove for the dimer model on
black-piecewise Temperleyan domains generalise in a straightforward manner to
the dimer model on $\mathcal{G}^{D}_{\delta}$ with the above law. We will not
comment further on this distinction except where necessary, and we will write
our proofs for concreteness in the setup of Section 2.1.

###  2.3 Coupling of flow lines, counterflow lines and GFF

In this section, we will recall some facts about imaginary geometry from
[MS16] and [MS17]. We will use them to establish the coupling of the Gaussian
free field (which will be the limit of the height function) and the continuous
tree (which will be the limit of the Temperleyan tree of the last section),
which will be described later.

We first fix some constants:

$\kappa=2;\quad\kappa^{\prime}=8;\quad\lambda=\frac{\pi}{\sqrt{2}};\quad\lambda^{\prime}=\frac{\pi}{\sqrt{8}};\quad\chi=\frac{1}{\sqrt{2}}.$

Let us start with the so-called counterflow line coupling. For any simply
connected domain $(\Omega;a,b)$, where $a,b$ are two points on the boundary of
$\Omega$ (understood as prime ends), we define the harmonic function
$u_{(\Omega;a,b)}$ as following. Fix a conformal map $\varphi$ from $\Omega$
onto the upper half plane $\mathbb{H}$, which maps $a$ to $0$ and $b$ to
$\infty$. Define $u_{(\mathbb{H};0,\infty)}$ to be the unique bounded harmonic
function, which equals $\lambda^{\prime}$ on $(-\infty,0)$ and equals
$-\lambda^{\prime}$ on $(0,\infty)$. Define

$u_{(\Omega;a,b)}:=u_{(\mathbb{H};0,\infty)}\circ\varphi-\chi\arg\varphi^{\prime}.$

It can be checked that this does not depend on the choice of the map
$\varphi$. Denote by $h_{\Omega}$ the Gaussian free field (GFF) with Dirichlet
boundary condition on $\Omega$, with two-point or covariance function given by
the Green function $G(x,y)$, normalised so that $G(x,y)=-\log|x-y|+O(1)$ as
$y\to x$. Denote by $\gamma$ a chordal $\operatorname{SLE}_{\kappa^{\prime}}$
in $\Omega$ from $a$ to $b$.

######  Theorem 2.3.

[MS16, Theorem 1.1, Theorem 1.2] Fix a simply connected domain $(\Omega;a,b)$.
There is a unique coupling of $h=h_{\Omega}+u_{(\Omega;a,b)}$ and $\gamma$,
such that for any stopping time $\tau$ of $\gamma$, the conditional law of
$h_{\Omega}+u_{(\Omega;a,b)}$ given $\gamma[0,\tau]$ restricted in
$(\Omega\setminus\gamma[0,\tau];\gamma(\tau),b)$ equals the law of

$h_{\Omega\setminus\gamma[0,\tau]}+u_{(\Omega\setminus\gamma[0,\tau];\gamma(\tau),b)},$

where $h_{\Omega\setminus\gamma[0,\tau]}$ is a on
$(\Omega\setminus\gamma[0,\tau];\gamma(\tau),b)$ which is independent of
$\gamma((0,\tau])$. Moreover, in this coupling, $\gamma$ is measurably
determined by $h_{\Omega}$.

Second, we introduce the flow line coupling. Fix a sequence
$\\{x_{1},\ldots,x_{2n}\\}$ such that $x_{1}<\ldots<x_{2n}$ and fix a sequence
$\\{z_{1},\ldots,z_{n-1}\\}$ such that $x_{2i}\leq z_{i}\leq x_{2i+1}$ for
$1\leq i\leq n-1$. We define
$u_{\mathbb{H}}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$ to be the
unique bounded harmonic function with the following boundary conditions:

$\displaystyle
u_{\mathbb{H}}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})=\begin{cases}0,&\text{on
}\cup_{i=1}^{n}(x_{2i-1},x_{2i}),\\\ -\lambda,&\text{on
}\cup_{i=1}^{n-1}(x_{2i},z_{i})\cup(x_{2n},\infty),\\\ \lambda,&\text{on
}\cup_{i=1}^{n-1}(z_{i},x_{2i+1})\cup(-\infty,x_{1}).\end{cases}$

We fix a simply connected domain
$(\Omega;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$, where
$x_{1},\ldots,x_{2n}$ are boundary points on $\partial\Omega$ in
counterclockwise orientation and $z_{i}$ is on the arc $[x_{2i}x_{2i+1}]$ for
$1\leq i\leq n-1$. We fix a conformal map $\varphi$ from $\Omega$ onto
$\mathbb{H}$ such that $\varphi(x_{1})<\ldots<\varphi(x_{2n})$. Define

$u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1}):=u_{\mathbb{H}}(\cdot;\varphi(x_{1}),\ldots,\varphi(x_{2n});\varphi(z_{1}),\ldots,\varphi(z_{n-1}))\circ\varphi-\chi\arg\varphi^{\prime}.$

Again, it can be checked this does not depend on the choice of the conformal
map $\varphi$. In [MS16], the authors established the coupling of $h_{\Omega}$
and the flow lines starting from the boundary points in the following sense
(this can be viewed as a recursive definition of the set of flow lines
emanating out of $z_{1},\ldots,z_{n-1})$.

######  Theorem 2.4.

[MS16, Theorem 1.1, Theorem 1.2] Fix a simply connected domain $\Omega$
together with marked boundary points
$(x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$ as above. There is a unique
coupling of curves $\\{\gamma_{1}^{I},\ldots,\gamma_{n-1}^{I}\\}$ and
$h_{\Omega}+u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$ such
that the following properties hold almost surely.

  * •

The law of the flow line $\gamma^{I}_{1}$ can be described as follows: we
denote by $W$ the driving function of $\varphi(\gamma^{I}_{1})$ and denote by
$\\{g_{t}:t\geq 0\\}$ the corresponding Loewner flow in the upper half plane
$\mathbb{H}$, then we have

$dW_{t}=\sqrt{2}dB_{t}+\sum_{i=1}^{2n}\frac{-1}{W_{t}-g_{t}(x_{i})}+\sum_{i=2}^{n-1}\frac{2}{W_{t}-g_{t}(z_{i})},\quad
W_{0}=z_{1},$ (2.2)

where $(B_{t}:t\geq 0)$ is the standard one-dimentional Brownian motion.

  * •

The curve $\gamma_{1}^{I}$ will end in $\cup_{j=2}^{n}(x_{2j}z_{j})$, where we
identify $z_{n}=x_{1}$. Given $\gamma^{I}_{1}$, and given the event that
$\gamma^{I}_{1}$ ends in $\cup_{j=2}^{n}(x_{2j}z_{j})$, we denote by
$\Omega_{L}$ the connected component of $\mathbb{H}\setminus\gamma^{I}_{1}$ in
the left side of $\gamma^{I}_{1}$ and denote by $\Omega_{R}$ the connected
component of $\mathbb{H}\setminus\gamma^{I}_{1}$ in the right side of
$\gamma^{I}_{1}$. The conditional law of
$h_{\Omega}+u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$
restricted in $\Omega_{L}$ equals

$h_{\Omega_{L}}+u_{\Omega_{L}}(\cdot;x_{1},x_{2},x_{2j+1},\ldots,x_{2n};z_{j},\ldots,z_{n-1})$

and the conditional law of
$h_{\Omega}+u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$
restricted in $\Omega_{R}$ equals

$h_{\Omega_{R}}+u_{\Omega_{R}}(\cdot;x_{3},\ldots,x_{2j-1},x_{2j};z_{2},\ldots,z_{j-1}).$

The conditional law of $\\{\gamma^{I}_{j},\ldots,\gamma^{I}_{n-1}\\}$ equals
the law of the flow lines of

$h_{\Omega_{L}}+u_{\Omega_{L}}(\cdot;x_{1},x_{2},x_{2j+1},\ldots,x_{2n};z_{j},\ldots,z_{n-1}).$

The conditional law of $\\{\gamma^{I}_{2},\ldots,\gamma^{I}_{j-1}\\}$ equals
the law of the flow lines of

$h_{\Omega_{R}}+u_{\Omega_{R}}(\cdot;x_{3},\ldots,x_{2j-1},x_{2j};z_{2},\ldots,z_{j-1}).$

  * •

Given $\cup_{i=1}^{n-1}\gamma_{i}$, there will be $n$ connected components of
$\Omega\setminus\cup_{i=1}^{n-1}\gamma_{i}$ and each componnent contains one
boundary arc $(x_{2i-1}x_{2i})$ for some $1\leq i\leq n$. We denote by
$\Omega_{i}$ the connected component which contains $(x_{2i-1}x_{2i})$ on its
boundary. The conditional law of
$h_{\Omega}+u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$
restricted in $\Omega_{i}$ equals the law of

$h_{\Omega_{i}}+u_{(\Omega_{i};x_{2i-1},x_{2i})}-\frac{\pi\chi}{2},$

See Figure 2.3 for an illustration.

Figure 2.3: This is an illustration of the boundary values of the
$\operatorname{GFF}$ and flow lines when $n=3$.

The counterflow line coupling of Theorem 2.3 can be augmented in a useful way
by considering the left and right boundaries of the counterflow line $\gamma$
when it reaches a point $z\in\Omega$. This leads to the following coupling of
counterflow and flow lines starting from interior points in imaginary
geometry, stated in [MS17, Theorem 1.1]. Fix a dense countable set
$\mathcal{Q}=\\{z_{i}\\}_{i\geq 1}$ in $\Omega$. Let $h$ and $\gamma$ be
coupled as in the counterflow line coupling of Theorem 2.3. For every
$z\in\Omega$, we denote by $\gamma_{z}$ the flow line of $h$ from $z$ with
angle $-\frac{\pi}{2}$ and denote by $\gamma^{*}_{z}$ the flow line of $h$
from $z$ with angle $\frac{\pi}{2}$. They can be viewed as the boundaries of
the counterflow line $\gamma$ when it reaches the point $z$. We will need the
following properties of this coupling:

######  Theorem 2.5.

[MS17, Theorem 1.1, Theorem 1.2, Theorem 1.7, Theorem 1.13] Fix a simply
connected domain $(\Omega;a,b)$. There exists a unique coupling of
$\\{\gamma_{z_{i}}\\}_{i\geq 1}$, $\\{\gamma^{*}_{z_{i}}\\}_{i\geq 1}$,
$\gamma$ and $h_{\Omega}$ such that the following properties hold almost
surely.

  * •

The curve $\gamma$ is the counterflow line of $h_{\Omega}$ in the sense of
Theorem 2.3. Let $n\geq 1$ and fix $z_{1},\ldots,z_{n}\in\mathcal{Q}$. The
conditional law of $h_{\Omega}+u_{(\Omega;a,b)}$ given
$\\{\gamma_{z_{i}}\\}_{1\leq i\leq n}$ equals to the law of

$h_{\Omega\setminus\cup_{i=1}^{n}\gamma_{z_{i}}}+u_{(\Omega\setminus\cup_{i=1}^{n}\gamma_{z_{i}};a,b)},$

where $h_{\Omega\setminus\cup_{i=1}^{n}\gamma_{z_{i}}}$ is an independent GFF.
The conditional law of $h_{\Omega}+u_{(\Omega;a,b)}$ given
$\\{\gamma_{z_{i}}^{*}\\}_{1\leq i\leq n}$ equals to the law of

$h_{\Omega\setminus\cup_{i=1}^{n}\gamma_{z}^{*}}+u_{(\Omega\setminus\cup_{i=1}^{n}\gamma_{z}^{*};a,b)},$

where $h_{\Omega\setminus\cup_{i=1}^{n}\gamma_{z}^{*}}$ is independent of
$\\{\gamma^{*}_{z_{i}}\\}_{1\leq i\leq n}$. Moreover,
$\\{\gamma_{z_{i}}\\}_{1\leq i\leq n}$ and $\\{\gamma^{*}_{z_{i}}\\}_{1\leq
i\leq n}$ are determined by $h_{\Omega}$.

  * •

For every $i\geq 1$, the flow lines $\gamma_{z_{i}}$ and $\gamma^{*}_{z_{i}}$
are simple curves. Moreover, $\gamma_{z_{i}}$ hits $\partial\Omega$ at $(ab)$
and $\gamma_{z_{i}}^{*}$ hits $\partial\Omega$ at $(ba)$ and
$\gamma_{z_{i}}\cap\gamma_{z_{i}}^{*}=\\{z_{i}\\}$. For $i\neq j$, the curves
$\gamma_{z_{i}}$ merges with $\gamma_{z_{j}}$ when they intersect each other
and the curves $\gamma^{*}_{z_{i}}$ merges with $\gamma^{*}_{z_{j}}$ when they
intersect each other.

  * •

Define $\tau_{z}$ to be the first time that $\gamma$ hits $z$. Almost surely,
the left boundary $\gamma^{*}[0,\tau_{z}]$ equals to $\gamma_{z}$ and the
right boundary of $\gamma[0,\tau_{z}]$ equals to $\gamma_{z}$.

The marginal law (and joint law) of $\gamma_{z}$ and $\gamma_{z}^{*}$ could
further be specified but we will not need this in the following, in fact we
will recover a description of these laws as part of our proof.

##  3 Coupling of the continuous tree and GFF when $n=2$

In this section, we will prove the convergence of the discrete Temperleyan
trees introduced in Lemma 2.1 and establish the coupling of the limiting tree
with the GFF with boundary conditions specified in Theorem 2.4, in the case
$n=2$. In this section, we will assume that $z_{1}=x_{3}$. (The proof in the
other case $z_{1}=x_{2}$ is almost the same). It is convenient to change our
notations slightly in the cas $n=2$, and denote the point $z_{1}$ by $b$,
while we denote the remaining points $x_{2},x_{4},x_{1}$ by $a,c,d$
respectively.

Suppose $(\Omega_{\delta};a_{\delta},b_{\delta},c_{\delta},d_{\delta})$ is a
discrete simply connected domain with four marked boundary points, where
$\Omega_{\delta}$ is a subset of $\delta\mathbb{Z}^{2}$ and
$a_{\delta},b_{\delta},c_{\delta}$ and $d_{\delta}$ are on
$\partial\Omega_{\delta}$. Consider the uniform spanning tree
($\operatorname{UST}$) on $\Omega_{\delta}$ on the graph where the boundary
$(a_{\delta}b_{\delta})$ is wired to be one vertex and the boundary arc
$(c_{\delta}d_{\delta})$ is wired to be another vertex. (This is the UST with
so-called _alternating boundary conditions_ introduced in [HLW20] and already
discussed above Theorem 1.4). Recall that there is a (single) branch,
$\gamma_{\delta}$, connecting the two arcs $(a_{\delta}b_{\delta})$ and
$(c_{\delta}d_{\delta})$ in this tree. We may thus define the event

$E_{\delta}:=\\{\text{the branch $\gamma_{\delta}$ starts from
}b_{\delta}b_{\delta}^{+}\\}.$

As already explained above Theorem 1.4, it is not hard to see that the tree
$\mathcal{T}_{\delta}$ obtained from Temperley’s bijection in the black
$4$-Temperleyan domain is nothing but a UST with alternating boundary
conditions, conditionally given the event $E_{\delta}$. Recall also
$\eta_{L,\delta}$ is the discrete Peano curve on the left side of
$\gamma_{\delta}$ starting from $d_{\delta}$ and ending in $a_{\delta}$, while
$\eta_{R,\delta}$ is the discrete Peano curve from $c_{\delta}$ to
$b_{\delta}$ on the right side of $\gamma_{\delta}$. For every
$z_{\delta}\in\Omega_{\delta}$, denote by $\gamma_{z_{\delta}}$ the path from
$z_{\delta}$ to
$(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})\cup\gamma_{\delta}$.

We now recall the Schramm topology (introduced in [Sch00]) which we use for
the convergence of trees.

  * •

Let $\mathcal{P}$ be the space of unparameterised paths in $\mathbb{C}$.
Define a metric on $\mathcal{P}$ as follows:

$\displaystyle
d(\gamma_{1},\gamma_{2}):=\inf\sup_{t\in[0,1]}\left|\hat{\gamma}_{1}(t)-\hat{\gamma}_{2}(t)\right|,$
(3.1)

where the infimum is taken over all the choices of parameterisations
$\hat{\gamma}_{1}$ and $\hat{\gamma}_{2}$ of $\gamma_{1}$ and $\gamma_{2}$.
Suppose $\\{{\gamma_{\delta}}\\}_{\delta>0}$ and $\gamma$ are continuous
curves in $\overline{\mathbb{H}}$ from $0$ to $\infty$ and their Loewner
driving functions are continuous. If $\gamma_{\delta}$ converges to $\gamma$
under the metric (3.1), then after parametrising by half-plane capacity,
$\gamma_{\delta}$ converges to $\gamma$ locally uniformly as continuous
functions on $[0,+\infty)$.

  * •

For any metric space $X$, denote by $\mathcal{H}(X)$ the space of closed
subsets of $X$ equipped with the Hausdorff metric. For every
$z,w\in\overline{\Omega}$, denote by $\mathcal{P}(z,w,\overline{\Omega})$ the
set of paths connecting $z$ and $w$ in $\overline{\Omega}$. We will view

$\\{z:z\in\overline{\Omega}\\}\times\\{w:w\in\overline{\Omega}\\}\times\cup_{z,w\in\overline{\Omega}}\mathcal{P}(z,w,\overline{\Omega})$

as a subset of
$\mathcal{H}(\overline{\Omega}\times\overline{\Omega}\times\mathcal{H}(\overline{\Omega}))$.
We will denote by $d_{\mathcal{H}}$ the corresponding metric and denote by
$\overline{\Omega}_{\mathcal{H}}$ the closure of
$\\{z:z\in\overline{\Omega}\\}\times\\{w:w\in\overline{\Omega}\\}\times\cup_{z,w\in\overline{\Omega}}\mathcal{P}(z,w,\overline{\Omega})$
under $d_{\mathcal{H}}$.

We now give the setup in which we will prove convergence of the discrete
Temperleyan trees. Fix a simply connected domain $(\Omega;a,b,c,d)$ such that
$\partial\Omega$ is locally connected and $(a,b,c,d)$ are four points on
$\partial\Omega$ in counterclockwise order, and suppose that there exists a
simply connected domain $\tilde{\Omega}$ whose boundary is a simple $C^{1}$
curve, such that $\Omega\subset\tilde{\Omega}$ and
$\partial\Omega\cap\partial\tilde{\Omega}$ equals $[bc]\cup[da]$. Suppose also
that a sequence of discrete simply connected domains
$\\{(\Omega_{\delta};a_{\delta},b_{\delta},c_{\delta},d_{\delta})\\}_{\delta>0}$
converges to $(\Omega;a,b,c,d)$ in the following sense: there exists $C>0$
such that

$d((b_{\delta}c_{\delta}),(bc))\leq C\delta,\quad
d((d_{\delta}a_{\delta}),(da))\leq C\delta\quad\text{and}\quad
d((a_{\delta}b_{\delta}),(ab))\to 0,\quad d((c_{\delta}d_{\delta}),(cd))\to
0\text{ as }\delta\to 0.$ (3.2)

For convenience, we require $\Omega_{\delta}\subset\Omega$. We emphasise that
the assumptions on $\partial\Omega$ and the convergence of discrete domains
are mainly needed to handle the reflecting boundaries: the assumption on
$\tilde{\Omega}$ amounts to saying that the reflecting arcs $(bc)$ and $(da)$
should be simple and smooth, while the wired arcs $(ab)$ and $(cd)$ can be
fairly arbitrary. We will view the discrete tree $\mathcal{T}_{\delta}$ as

$\\{(x_{\delta},y_{\delta},\gamma_{x_{\delta},y_{\delta}})|x_{\delta},y_{\delta}\in\Omega_{\delta}\text{
and }\gamma_{x_{\delta},y_{\delta}}\text{ is the unique path connecting
}x_{\delta}\text{ and }y_{\delta}\text{ in }\mathcal{T}_{\delta}\\},$

which is an element in $\overline{\Omega}_{\mathcal{H}}$.

######  Theorem 3.1.

As $\delta\to 0$, the discrete tree $\mathcal{T}_{\delta}$ converges in
distribution (with respect to the metric $d_{\mathcal{H}}$). Let $\mathcal{T}$
denote the limit, which we call a continuous tree. The continuous tree
$\mathcal{T}$ can be constructed as follows:

Fix any dense set $\mathcal{Q}=\\{z_{i}\\}_{i\geq 1}$ in $\Omega$. Let
$\gamma$ be a chordal $\operatorname{SLE}_{2}(-1,-1;-1,-1)$ curve in $\Omega$
with force points $a,d$ and $b,c$ starting from $b$, and suppose it is coupled
with $h_{\Omega}+u_{\Omega}(\cdot;a,b,c,d;b)$ as in the flow line coupling of
Theorem 2.4. Given $\gamma$, let $\Omega_{L}$ and $\Omega_{R}$ be the two
domains on either side of $\gamma$, as in Theorem 2.4. Denote by
$\mathcal{Q}_{L}$ (resp. $\mathcal{Q}_{R}$) the points of $\mathcal{Q}$ that
fall in $\Omega_{L}$ (resp. $\Omega_{R}$). Then, we sample
$\big{\\{}\gamma_{z_{i}^{L}}\big{\\}}_{i\geq 1}$ and sample
$\big{\\{}\gamma^{*}_{z_{i}^{R}}\big{\\}}_{i\geq 1}$ as in Theorem 2.5,
applied in $\Omega_{L}$ and $\Omega_{R}$ respectively. Then $\mathcal{T}$,
viewed as a random variable in the space $\mathcal{H}$, is simply the closure
of the union of $\big{\\{}\gamma_{z_{i}^{L}}\big{\\}}_{i\geq 1}$,
$\big{\\{}\gamma^{*}_{z_{i}^{R}}\big{\\}}_{i\geq 1}$, and $\gamma$.

It is clear that Theorem 1.3 follows from Theorem 3.1 and the counterflow line
coupling of Theorem 2.3. The rest of Section 3.2 is devoted to a proof of
Theorem 3.1. We will frequently use the following discrete Beurling estimate
for random walk on $\Omega_{\delta}$ the reflecting boundaries
$(b_{\delta}c_{\delta})$ and $(d_{\delta}a_{\delta})$.

######  Lemma 3.2.

There exists constants $C>0,\delta_{0}>0,0<\epsilon<1$ such that the following
estimate hold. For every $\delta\leq r\leq\delta_{0}$, for every connected
subgraph $A$ of $\Omega_{\delta}$ with $\operatorname{diam}(A)>r$ and for
every $z_{\delta}\in\Omega_{\delta}$ with
$d=d(z_{\delta},A\cup(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta}))\leq r$,
we have

$\mathbb{P}\left[\mathcal{R}\text{ hits }\partial B(z_{\delta},Cr)\text{
before }A\cup(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})\right]\leq
C(d/r)^{\epsilon},$

where $\mathcal{R}$ is the random walk starting from $z_{\delta}$, reflecting
at $(b_{\delta}c_{\delta})\cup(d_{\delta}a_{\delta})$, and stopped when it
hits $(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})$.

###### Proof.

If $\Omega_{\delta}$ is a subgraph of $\delta\mathbb{Z}^{2}$, the proof is
same as the proof of [Sch00, Lemma 11.2]. To generalize the discrete Beurling
estimate to the general setting mentioned in Section 2.2, we will prove the
result based on the assumptions there. To prove it, it suffices to prove the
result in the case $d=r$ and the right hand side an absolute constant strictly
less than 1, say $\epsilon$. Hence we may assume that
$d(z_{\delta},A\cup(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta}))=r$.

We will proceed by contradiction. Suppose for a sequence of
$\\{\epsilon_{n},\delta_{n},r_{n},C_{n}\\}$ and
$\\{\Omega_{\delta_{n}},z_{\delta_{n}},A_{\delta_{n}}\\}_{n\geq 1}$ such that
$\delta_{n}\to 0$, $\epsilon_{n}\to 0$, $C_{n}\to\infty$, and
$\operatorname{diam}(A_{\delta_{n}})>r_{n}$; but

$\mathbb{P}\left[\mathcal{R}\text{ hits }\partial
B(z_{\delta_{n}},C_{n}r_{n})\text{ before
}A_{\delta_{n}}\cup(a_{\delta_{n}}b_{\delta_{n}})\cup(c_{\delta_{n}}d_{\delta_{n}})\right]>1-\epsilon_{n}.$
(3.3)

We may assume that $r_{n}$ tends to zero. (Otherwise, if
$C_{n}>2\operatorname{diam}(\Omega)$, then (3.3) trivially fails.) We denote
$A_{\delta_{n}}\cup(a_{\delta_{n}}b_{\delta_{n}})\cup(c_{\delta_{n}}d_{\delta_{n}})$
by $\overline{A}_{\delta_{n}}$.

Since $\partial\Omega$ is $C^{1}$ and simple, if we denote this curve by
$l:[0,1]\to\mathbb{C}$, there exists a constant $M>0$, such that for $0\leq
s<t<1$, we have

$M^{-1}\leq\frac{|l(t)-l(s)|}{(t-s)\wedge(1-t+s)}\leq M.$ (3.4)

We denote by $w_{\delta_{n}}\in\overline{A}_{\delta_{n}}$ a point such that

$d(z_{\delta_{n}},w_{\delta_{n}})=d(z_{\delta_{n}},\overline{A}_{\delta_{n}}).$

If $z_{\delta_{n}}$ and $\overline{A}_{\delta_{n}}$ is separated by $l$, then
there exist $s<t$ such that $l(s)$ and $l(t)$ belong to the straight line
connecting $z_{\delta_{n}}$ and $w_{\delta_{n}}$ and $l[s,t]$ hits $\partial
B(z_{\delta_{n}},2M^{2}r_{n})$. In particular, we have

$d(l(s),l(t))\leq r_{n}\quad{\text{ and }}\operatorname{diam}(l[s,t])\geq
M^{2}r_{n},$

which is a contradiction with (3.4).

Suppose first there exists $M^{\prime}>0$, such that
$\frac{r_{n}}{\delta_{n}}\leq M^{\prime}$. In $B(z_{\delta_{n}},4M^{2}r_{n})$,
by our assumptions (1) and (3), the probability that the random walk starting
from $z_{\delta_{n}}$ hits $\overline{A}_{\delta_{n}}$ before $\partial
B(z_{\delta_{n}},C_{n}r_{n})$ is uniformly bounded away from zero. For $n$
large enough, we have $C_{n}>4M^{2}$, and this leads to a contradiction in
conjunction with (3.3).

Now suppose instead that $\frac{\delta_{n}}{r_{n}}$ tends to zero. By
translation, we may assume $z_{\delta_{n}}=0$ for $n\geq 1$. We will consider

$\left\\{\frac{1}{r_{n}}\left(\Omega_{\delta_{n}}\cap
B(z_{\delta_{n}},4M^{2}r_{n})\right),\frac{1}{r_{n}}(\overline{A}_{\delta_{n}}\cap
B(z_{\delta_{n}},4M^{2}r_{n}))\right\\}_{n\geq 1}.$

There are two cases:

  * •

In the first case, if the free boundary
$(b_{\delta_{n}}c_{\delta_{n}})\cup(d_{\delta_{n}}a_{\delta_{n}})$ does not
intersect $B(z_{\delta_{n}},4M^{2}r_{n})$, then we can apply directly the
usual Beurling estimate (see e.g., [BLR20, Lemma 4.3]), hence we derive a
contradiction directly.

  * •

In the second case, we may assume that the free boundary intersect
$B(z_{\delta_{n}},4M^{2}r_{n})$ for all $n\geq 1$. Then, as $n\to\infty$, the
intersection of $B(z_{\delta_{n}},4M^{2}r_{n})\cap\partial\Omega_{\delta_{n}}$
converges to a straight line (under the metric (3.1)) since the arc is
$C^{1}$. (Recall that under our assumption
$\partial\Omega_{\delta}=\partial\Omega$ in the setup of Section 2.2.) Thus,
without loss of generality, we may assume that
$\frac{1}{r_{n}}\left(\Omega_{\delta_{n}}\cap
B(z_{\delta_{n}},4M^{2}r_{n})\right)$ converges to one component of
$B(0,4M^{2})\setminus l$ (in the sense that the boundary converges under the
metric (3.1)), where $l$ is a straight segment. Moreover, we will assume
$\frac{1}{r_{n}}(\overline{A}_{\delta_{n}}\cap B(z_{\delta_{n}},4M^{2}r_{n}))$
converges to a compact set $\overline{A}$ under the Hausdorff metric. Note
that $l$ does not separate $\overline{A}$ and $z=0$. In general the segment
$l$ might contain both some wired and reflecting parts. Without loss of
generality we only consider the case where $l$ consists exclusively of the
limit of a free boundary arc (in fact the argument is only easier if $l$ also
contains some wired portion).

For $v_{\delta_{n}}\in V(\Omega_{\delta_{n}})$, We define

$h_{\delta_{n}}(v_{\delta_{n}}):=\mathbb{P}\left[\mathcal{R}\text{ hits
}\partial B(z_{\delta_{n}},C_{n}r_{n})\text{ before
}\overline{A}_{\delta_{n}}\right],$

where $\mathcal{R}$ is the random walk starting from $v_{\delta_{n}}$. By our
assumption (4), $h_{\delta_{n}}$ converges to a harmonic function $h$
uniformly on $\frac{1}{r_{n}}\left(\Omega_{\delta_{n}}\cap
B(z_{\delta_{n}},4M^{2}r_{n})\right)$. Here $h$ is the unique bounded harmonic
function with the following boundary conditions: $h=1$ on $\overline{A}$ and
$h=0$ on $\partial B(0,4M^{2})$, and $\partial_{n}h=0$ on $l$. By the minimum
principle, we know that $h(0)>0$ (note that here we allow $0\in l$) and hence
$h_{\delta_{n}}(z_{\delta_{n}})$ is uniformly bounded away from zero. Since
for $n$ large enough, we have $C_{n}>4M^{2}$, this leads to a contradiction in
conjunction with (3.3).

This completes the proof of the lemma. ∎

###  3.1 Convergence of $\gamma_{\delta}$ and discrete Peano curves

In this subsection, we will prove the convergence of
$\\{\gamma_{\delta}\\}_{\delta>0}$ under the metric (3.1) and discuss the
convergence of the discrete Peano curves $\\{\eta_{L}^{\delta}\\}_{\delta>0}$
and $\\{\eta_{R}^{\delta}\\}_{\delta>0}$. Similar results have been proved in
[HLW20], but the setup is different. In that paper, the authors considered the
$\operatorname{UST}$ without conditioning. The line of the proof is almost the
same and we will point out how to modify the proof there to the setting in
this paper. We denote by $K$ the conformal modulus of $(\Omega;a,b,c,d)$ and
denote by $\varphi$ the conformal map from $(\Omega;a,b,c,d)$ onto the
rectangle $(0,1)\times(0,K)$ which sends $(a,b,c,d)$ to
$(0,1,1+\mathfrak{i}K,\mathfrak{i}K)$.

We will deal with the convergence of $\\{\gamma_{\delta}\\}_{\delta>0}$ at
first. First of all, we need the following lemmas to get rid of the
conditioning. Suppose $A$ is a circle such that $b\notin A$ and
$A\cap\partial\Omega\neq\emptyset$. We denote by $\Omega_{A}$ the connected
component of $\Omega\setminus A$ whose boundary contains $b$. We denote by
$A_{\delta}$ the corresponding discrete approximations on $\Omega_{\delta}$
and denote by $\Omega_{\delta,A_{\delta}}$ the connected component of
$\Omega_{\delta}\setminus A_{\delta}$ which contains $b_{\delta}$. Define

$h_{A_{\delta}}(v):=\mathbb{P}[\mathcal{R}\text{ starting from }v\text{ hits
}A_{\delta}\text{ before exits }\Omega_{\delta}],$

where $\mathcal{R}$ has the same law as the random walk reflecting at
$(b_{\delta}c_{\delta})\cup(d_{\delta}a_{\delta})$, and ending at
$(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})$. Denote by $h_{A}$ the
unique bounded harmonic function on $\Omega_{A}$ with the following boundary
conditions: $h_{A}$ equals one on $A$ and equals zero on
$((ab)\cup(cd))\cap\partial\Omega_{A}$ and
$\partial_{n}(h_{A}\circ\varphi^{-1})$ equals zero on

$([0,\mathfrak{i}K]\cup[1,1+\mathfrak{i}K])\cap\partial\varphi(\Omega_{A}),$

where $n$ is the outer normal, and $\varphi$ is the unique conformal from
$\Omega$ to a rectangle $(0,1)\times(0,K)$ (when we identify $\mathbb{C}$ with
$\mathbb{R}^{2}$) sending $a,b,c,d$ to the four corners of the rectangle in
counterclockwise order starting from 0.

######  Lemma 3.3.

For every $r>0$, we define $\Omega_{A,r}$ to be
$\Omega_{A}\setminus\\{z:d(z,A\cap((ab)\cup(cd)))<r\\}$ and define
$\Omega_{A,r,\delta}$ to be the discrete approximation of $\Omega_{A,r}$ on
$\Omega_{\delta}$. Then, $h_{A_{\delta}}|_{\Omega_{A,r,\delta}}$ converges to
$h_{A}|_{\Omega_{A,r}}$ uniformly.

###### Proof.

For the setup considered in Section 2.2, we simply observe that random walk
$\mathcal{R}$ on $\Omega_{\delta}$ will converge to Brownian motion
orthogonally reflected on $(bc)$ and $(da)$ and killed on $(ab)$ and $(cd)$
(orthogonal reflection is defined e.g. by doing the reflection in the
rectangle and applying the conformal map $\varphi^{-1}$).

We also give a proof when we are considering the lattice $\mathbb{Z}^{2}$. We
first show the equicontinuity of
$\\{h_{A_{\delta}}|_{\Omega_{A,r,\delta}}\\}_{\delta>0}$. We fix
$0<\epsilon<r$ sufficiently small, and fix
$w_{\delta},z_{\delta}\in\Omega_{A,r,\delta}$ such that
$|w_{\delta}-z_{\delta}|<\epsilon$. There are three cases:

  * •

If $d(z_{\delta},A_{\delta})\leq\sqrt{\epsilon}$, then by Lemma 3.2, there
exist constants $C>0$ and $c>0$ such that

$1-h_{A_{\delta}}(z_{\delta})\leq C\epsilon^{c}\quad\text{ and }\quad
1-h_{A_{\delta}}(w_{\delta})\leq C\epsilon^{c}.$

In this case, we have

$|h_{A_{\delta}}(w_{\delta})-h_{A_{\delta}}(z_{\delta})|\leq C\epsilon^{c}.$

  * •

If
$d(z_{\delta},(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta}))\leq\sqrt{\epsilon}$,
by the similar argument above, we also have

$|h_{A_{\delta}}(w_{\delta})-h_{A_{\delta}}(z_{\delta})|\leq C\epsilon^{c}.$

  * •

If $d(z_{\delta},(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})\cup
A_{\delta})>\sqrt{\epsilon}$, from the discrete harmonicity of
$h_{A_{\delta}}$, we can draw a discrete path $l_{\delta}$ from $z_{\delta}$
to $A_{\delta}$, such that for every $v_{\delta}\in l_{\delta}$, we have
$h_{A_{\delta}}(v_{\delta})\geq h_{A_{\delta}}(z_{\delta})$. Thus, by Lemma
3.2, there exist constants $C>0$ and $c>0$, we have

$h_{A_{\delta}}(w_{\delta})\geq\left(1-C\epsilon^{c}\right)h_{A_{\delta}}(z_{\delta}).$

This implies that

$h_{A_{\delta}}(z_{\delta})-h_{A_{\delta}}(w_{\delta})\geq-C\epsilon^{c}.$

By symmetry, we have

$|h_{A_{\delta}}(z_{\delta})-h_{A_{\delta}}(w_{\delta})|\leq C\epsilon^{c}.$

The estimates in these three cases give the equicontinuity of
$\\{h_{A_{\delta}}|_{\Omega_{A,r,\delta}}\\}_{\delta>0}$.

To identify uniquely the limit, we can use argument similar to [HLW20, Lemma
B.2]. ∎

We also need a similar result when $A$ is a subarc of $(cd)$. Suppose
$(\tilde{c}\tilde{d})$ is a subarc of $(cd)$ and we denote by
$(\tilde{c}_{\delta}\tilde{d}_{\delta})$ a discrete approximation of
$(\tilde{c}\tilde{d})$ on $\Omega_{\delta}$. Define

$h_{(\tilde{c}_{\delta}\tilde{d}_{\delta})}(v):=\mathbb{P}[\mathcal{R}\text{
starting from }v\text{ hits
}(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})\text{ at
}(\tilde{c}_{\delta}\tilde{d}_{\delta})],$ (3.5)

where $\mathcal{R}$ has the same law as the random walk reflecting at
$(b_{\delta}c_{\delta})\cup(d_{\delta}a_{\delta})$, and ending at
$(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})$.

Denote by $h_{(\tilde{c}\tilde{d})}$ the unique bounded harmonic function on
$\Omega$ with the following boundary conditions: $h_{(\tilde{c}\tilde{d})}$
equals one on $(\tilde{c}\tilde{d})$ and equals zero on
$(ab)\cup(c\tilde{c})\cup(\tilde{d}d)$ and
$\partial_{n}(h_{(\tilde{c}\tilde{d})}\circ\varphi^{-1})$ equals zero on
$[0,\mathfrak{i}K]\cup[1,1+\mathfrak{i}K]$, where $n$ is the outer normal. It
is straightforward to check that $h_{(\tilde{c}\tilde{d})}$ is nowhere zero on
the arc $(bc)$ or $(da)$, i.e., $h_{(\tilde{c}\tilde{d})}(z)>0$ for
$z\in(bc)\cup(da)$.

######  Lemma 3.4.

For every $r>0$, we define the domain

$\Omega_{\tilde{c},\tilde{d},r}:=\Omega\setminus\\{z:d(z,\\{\tilde{c},\tilde{d}\\})<r\\}$

and define $\Omega_{\tilde{c},\tilde{d},r,\delta}$ to be the discrete
approximation of $\Omega_{\tilde{c},\tilde{d},r}$ on $\Omega_{\delta}$. Then,
the discrete harmonic function
$h_{(\tilde{c}_{\delta}\tilde{d}_{\delta})}|_{\Omega_{\tilde{c},\tilde{d},r,\delta}}$
converges to $h_{(\tilde{c}\tilde{d})}|_{\Omega_{\tilde{c},\tilde{d},r}}$
uniformly.

###### Proof.

The proof is similar to the proof of Lemma 3.3. ∎

######  Lemma 3.5.

We assume $A$ is subarc of $(cd)$ or $A$ is a circle such that
$A\cap\partial\Omega\neq\emptyset$. Then, we have

$\lim_{\delta\to
0}\frac{h_{A_{\delta}}(b^{+}_{\delta})}{h_{(c_{\delta}d_{\delta})}(b^{+}_{\delta})}=\frac{\partial_{n}(h_{A}\circ\varphi^{-1})(b)}{\partial_{n}(h_{(cd)}\circ\varphi^{-1})(b)},$

where $n$ is the outer normal perpendicular to $(0,1)$.

###### Proof.

We define the function $g$ on a (subset of the) rectangle as follows:

$\displaystyle
g(z):=\begin{cases}\frac{h_{A}\circ\varphi^{-1}(z)}{h_{(cd)}\circ\varphi^{-1}(z)}&\text{
if }z\in\varphi(\Omega_{A})\\\
\frac{\partial_{n}h_{A}\circ\varphi^{-1}(z)}{\partial_{n}h_{(cd)}\circ\varphi^{-1}(z)},&\text{
if }z\in(0,1)\cap\partial\varphi(\Omega_{A}).\end{cases}$ (3.6)

Note that $g\circ\varphi$ is continuous near $b$. Now, we consider a circle
$\partial B(b,r)$ and we denote by $l_{b,r}$ the arc of that circle separating
$b$ and $(cd)$, and denote by $x:=l_{b,r}\cap(ab)$. We denote by
$l_{b,r,\delta}$ and $x_{\delta}$ some discrete approximation on
$\Omega_{\delta}$ of $l_{b,r}$ and $x$ respectively. For every $\epsilon>0$,
we may choose $r$ small enough, such that for every $w\in B(b,r)\cap\Omega$,
we have

$|g\circ\varphi(w)-g\circ\varphi(b)|\leq\epsilon.$

By uniform convergence in Lemma 3.3 and Lemma 3.4, combining with [CW21,
Corollary 3.8], there exists $\eta>0$, such that if
$v_{\delta}\in\Omega_{\delta}$ with $|v_{\delta}-x_{\delta}|<\eta$, then we
have

$\left|\frac{h_{A_{\delta}}(v_{\delta})}{h_{(c_{\delta}d_{\delta})}(v_{\delta})}-g\circ\varphi(x)\right|\leq\epsilon,\quad\text{
for }\delta\text{ small enough}.$

By the uniform convergence in Lemma 3.3 and Lemma 3.4 we have
$\frac{h_{A_{\delta}}}{h_{(c_{\delta}d_{\delta})}}\big{|}_{l_{b,r,\delta}\cap
B^{c}(x_{\delta},\eta)}$ converges to $g\circ\varphi|_{l_{b,r}\cap
B^{c}(x,\eta)}$ uniformly (by positivity of $h_{(\tilde{c}\tilde{d})}$ on
$l_{b,r,}\cap B^{c}(x,\eta)$). Thus, we have

$\left|\frac{h_{A_{\delta}}(v_{\delta})}{h_{(c_{\delta}d_{\delta})}(v_{\delta})}-g\circ\varphi(b)\right|\leq\epsilon,\quad\text{
for all }v_{\delta}\in l_{b,r,\delta}.$ (3.7)

Moreover, we can assume without loss of generality that $r$ is small enough
that $l_{b,r,\delta}$ seperates $A_{\delta}$ and $b_{\delta}$ in
$\Omega_{\delta}$. We denote by $\tau_{\delta}(l_{b,r,\delta})$ the hitting
time of random walk $\mathcal{R}_{\delta}$ at $l_{b,r,\delta}$ and denote by
$\tau_{\delta}((a_{\delta}b_{\delta}))$ and $\tau_{\delta}(A_{\delta})$
similarly. Note that

$\displaystyle\frac{h_{A_{\delta}}(b^{+}_{\delta})}{h_{(c_{\delta}d_{\delta})}(b^{+}_{\delta})}$
$\displaystyle=\frac{\mathbb{P}_{b^{+}_{\delta}}[\mathbb{1}_{\tau_{\delta}(l_{b,r,\delta})<\tau_{\delta}(A_{\delta})}h_{A_{\delta}}(\tau_{\delta}(l_{b,r,\delta}))]}{\mathbb{P}_{b^{+}_{\delta}}[\mathbb{1}_{\tau_{\delta}(l_{b,r,\delta})<\tau_{\delta}((c_{\delta}d_{\delta}))}h_{(c_{\delta}d_{\delta})}(\tau_{\delta}(l_{b,r,\delta}))]}$
$\displaystyle=\frac{\mathbb{P}_{b^{+}_{\delta}}\left[\mathbb{1}_{\tau_{\delta}(l_{b,r,\delta})<\tau_{\delta}(A_{\delta})}\frac{h_{A_{\delta}}(\tau_{\delta}(l_{b,r,\delta}))}{h_{(c_{\delta}d_{\delta})}(\tau_{\delta}(l_{b,r,\delta}))}h_{(c_{\delta}d_{\delta})}(\tau_{\delta}(l_{b,r,\delta}))\right]}{\mathbb{P}_{b^{+}_{\delta}}[\mathbb{1}_{\tau_{\delta}(l_{b,r,\delta})<\tau_{\delta}((c_{\delta}d_{\delta}))}h_{(c_{\delta}d_{\delta})}(\tau_{\delta}(l_{b,r,\delta}))]}.$

Combining with (3.7), we have

$\lim_{\delta\to
0}\frac{h_{A_{\delta}}(b^{+}_{\delta})}{h_{(c_{\delta}d_{\delta})}(b^{+}_{\delta})}=\frac{\partial_{n}h_{A}\circ\varphi^{-1}(b)}{\partial_{n}h_{(cd)}\circ\varphi^{-1}(b)}.$

This completes the proof. ∎

######  Remark 3.6.

Though [CW21, Corollary 3.8] is stated for the square lattice, the proof is
based on [Che16]. The assumptions on the graph is much weaker in [Che16] and
we have included them in our assumptions in Section 2.2.

We now explain how this can be used to prove the convergence of
$\\{\gamma_{\delta}\\}_{\delta>0}$. With Lemma 3.5, it is relatively easy to
modify the proof in [HLW20] to this setting. We will now explain it in detail.
The tightness of $\\{\gamma_{\delta}\\}_{\delta>0}$ will be a consequence of
the following Lemma, which is similar to [LSW04, Theorem 3.9]. See also
[HLW20, Lemma C.5] where a similar argument is used. Recall that
$\mathcal{R}_{\delta}$ denotes the random walk in $\Omega_{\delta}$ with
reflecting boundary conditions on
$(b_{\delta}c_{\delta})\cup(d_{\delta}a_{\delta})$ and killed on
$(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})$. Denote by
$\mathbb{P}_{v_{\delta}}$ the law of the random walk starting from
$v_{\delta}$, for every $v_{\delta}\in\Omega_{\delta}$.

######  Lemma 3.7.

Let $z_{0}\in\Omega$. For any $\alpha>0$ and
$0<\beta\leq(1/4)\operatorname{dist}((ab),(cd))$, let
$\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\gamma_{\delta})$ denote the event
that there are two points $o_{1},o_{2}\in\gamma_{\delta}$ with $o_{1},o_{2}\in
B(z_{0},\beta/4)$ with $|o_{1}-o_{2}|\leq\alpha$, and such that the subarc of
$\gamma_{\delta}$ between $o_{1}$ and $o_{2}$ is not contained in
$B(z_{0},\beta)$. For every $\epsilon>0$ and every $\beta>0$, we can choose
$\alpha=\alpha(\beta,\epsilon)$, small enough that for every $\delta>0$, we
have

$\mathbb{P}[\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\gamma_{\delta})]\leq\epsilon.$

###### Proof.

Denote by $B_{\delta}(z_{0},\beta)$ the discrete approximation of
$B(z_{0},\beta)$. By Wilson’s algorithm, we know that the law of
$\gamma_{\delta}$ is same as the law of loop-erasure of the random walk
$\mathcal{R}_{\delta}$, starting from $b_{\delta}^{+}$, conditional on hitting
$(c_{\delta}d_{\delta})$ before $(a_{\delta}b_{\delta})$. We will use the same
notations in [HLW20, Lemma C.5]. Let $t_{0}=0$. For $j\geq 1$, define
inductively

$s_{j}:=\inf\left\\{r\geq t_{j-1}:\mathcal{R}_{\delta}\in
B\left(z_{0},\beta/4\right)\right\\}\quad\text{and}\quad t_{j}:=\inf\\{r\geq
s_{j}:\mathcal{R}_{\delta}\notin B(z_{0},\beta)\\}.$

Let $\tau_{\delta}(c_{\delta}d_{\delta})$ be the hitting time of
$\mathcal{R}_{\delta}$ at $(c_{\delta}d_{\delta})$. For every $r>0$, define
$\text{LE}(\mathcal{R}_{\delta}[0,r])$ to be the loop erasure of
$\mathcal{R}_{\delta}[0,r]$. Define the event
$\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\text{LE}(\mathcal{R}_{\delta}[0,t_{j}]))$
similarly as $\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\gamma_{\delta})$ and
define the event $T_{j}:=\\{t_{j}\leq\tau_{\delta}(c_{\delta}d_{\delta})\\}$
that there are $j$ crossings. Denote
$\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\text{LE}(\mathcal{R}_{\delta}[0,t_{j}]))$
by $Y_{j}$ for simplicity. Note that $Y_{1}=\emptyset$ and

$\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\gamma_{\delta})\subset\\{\cup_{j=2}^{\infty}(Y_{j}\cap
T_{j})\\}\subset\\{T_{m+1}\cup(\cup_{j=2}^{m}Y_{j})\\},\quad\text{ for every
}m>1.$

Thus, we only need to control

$\mathbb{P}_{b_{\delta}^{+}}[\cup_{j=1}^{m}Y_{j}\,|\,\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]\text{ and
}\mathbb{P}_{b_{\delta}^{+}}[T_{m+1}\,|\,\mathcal{R}_{\delta}\text{ hits
}(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})].$

For the first estimate, we have

$\displaystyle\mathbb{P}_{b_{\delta}^{+}}[\cup_{j=1}^{m}Y_{j}\,|\,\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]$
$\displaystyle=$
$\displaystyle\frac{\mathbb{P}_{b_{\delta}^{+}}[\mathbb{1}_{\\{\mathcal{R}_{\delta}\text{
hits }\partial
B_{\delta}(z_{0},\beta/4)\\}}\mathbb{P}_{\mathcal{R}_{\delta}(s_{1})}[\\{\cup_{j=1}^{m}Y_{j}\\}\cap\\{\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before
}(a_{\delta}b_{\delta})\\}]]}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}$
$\displaystyle\leq$ $\displaystyle\frac{h_{\partial
B_{\delta}(z_{0},\beta/4)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\left(\max_{v_{\delta}\in\partial
B_{\delta}(z_{0},\beta/4)}\mathbb{P}_{v_{\delta}}[\cup_{j=1}^{m}Y_{j}]\right).$

By the Beurling estimate of Lemma 3.2, there exist $C>0$ and $k>0$, such that

$\max_{v_{\delta}\in\partial
B_{\delta}(z_{0},\beta/4)}\mathbb{P}_{v_{\delta}}[\cup_{j=1}^{m}Y_{j}]\leq
Cm^{2}\left(\frac{\alpha}{\beta}\right)^{k}.$

If $b_{\delta}^{+}\in B(z_{0},\beta/4)$, then combining with Lemma 3.5, there
exists $C>0$, such that

$\frac{h_{\partial
B_{\delta}(z_{0},\beta/4)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\leq
C.$

If $b_{\delta}^{+}\notin B(z_{0},\beta/4)$, then if we denote by
$d=d(b_{\delta}^{+},\partial B_{\delta}(z_{0},\beta/4))$, we have

$\frac{h_{\partial
B_{\delta}(z_{0},\beta/4)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\leq\frac{h_{\partial
B_{\delta}(b_{\delta}^{+},d)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\leq
C,$

where the last inequality is from Lemma 3.5. Thus, we have

$\mathbb{P}_{b_{\delta}^{+}}[\cup_{j=1}^{m}Y_{j}\,|\,\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]\leq
Cm^{2}\left(\frac{\alpha}{\beta}\right)^{k}.$

For the second estimate, we have

$\displaystyle\mathbb{P}_{b_{\delta}^{+}}[T_{m+1}\,|\,\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]$
$\displaystyle=$
$\displaystyle\frac{\mathbb{P}_{b_{\delta}^{+}}[\mathbb{1}_{\\{\mathcal{R}_{\delta}\text{
hits }\partial
B_{\delta}(z_{0},\beta)\\}}\mathbb{P}_{\mathcal{R}_{\delta}(t_{1})}[\\{t_{j}\leq\tau\\}\cap\\{\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before
}(a_{\delta}b_{\delta})\\}]]}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}$
$\displaystyle\leq$ $\displaystyle\frac{h_{\partial
B_{\delta}(z_{0},\beta)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\left(\max_{v_{\delta}\in\partial
B_{\delta}(z_{0},\beta)}\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
hits }\partial B_{\delta}(z_{0},\beta/4)\text{ before
}(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})]\right)^{j}.$

By the same argument in [HLW20, Equation C.2] (or by using a uniform crossing
estimate), we have that there exists $0<q=q(\beta)<1$ such that for every
$\delta>0$,

$\min_{v_{\delta}\in\partial
B_{\delta}(z_{0},\beta)}\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
hits }(a_{\delta}b_{\delta})\cup(c_{\delta}d_{\delta})\text{ before }\partial
B_{\delta}(z_{0},\beta/4)]\geq q.$

By the same argument in the first estimate, we have that there exists $C>0$,
such that

$\mathbb{P}_{b_{\delta}^{+}}[T_{m+1}\,|\,\mathcal{R}_{\delta}\text{ hits
}(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]\leq C(1-q)^{j}.$

(See also Lemma 4.13 in [BLR20] for a similar estimates). Thus, for every
$\epsilon>0$, by first choosing $m$ large enough and then choosing appropriate
$\alpha=\alpha(\beta,\epsilon)$, we have

$\mathbb{P}[\mathcal{A}_{\delta}(z_{0},\beta,\alpha;\gamma_{\delta})]\leq\epsilon.$

This completes the proof. ∎

By Lemma 3.7, we can deduce the tightness of
$\\{\gamma_{\delta}\\}_{\delta>0}$.

######  Lemma 3.8.

The sequence $\\{\gamma_{\delta}\\}_{\delta>0}$ is tight. Moreover, any
subsequential limit is a simple curve in $\overline{\Omega}$ which intersects
$[ab]\cup[cd]$ only at two ends.

###### Proof.

The tightness of $\gamma_{\delta}$ follows from Lemma 3.7 using the ingenious
argument given in [LSW04, Lemma 3.12]: essentially, a family of curves without
bubbles forms a compact set. See also the proof of [HLW20, Lemma C.2] where
this argument is also discussed. Since the proof of tightness is identical, we
do not include here.

Thus let us suppose that $\gamma$ is any subsequential limit. Without loss of
generality and with a slight abuse of notation, we will use
$\\{\gamma_{\delta}\\}_{\delta>0}$ for the subsequence converging to $\gamma$.
We begin with the proof that $\gamma$ intersects $(ab)\cup(cd)$ at exactly two
ends. For this we modify the argument in [HLW20, Lemma C.2]. It suffices to
show that almost surely $\gamma$ intersects $[ab]$ at one end, since the proof
for $[cd]$ is identical.

For every $s>0$, we define $\tau_{s,\delta}$ to be the first time that
$\gamma_{\delta}$ hits $\partial B_{\delta}(b^{+}_{\delta},s)$ and define

$\tau_{s,\delta}(\epsilon):=\inf\\{t\geq\tau_{s,\delta}:d(\gamma_{\delta}(t),(a_{\delta}b_{\delta}))<\epsilon\\}.$

Note that

$\displaystyle\mathbb{P}[\tau_{s,\delta}(\epsilon)<\infty]\leq$
$\displaystyle\frac{\mathbb{P}_{b_{\delta}^{+}}[\mathcal{R}_{\delta}\text{
hits }\partial B_{\delta}(b_{\delta}^{+},s)\text{ before
}(a_{\delta}b_{\delta})]}{\mathbb{P}_{b_{\delta}^{+}}[\mathcal{R}_{\delta}\text{
hits }(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]}$
$\displaystyle\times\max_{v_{\delta}\in\partial
B_{\delta}(b_{\delta}^{+},s)}\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
hits the }\epsilon\text{-neighbour of }(a_{\delta}b_{\delta})\text{ and hits
}(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})]$
$\displaystyle=$ $\displaystyle\frac{h_{\partial
B_{\delta}(b_{\delta}^{+},s)}(b_{\delta}^{+})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}$
$\displaystyle\times\max_{v_{\delta}\in\partial
B_{\delta}(b_{\delta}^{+},s)}\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
hits the }\epsilon\text{-neighbour of }(a_{\delta}b_{\delta})\text{ and hits
}(c_{\delta}d_{\delta})\text{ before }(a_{\delta}b_{\delta})].$

Combining with the Beurling estimate of Lemma 3.2 and Lemma 3.5 on the
limiting behaviour of the ratio of harmonic functions on the right hand side,
we have that there exist $C=C(s)>0$ and $c>0$, such that

$\mathbb{P}[\tau_{s,\delta}\leq\tau_{s,\delta}(\epsilon)<\infty]\leq
C\epsilon^{c}.$

Define $\tau_{s}$ and $\tau_{s}(\epsilon)$ for $\gamma$ similarly as defining
$\tau_{s,\delta}$ and $\tau_{s,\delta}(\epsilon)$ for $\gamma_{\delta}$. By
letting $\delta\to 0$, we have

$\mathbb{P}[\tau_{s}\leq\tau_{s}(\epsilon)<\infty]\leq C\epsilon^{c}.$

By letting $\epsilon\to 0$, we know that almost surely, $\gamma$ will never
hit $(ab)$ after the time $\tau_{s}$. By letting $s\to 0$, we know that almost
surely, $\gamma$ intersects $[ab]$ only at one end. This completes the proof.
∎

Now, we begin to identify the limiting curve. We fix a subsequential limit
$\gamma$ and we suppose $\\{\gamma_{\delta_{n}}\\}_{n\geq 0}$ converge to
$\gamma$ in law. We parametrise $\gamma_{\delta_{n}}$ and $\gamma$ by the time
interval $[0,1]$. We will show that $\gamma$ is a chordal
$\operatorname{SLE}_{2}(-1,-1;-1,-1)$ curve in $\Omega$ from $b$ to $(cd)$
with the marked points given by $(a,d;b,c)$ (recall this curve is target
invariant, so we do not need to specify the target point of $\gamma$ in
$\Omega$, and just stop it upon hitting the arc $(cd)$).

The proof is almost the same as the proof of [HLW20, Theorem 1.6]. The main
difference is that because of the conditioning, we can not deduce that
$\gamma\cap((bc)\cup(da))=\emptyset$ at first. We will explain how to modify
the proof in our setting in detail.

######  Proposition 3.9.

The discrete curves $\\{\gamma_{\delta}\\}_{\delta>0}$ converges to
$\operatorname{SLE}_{2}(-1,-1;-1,-1)$ curve from $b$ to $(cd)$ with the marked
points $(a,d;b,c)$ in law, when $\delta\to 0$.

###### Proof.

For every $s>0$, we define
$\tau_{n}(s):=\inf\\{t>0:d(\gamma_{\delta_{n}}(t),b_{\delta_{n}}^{+})\geq
s\\}$ and define $\tau(s):=\inf\\{t>0:d(\gamma(t),b)\geq s\\}$. We may couple
$\\{\gamma_{\delta_{n}}\\}_{n\geq 0}$ and $\gamma$ together such that
$\gamma_{\delta_{n}}$ converges to $\gamma$ almost surely. By considering the
continuous modification, we may also assume that $\tau_{n}(s)$ converges to
$\tau(s)$ almost surely. See for instance [Kar19] for details.

First of all, we will prove that $\gamma(\tau(s))$ does not belong to $(bc)$
almost surely. Denote by $l_{s}$ the connected component of $\partial
B(b,s)\cap\Omega$ seperating $b$ and $(cd)$. We define $y:=l_{s}\cap(bc)$ and
define $y_{\delta_{n}}$ to be a discrete approximation of $y$ on
$(b_{\delta_{n}}c_{\delta_{n}})$. For every $\epsilon>0$, we have that

$\mathbb{P}[\gamma_{\delta_{n}}\text{ hits }\partial
B_{\delta}(y_{\delta},2\epsilon)]\leq\frac{h_{\partial
B_{\delta_{n}}(y_{\delta_{n}},2\epsilon)}(b_{\delta_{n}}^{+})}{h_{(c_{\delta_{n}}d_{\delta_{n}})}(b_{\delta_{n}}^{+})}.$

By letting $n\to\infty$, by Lemma 3.5, we have that

$\mathbb{P}[y\in\gamma]\leq\mathbb{P}[\gamma\text{ hits }\partial
B(y,\epsilon)]\leq\frac{\partial_{n}(h_{\partial
B(y,2\epsilon)}\circ\varphi^{-1})(b)}{\partial_{n}(h_{(cd)}\circ\varphi^{-1})(b)},$

where $\varphi$ is the conformal map from $\Omega$ onto the rectangle. By
letting $\epsilon\to 0$, we have

$\mathbb{P}[y\in\gamma]=0.$ (3.8)

This implies that $\gamma(\tau(s))$ does not belong to $(bc)$ almost surely.

Second, we define
$T(\epsilon):=\inf\\{t>\tau(s):d(\gamma(\tau(s)),\partial\Omega)\leq\epsilon\\}$.
We denote by $\Omega(t)$ the connected component of
$\Omega\setminus\gamma[0,(t+\tau(s))\wedge T(\epsilon)]$ which contains $d$
and denote by $b(0)$ the last hitting point of $\gamma[0,(t+\tau(s))\wedge
T(\epsilon)]$ on $[bc]$ (if there is one). We define $T_{n}(\epsilon)$ and
$\Omega_{\delta_{n}}(t)$ and $b_{\delta_{n}}(0)$ similarly in the discrete
setting. By considering the continuous modification, we may also assume that
$T_{n}(\epsilon)$ converges to $T(\epsilon)$ almost surely. Now, we explain
briefly how to deduce that given $\gamma[0,\tau(s)]$, the conditional law of
$\gamma$ is the same as the law of $\operatorname{SLE}_{2}(-1,-1;-1,-1)$ in
$\Omega(0)$, which starts from $\gamma(\tau(s))$ with marked points
$(a,d;b(0),c)$.

We fix $\tilde{c},\tilde{d}\in[cd]$. We denote by
$\tilde{c}_{\delta_{n}},\tilde{d}_{\delta_{n}}$ some discrete approximations
of $(c_{\delta_{n}}d_{\delta_{n}})$. Given $\Omega_{\delta_{n}}(t)$, for
$v_{\delta_{n}}\in\Omega_{\delta_{n}}(t)$, we define

$h_{t,(\tilde{c}_{\delta_{n}}\tilde{d}_{\delta_{n}})}(v_{\delta_{n}}):=\mathbb{P}[\mathcal{R}_{\delta_{n}}\text{
exits }\Omega_{\delta_{n}}(t)\text{ at
}(\tilde{c}_{\delta_{n}}\tilde{d}_{\delta_{n}})],$

where $\mathcal{R}_{\delta_{n}}$ has the same law as the random walk on
$\Omega_{\delta_{n}}(t)$, which starts from $v_{\delta_{n}}$ and is killed on
$(a_{\delta_{n}}b_{\delta_{n}}(t))\cup(c_{\delta_{n}}d_{\delta_{n}})$, with
the reflecting boundaries
$(b_{\delta_{n}}(t)c_{\delta_{n}})\cup(d_{\delta_{n}}a_{\delta_{n}})$.

Note that for $r<t$, by the domain Markov property for loop-erased random
walk,

$\displaystyle\mathbb{P}[\gamma_{\delta_{n}}\text{ hits
}(\tilde{c}_{\delta_{n}}\tilde{d}_{\delta_{n}})\,|\,\gamma_{\delta_{n}}[0,(t+\tau_{n}(s))\wedge
T_{n}(\epsilon)]]=$
$\displaystyle\frac{\sum_{v_{\delta_{n}}\sim\gamma_{\delta_{n}}((t+\tau_{n}(s))\wedge
T_{n}(\epsilon))}h_{t,(\tilde{c}_{\delta_{n}}\tilde{d}_{\delta_{n}})}(v_{\delta_{n}})}{\sum_{v_{\delta_{n}}\sim\gamma_{\delta_{n}}((t+\tau_{n}(s))\wedge
T_{n}(\epsilon))}h_{t,(c_{\delta_{n}}d_{\delta_{n}})}(v_{\delta_{n}})}.$ (3.9)

Recall that $\varphi$ is the conformal map from $(\Omega;a,b,c,d)$ onto the
rectangle, which maps $(a,b,c,d)$ to $(0,1,1+\mathfrak{i}K,\mathfrak{i}K)$. We
fix the conformal map $f$ from the rectangle onto $\mathbb{H}$, which maps
$(0,1,\mathfrak{i}K)$ to $(0,1,\infty)$. We denote $f\circ\varphi(c)$ by $x$
and denote $f\circ\varphi(\tilde{c})$ by $\tilde{x}$ and denote
$f\circ\varphi(\tilde{d})$ by $\tilde{y}$. We fix the conformal map
$h_{\tau(s)}$ from $f\circ\varphi(\Omega(0))$ onto $\mathbb{H}$ such that
$h_{\tau(s)}(z)=z+o(1)$ when $z$ tends to $\infty$. We denote by
$W_{\tau(s),\cdot}$ the driving function of $h_{\tau(s)}\circ
f\circ\varphi(\gamma[\tau(s),T(\epsilon)])$ and denote by
$(h_{\tau(s),t};0\leq t\leq T(\epsilon)-\tau(s))$ the corresponding Loewner
flow. Then, by using results from [HLW20] (see in particular the start of
proof of Theorem 1.6), it follows that the conditional law of
$h_{\tau(s)}\circ f\circ\varphi(\gamma[\tau(s),T(\epsilon)])$ given
$\gamma[0,\tau(s)]$ must be an SLE${}_{2}(-1,-1,-1,-1)$. (Indeed, given
$\gamma[0,\tau(s)]$ the rest of the curve is the loop-erasure of a chordal SLE
in which the boundary conditions alternate between Dirichlet and Neumann, and
the change of boundary condition does not occur at the tip of the curve, so
this is exactly the setup of [HLW20]). Briefly, we recall the argument: first,
using (3.9) and the same argument as in [HLW20, Corollary 5.5], we have

$\displaystyle\lim_{n\to\infty}\mathbb{P}[\gamma_{\delta_{n}}\text{ hits
}(\tilde{c}_{\delta_{n}}\tilde{d}_{\delta_{n}})\,|\,\gamma_{\delta_{n}}[0,(t+\tau_{n}(s))\wedge
T_{n}(\epsilon)]]$ $\displaystyle=$
$\displaystyle\int_{h_{\tau(s),t}(\tilde{x})}^{h_{\tau(s),t}(\tilde{y})}\partial_{n}P(z;h_{\tau(s),t}(0),W_{\tau(s),t},h_{\tau(s),t}\circ
h_{\tau(s)}\circ f\circ\varphi(b(0)),h_{\tau(s),t}\circ
h_{\tau(s)}(x))|_{z=w}dw,$

where $P$ is the Poisson kernel corresponding to our boundary conditions,
which is given in [HLW20, Equation 5.10]. By the same argument in [HLW20,
Lemma 5.10] (similar to [LSW04]), we know that for $w\in(x,\infty)$,

$h_{\tau(s),t}^{\prime}(w)\partial_{n}P(z;h_{\tau(s),t}(0),W_{\tau(s),t},h_{\tau(s),t}\circ
h_{\tau(s)}\circ f\circ\varphi(b(0)),h_{\tau(s),t}\circ
h_{\tau(s)}(x))|_{z=w}$

is a martingale for $h_{\tau(s)}\circ
f\circ\varphi(\gamma[\tau(s),T(\epsilon)])$. This can be used to identify
uniquely the law of the driving function.

In particular $\gamma$ will not hit $[bc]\cup[da]$ after time $\tau(s)$. By
letting $\epsilon\to 0$, we know that given $\gamma[0,\tau(s)]$, the
conditional law of $\gamma$ is the same as the law of
$\operatorname{SLE}_{2}(-1,-1;-1,-1)$ in $\Omega(0)$, which starts from
$\gamma(\tau(s))$ with marked points $(a,d;b(0),c)$. Finally, by letting $s\to
0$, this implies that $\gamma$ only hits $\partial\Omega$ at its two ends.
This implies that we always have $b(0)=b$. We denote by $W$ the driving
function of $f\circ\varphi(\gamma)$. We denote by $(g_{t};t\geq 0)$ the
corresponding Loewner flow. Then, we have

$W_{\tau(s)+t}=W_{\tau(s),t}\quad\text{and}\quad
g_{\tau(s)+t}=h_{\tau(s),t}\circ h_{\tau(s)}.$

By definition of $\operatorname{SLE}_{2}(-1,-1;-1,-1)$, after parameterizing
$f\circ\varphi(\gamma)$ by its half-plane capacity, we have

$W_{t+\tau(s)}-W_{\tau(s)}=\sqrt{2}(B_{t+\tau(s)}-B_{\tau(s)})+\int_{\tau(s)}^{\tau(s)+t}\left(\frac{-1}{W_{r}-g_{r}(0)}+\frac{-1}{W_{r}-g_{r}(1)}+\frac{-1}{W_{r}-g_{r}(x)}\right)dr,$

for $t\leq T(\epsilon)-\tau(s)$. By letting $\epsilon\to 0$ and then letting
$s\to 0$, we have that

$W_{t}-W_{0}=\sqrt{2}B_{t}+\int_{0}^{t}\left(\frac{-1}{W_{r}-g_{r}(0)}+\frac{-1}{W_{r}-g_{r}(1)}+\frac{-1}{W_{r}-g_{r}(x)}\right)dr,$

where $B$ is the standard $1$-dimensional Brownian motion. Thus, we have that
$\gamma$ has the same law as the law of $\operatorname{SLE}_{2}(-1,-1;-1,-1)$
in $\Omega$, which starts from $b$ with marked points $(a,d;b,c)$. This
completes the proof. ∎

Now, we discuss the convergence of $\\{\eta_{L}^{\delta}\\}_{\delta>0}$ and
$\\{\eta_{R}^{\delta}\\}_{\delta>0}$. In [HLW20], the authors considered the
$\operatorname{UST}$ on
$(\Omega_{\delta};a_{\delta},b_{\delta},c_{\delta},d_{\delta})$ where
$(a_{\delta}b_{\delta})$ and $(c_{\delta}d_{\delta})$ are wired. As mentioned
before, this tree has a unique branch connecting the two wired vertices to one
another, which we denote by $\tilde{\gamma}_{\delta}$ (note that this curve is
not conditioned on its starting point). These authors proved that the discrete
Peano curves on either side of $\tilde{\gamma}_{\delta}$, which we denote by
$\tilde{\eta}_{L}^{\delta}$ and $\tilde{\eta}_{R}^{\delta}$, converge to a
pair of $\operatorname{hSLE}_{8}$. In our setting, the tightness of discrete
Peano curves may not been proved as in [HLW20], since we condition on an event
with very small probability. But assuming the convergence of
$\tilde{\eta}_{L}^{\delta}$ and $\tilde{\eta}_{R}^{\delta}$, we can prove the
convergence of $\eta_{L}^{\delta}$ and $\eta_{R}^{\delta}$ before hitting
$(a_{\delta}b_{\delta})$. In the remaining part, we will only consider the
convergence of $\\{\eta_{R}^{\delta}\\}_{\delta>0}$, since the convergence for
$\\{\eta_{L}^{\delta}\\}_{\delta>0}$ is similar.

We denote by $\tau_{\epsilon}^{\delta}$ to be the hitting time by
$\eta_{R}^{\delta}$ of the $\epsilon$-neighbourhood of
$(d_{\delta}b_{\delta})$ and denote by $\tilde{\tau}^{\delta}_{\epsilon}$ for
$\tilde{\eta}_{R}^{\delta}$ similarly. We denote by $\tilde{\eta}_{R}$ the
$\operatorname{hSLE}_{8}$ curve from $c$ to $b$ with marked points $d,a$. We
first recall the driving function of this curve. Let us fix a conformal map
$\phi$ from $\Omega$ onto $\mathbb{H}$, which maps $(c,d,b)$ to
$(0,1,\infty)$. Suppose $\tilde{W}$ is the driven function of
$\phi(\tilde{\eta}_{R})$ and $(\tilde{g}_{t}:t\geq 0)$ the corresponding
conformal maps. Then, before hitting $(\phi(a),+\infty)$, by definition, the
law of $\tilde{W}$ can be described as that of the solution to the following
hypergeometric stochastic differential equation:

$d\tilde{W}_{t}=\sqrt{8}dB_{t}+\left(\frac{2}{\tilde{W}_{t}-\tilde{g}_{t}(\phi(d))}-\frac{2}{\tilde{W}_{t}-\tilde{g}_{t}(\phi(a))}-8\frac{\tilde{g}_{t}(\phi(a))-\tilde{g}_{t}(\phi(d))}{\left(\tilde{g}_{t}(\phi(a))-\tilde{W}_{t}\right)^{2}}\frac{F^{\prime}\left(\frac{\tilde{g}_{t}(\phi(d))-\tilde{W}_{t}}{\tilde{g}_{t}(\phi(a))-\tilde{W}_{t}}\right)}{F\left(\frac{\tilde{g}_{t}(\phi(d))-\tilde{W}_{t}}{\tilde{g}_{t}(\phi(a))-\tilde{W}_{t}}\right)}\right)dt,$
(3.10)

where $F(\cdot)={}_{2}F_{1}(1/2,1/2,1;\cdot)$. See [HLW20] for more details.
For every
$v_{\delta}\in\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t]$, we
define

$h_{t}(v_{\delta}):=\mathbb{P}[\mathcal{R}_{\delta}\text{ hits
}(\tilde{\eta}_{R}^{\delta}(t)d_{\delta})\text{ before
}(a_{\delta}b_{\delta})],$

where $\mathcal{R}_{\delta}$ has the same law as as before, namely a random
walk starting from $v_{\delta}$ and ending at
$(a_{\delta}b_{\delta})\cup(\tilde{\eta}_{R}^{\delta}(t)d_{\delta})$, with
reflecting boundaries $(b_{\delta}\tilde{\eta}_{R}^{\delta}(t))$ and
$(d_{\delta}a_{\delta})$.

We still need two notations. We denote by $SF_{2}^{\delta}(\Omega_{\delta})$
the set of spanning forests on $\Omega_{\delta}$ where
$(a_{\delta}b_{\delta})$ and $(c_{\delta}d_{\delta})$ are wired but belong to
different components. We denote by $ST^{\delta}(\Omega_{\delta})$ the set of
spanning trees on $\Omega_{\delta}$ (where, once again,
$(a_{\delta}b_{\delta})$ and $(c_{\delta}d_{\delta})$ are wired).

######  Lemma 3.10.

Suppose $r$ is a continuous function on the curves space $(\mathcal{P},d)$.
Then, for every $t>0$, we have

$\mathbb{E}[r(\eta_{R}^{\delta}[0,t\wedge\tau^{\delta}_{\epsilon}])]=\mathbb{E}\left[r(\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])\frac{h_{t\wedge\tilde{\tau}^{\delta}_{\epsilon}}(b^{+}_{\delta})\times\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})\times\frac{|SF_{2}^{\delta}(\Omega_{\delta})|}{|ST^{\delta}(\Omega_{\delta})|}}\right],$

where
$SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])$
and
$ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])$
are defined for
$\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}]$
similarly to $SF_{2}^{\delta}(\Omega_{\delta})$ and
$ST^{\delta}(\Omega_{\delta})$, and $h_{(c_{\delta}d_{\delta})}$ is defined in
(3.5).

###### Proof.

We denote by $|S|$ the cardinality of a set $S$. Note that

$\displaystyle\mathbb{P}[\text{ the first step of
}\tilde{\gamma}_{\delta}\text{ is }b_{\delta}b_{\delta}^{+}]$ $\displaystyle=$
$\displaystyle\frac{|\\{t^{\delta}\in ST^{\delta}(\Omega_{\delta}):\text{ the
first step of the branch connecting }a_{\delta}b_{\delta}\text{ to
}c_{\delta}d_{\delta}\text{ is
}b_{\delta}b_{\delta}^{+}\\}|}{|ST^{\delta}(\Omega_{\delta})|}$
$\displaystyle=$
$\displaystyle\frac{|SF_{2}^{\delta}(\Omega_{\delta})|}{|ST^{\delta}(\Omega_{\delta})|}\times\frac{|\\{f^{\delta}\in
SF_{2}^{\delta}(\Omega_{\delta}):b_{\delta}^{+}\text{ connects to
}(c_{\delta}d_{\delta})\\}|}{|SF_{2}^{\delta}(\Omega_{\delta})|}$
$\displaystyle=$
$\displaystyle\frac{|SF_{2}^{\delta}(\Omega_{\delta})|}{|ST^{\delta}(\Omega_{\delta})|}\times
h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+}),$

where the last equality is from Wilson’s algorithm. The conclusion is obtained
from the domain Markov property of the $\operatorname{UST}$. ∎

Now, we prove the convergence of
$\\{\eta_{R}^{\delta}[0,\tau^{\delta}_{\epsilon}]\\}_{\delta>0}$ for every
$\epsilon>0$. We define $\tilde{\tau}_{\epsilon}$ to be the hitting time by
$\tilde{\eta}_{R}$ of the $\epsilon$-neighbourhood of $(db)$.

######  Proposition 3.11.

The discrete Peano curves
$\\{\eta_{R}^{\delta}[0,\tau^{\delta}_{\epsilon}]\\}_{\delta>0}$ converge in
law to $\operatorname{SLE}_{8}(2,2)$ with marked points $d,a$ stopped at
$\tau_{\epsilon}$. The precise formula of the driven function is given in
(3.13).

###### Proof.

By [HLW20, Theorem 1.5], we have $\tilde{\eta}_{R}^{\delta}$ converges to
$\tilde{\eta}_{R}$ in law. We couple
$\\{\tilde{\eta}_{R}^{\delta}\\}_{\delta>0}$ and $\tilde{\eta}_{R}$ together,
such that $\tilde{\eta}_{R}^{\delta}$ converges to $\tilde{\eta}_{R}$ almost
surely. If we define
$\tilde{\tau}_{\epsilon,+}:=\inf\\{\tilde{\tau}_{\epsilon^{\prime}}:\epsilon^{\prime}<\epsilon\\}$,
from the absolute continuity of $\tilde{\eta}_{R}$ with respect to
$\operatorname{SLE}_{8}$, we know that almost surely,
$\tilde{\tau}_{\epsilon}=\tilde{\tau}_{\epsilon,+}$. Thus, under this
coupling, we have $\tilde{\tau}^{\delta}_{\epsilon}$ converges to
$\tilde{\tau}_{\epsilon}$ almost surely.

By the same argument as in Lemma 3.4, we can prove that
$h_{t\wedge\tilde{\tau}^{\delta}_{\epsilon}}$ converges a.s. locally uniformly
(under this coupling distribution). We denote the limit by
$h_{t\wedge\tilde{\tau}_{\epsilon}}$, and we note that it is a bounded
harmonic function which satisfies the following boundary conditions:

$h_{t\wedge\tilde{\tau}_{\epsilon}}\circ\phi^{-1}\circ\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}^{-1}\begin{cases}=1&\text{
on
}(\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}},\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d)))\\\
=0&\text{ on }(\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a)),\infty)\\\
\end{cases}$

and

$\partial_{n}\left(h_{t\wedge\tilde{\tau}_{\epsilon}}\circ\phi^{-1}\circ\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}^{-1}\right)=0\text{
on
}(-\infty,\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}})\cup(\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d)),\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))).$

By [HLW20, Lemma 4.6], we have that
$\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}$
converges to the conformal modulus of the domain
$(\Omega\setminus\tilde{\eta}_{R}[0,t\wedge\tilde{\tau}_{\epsilon}])$ with
marked points $(a,b,\tilde{\eta}(t\wedge\tilde{\tau}_{\epsilon}),d)$, as
$\delta\to 0$ (as can be seen from that proof, the only requirement for this
convergence is the Carathéodory convergence of the domains). Thus, if we
denote by $f_{t\wedge\tilde{\tau}_{\epsilon}}$ the conformal map from
$(\mathbb{H};\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}},\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d)),\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a)),\infty)$
onto $(R;0,1,1+\mathfrak{i}K,K)$, where $R$ is a rectangle with unit width,
then we have

$h_{t\wedge\tilde{\tau}^{\delta}_{\epsilon}}(\cdot)\times\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}$

converges to
$K-\mathrm{Im}f_{t\wedge\tilde{\tau}_{\epsilon}}\circ\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}\circ\phi$
locally uniformly on
$\Omega\setminus\tilde{\eta}_{R}[0,t\wedge\tilde{\tau}_{\epsilon}]$ (as this
is the only bounded harmonic function satisfying the required boundary
conditions on $R$). From the Schwarz–Christorffel formula, we have

$f_{t\wedge\tilde{\tau}_{\epsilon}}(z)=\frac{\int_{0}^{\frac{z-\tilde{W}_{t}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}}\left(s(s-1)\left(s-\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right)\right)^{-1/2}ds}{\int_{0}^{1}\left(s(s-1)\left(s-\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right)\right)^{-1/2}ds}.$

For the denominator, if we change the variable by setting $s=\sin^{2}\theta$,
we have

$\displaystyle\int_{0}^{1}\left(s(s-1)\left(s-\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right)\right)^{-1/2}ds$
$\displaystyle=$ $\displaystyle
2\sqrt{\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}}\times\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\sin^{2}\theta}}$
$\displaystyle=$ $\displaystyle
2\sqrt{\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}}\times{}_{2}F_{1}\left(1/2,1/2,1;\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right),$

where the last equality is obtained from the standard relation between
elliptic function and hypergeometric function. In particular, as $z\to\infty$,

$f^{\prime}_{t\wedge\tilde{\tau}_{\epsilon}}(z)\sim\sqrt{{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}}\times
z^{-3/2}\times\frac{1}{{}_{2}F_{1}\left(1/2,1/2,1;\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right)}.$

Furthermore, since

$g_{t\wedge\tilde{\tau}_{\epsilon}}(z)=z+o(1)\quad\text{as }z\to\infty,$

we obtain by Lemma 3.5,

$\displaystyle\lim_{\delta\to
0}\frac{h_{t\wedge\tilde{\tau}^{\delta}_{\epsilon}}(b^{+}_{\delta})\times\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})\times\frac{|SF_{2}^{\delta}(\Omega_{\delta})|}{|ST^{\delta}(\Omega_{\delta})|}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\phi(a)-\tilde{W}_{0}}}\times\frac{{}_{2}F_{1}\left(1/2,1/2,1;\frac{\phi(d)-\tilde{W}_{0}}{\phi(a)-\tilde{W}_{0}}\right)}{{}_{2}F_{1}\left(1/2,1/2,1;\frac{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}{\tilde{g}_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))-\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}}\right)}.$
(3.11)

We denote the limit by $M_{t\wedge\tilde{\tau}_{\epsilon}}$. Note that

$\frac{h_{t\wedge\tilde{\tau}_{b_{\delta},\epsilon},(c_{\delta}d_{\delta})}(b^{+}_{\delta})}{h_{(c_{\delta}d_{\delta})}(b_{\delta}^{+})}\leq
C$ (3.12)

where $C=C(\epsilon)$ depends only on $\epsilon>0$. Indeed, if the walk
$\mathcal{R}$ reaches
$\tilde{\eta}^{\delta}([0,\tilde{\tau}_{\epsilon}^{\delta}])$ before touching
$(a_{\delta}b_{\delta})$, then it can hit $(d_{\delta},c_{\delta})$ with
uniformly positive probability (depending only on $\epsilon$).

Moreover, we claim that the ratio

$\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}$

is uniformly bounded by a constant which may depend on $\epsilon$ but not on
$\tilde{\eta}$. Indeed, let us argue by contradiction. Suppose this was not
the case. Then we would find a sequence $\delta_{n}\to 0$ and a a sequences of
paths $\eta^{n}$ such that the ratio above tends to $\infty$ along the
sequence $\delta_{n}$. But arguing by compactness, we can always find a
furthersubsequence (call it $\delta_{n}$ again with an abuse of notations)
such that $\Omega\setminus\eta^{n}[0,\tau^{\delta_{n}}_{\epsilon}]$ converges
in the Carathéodory sense. But, as already mentioned (and see Lemma 4.6 in
[HLW20] for a proof), such a convergence is sufficient to guarantee the
convergence of the above ratio to conformal modulus of the limiting domain,
which gives the desired contradiction.

This implies that there exists a constant $C^{\prime}$ depending only on
$\epsilon$ and $\Omega$, such that for every $\delta>0$,

$\frac{|SF_{2}^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}{|ST^{\delta}(\Omega_{\delta}\setminus\tilde{\eta}_{R}^{\delta}[0,t\wedge\tilde{\tau}^{\delta}_{\epsilon}])|}\leq
C^{\prime}.$

Therefore, the martingale $M_{t\wedge\tilde{\tau}_{\epsilon}^{\delta}}$ is
uniformly bounded by a constant (which may depend on $\epsilon$ but not on
anything else). Thus, if $r$ is a continuous function on the curves space
$(\mathcal{P},d)$, by dominated convergence theorem, we have

$\lim_{\delta\to
0}\mathbb{E}[r(\eta_{R}^{\delta})[0,t\wedge\tau_{b_{\delta},\epsilon}]]=\mathbb{E}[r(\tilde{\eta}_{R}[0,t\wedge\tilde{\tau}_{\epsilon}])M_{t\wedge\tilde{\tau}_{\epsilon}}].$

This complete the proof of the convergence of
$\\{\eta_{R}^{\delta}[0,\tau_{\epsilon}^{\delta}]\\}_{\delta>0}$ in law. We
denote by $\eta_{R}[0,\tau_{\epsilon}]$ the limit.

Now, we derive the explicit formula of the law of
$\phi(\eta_{R}[0,\tau_{\epsilon}])$. Since $(\mathcal{P},d)$ is a metric
space, for any open set $O$, the indicator function $\mathbb{1}_{O}$ can be
approximated by bounded continuous functions $\\{r_{n}\\}_{n\geq 0}$ such that
$r_{n}\nearrow\mathbb{1}_{O}$. Thus, we have

$\mathbb{E}[\mathbb{1}_{O}(\eta_{R}[0,t\wedge\tau_{\epsilon}])]=\mathbb{E}\left[\mathbb{1}_{O}(\tilde{\eta}_{R}[0,t\wedge\tilde{\tau}_{\epsilon}])M_{t\wedge\tilde{\tau}_{\epsilon}}\right].$

This implies that the law of $(\eta_{R}[0,t\wedge\tau_{\epsilon}]:t\geq 0)$
equals the law of $(\tilde{\eta}_{R}[0,t\wedge\tilde{\tau}_{\epsilon}]:t\geq
0)$ weighted by the martingale $(M_{t\wedge\tilde{\tau}_{\epsilon}}:t\geq 0)$.
If we denote by $W$ the driving function of
$\phi(\eta_{R}[0,\tau_{\epsilon}])$ and denote by
$(g_{t\wedge\tau_{\epsilon}}:t\geq 0)$ the corresponding conformal maps, by
Girsanov’s theorem and (3.11), we have

$dW_{t\wedge\tilde{\tau}_{\epsilon}}=d\tilde{W}_{t\wedge\tilde{\tau}_{\epsilon}}+\frac{d\langle
M,\tilde{W}\rangle_{t\wedge\tilde{\tau}_{\epsilon}}}{M_{t\wedge\tilde{\tau}_{\epsilon}}}=\sqrt{8}dB_{t\wedge\tilde{\tau}_{\epsilon}}+\left(\frac{2}{W_{t\wedge\tilde{\tau}_{\epsilon}}-g_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(d))}+\frac{2}{W_{t\wedge\tilde{\tau}_{\epsilon}}-g_{t\wedge\tilde{\tau}_{\epsilon}}(\phi(a))}\right)d{t\wedge\tilde{\tau}_{\epsilon}}.$
(3.13)

This completes the proof. ∎

######  Remark 3.12.

  * •

By a similar proof, we have that for every $\epsilon>0$, the discrete Peano
curve $\eta_{L}^{\delta}$ stopped at the hitting time of the
$\epsilon$-neighbourhood of $(b_{\delta}d_{\delta})$ converges to chordal
$\operatorname{SLE}_{8}(2,-2)$ from $d$ to $a$ with force points $b,c$ stopped
at the hitting time of the $\epsilon$-neighbourhood of $(ab)$.

  * •

In the discrete setting, $\gamma_{\delta}$ is the left boundary of
$\eta_{L}^{\delta}$ and is also the right boundary of $\eta_{R}^{\delta}$. The
convergence of $\\{\eta_{L}^{\delta}\\}_{\delta>0}$ and
$\\{\eta_{R}^{\delta}\\}_{\delta>0}$ as well as that of $\gamma_{\delta}$ in
Proposition 3.9 imply that the left boundary of $\operatorname{SLE}_{8}(2,-2)$
and the right boundary of $\operatorname{SLE}_{8}(2,2)$ are both given by a
chordal $\operatorname{SLE}_{2}(-1,-1;-1,-1)$ starting from the appropriate
boundary point. This identity is equivalent to the coupling between the flow
line and counterflow lines given in [MS16]. In Section 5, we will further show
the convergence of the discrete winding field (i.e., dimer height function) to
the corresponding Gaussian free field under which flow and counterflow lines
are coupled.

  * •

Recall that the law of the $\operatorname{UST}$ we consider in this paper is
equivalent to the law of $\operatorname{UST}$ with alternating boundary
conditions (as described e.g. below Theorem 1.2), conditional on the discrete
Peano curve $\eta_{R}^{\delta}$ hitting $(a_{\delta}b_{\delta})$ at
$b_{\delta}$. By almost the same argument as above, it can be shown that if we
consider the $\operatorname{UST}$ with alternating boundary conditions, if we
now condition on the event that the discrete Peano curve $\eta_{R}^{\delta}$
hits $(a_{\delta}b_{\delta})$ at $z_{\delta}$ and assume $z_{\delta}$
converges to $z\in(ab)$, then the conditional law of $\eta_{R}^{\delta}$
converges to $\operatorname{SLE}_{8}(2,2,-4)$ with marked points $d$, $a$ and
$z$. This implies that given the hitting point of $\operatorname{hSLE}_{8}$ at
$(ab)$, which we denote by $z$, the conditional law will be
$\operatorname{SLE}_{8}(2,2;-4)$ with marked points $d$, $a$ and $z$. From
this Theorem 1.4 follows easily.

  * •

It is natural to ask whether the identities described above hold more
generally for chordal $\operatorname{hSLE}_{\kappa}(\nu)$ process for
$\kappa\in(4,8)$ and $\nu>\frac{\kappa}{2}-4$. In fact, in the forthcoming
paper [Liu23], the following result will be shown. Suppose $\eta$ has the same
law as a chordal $\operatorname{hSLE}_{\kappa}(\nu)$ curve from $b$ to $c$
with marked points $d$ and $a$, given the hitting point of $\eta$ at $(ab)$
(which we denote by $z$), the conditional law of $\eta$ equals to the law of
$\operatorname{SLE}_{\kappa}(2,\kappa-6,-4)$ with marked points $d,a$ and $z$.
As a consequence, using tools from imaginary geometry, this decomposition
implies the time-reversibility of $\operatorname{hSLE}_{\kappa}(\nu)$ for
$\kappa\in(4,8)$ and $\nu>-2$.

###  3.2 Convergence of the discrete trees

In this section, we will prove Theorem 3.1. Recall that at this stage we have
proved convergence of the interface $\gamma_{\delta}$ towards an SLE2 type
chordal curve, and we have proved convergence of the Peano curves on either
side of it, but only up until they hit $(ab)$. This is not completely
sufficient to get convergence of the discrete trees in the Schramm sense,
because in order to describe all the branches of the tree we would need to
know the convergence of the full Peano curves not just stopped when they hit
$(ab)$. On the other hand, it is clear that, at the discrete level, given
$\gamma_{\delta}$, the two components describing the rest of the tree can be
viewed as a uniform spanning trees in their respective domain with Dobrushin
boundary conditions, with the arcs $(da)$ and $(bc)$ reflecting, and every
other part of the boundary being wired. This is very close to the setup of the
original paper of Lawler, Schramm and Werner [LSW04], but there is a technical
subtlety: namely, in order to get a strong enough form of convergence (say
uniform) the authors of [LSW04] require the boundary to be smooth, which
obviously is not the case here (convergence in the sense of driving function
and hence Carathéodory convergence does not require smoothness in their paper,
unfortunately this is not sufficient here).

However, in order to upgrade the form of convergence from driving function
convergence to strong (or uniform up to reparametrisation) convergence, it
suffices to prove tightness with respect to this strong topology, as then the
limit law is uniquely identified by the convergence of the driving function
(which, we recall, holds without assumption on the domain). But tightness in
this sense is in fact not hard to show: it follows along the same line as
already argued in Lemma 3.8, and in fact is considerably simpler since there
is no need to condition on an event of asymptotically degenerate probability.
We leave the details to the reader; and obtain the convergence of the branches
of the tree $\mathcal{T}^{\delta}$ to the brances of $\mathcal{T}$ in the
sense of finite-dimensional distribution of the branches. To conclude the
proof of Theorem 3.1, it remains to apply the following well known compactness
argument, due to Schramm [Sch00, Theorem 10.2] (this is sometimes known as
Schramm’s lemma). Its proof is unchanged, and so we can simply quote the
result here:

######  Lemma 3.13.

For every $\epsilon>0$, there exist $\delta_{0}>0$ and $\epsilon_{0}>0$, such
that the following holds. For any set $\\{z_{1},\ldots,z_{n}\\}$ with the
property that every $z\in\Omega$, there exists $z_{i}$ such that there exists
a curve connecting $z$ and $z_{i}$ whose length is less than $\epsilon_{0}$.
We view $(ab)$ and $(cd)$ as two points and define
$V:=\\{z_{1},\ldots,z_{n},(ab),(cd)\\}$. Denote by $V_{\delta}$ the discrete
approximation on $\Omega_{\delta}$. Denote by $\mathcal{T}_{\delta}(V)$ the
minimal subtree containing $V_{\delta}$. Then,we have

$\mathbb{P}[d_{\mathcal{H}}(\mathcal{T}_{\delta},\mathcal{T}_{\delta}(V))>\delta_{0}]\leq\epsilon.$

This concludes the proof of Theorem 3.1.

##  4 General case

In this section, we will prove of the convergence of the discrete trees
considered in Lemma 2.1 and give the coupling of the limiting tree with a GFF
for $n\geq 3$ in the sense of Theorem 2.4. Fix a simply connected domain

$(\Omega;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1}),$

where $x_{1},\ldots,x_{2n}$ are marked boundary points on $\partial\Omega$
(understood as prime ends) in counterclockwise order and $z_{i}$ equals
$x_{2i}$ or $x_{2i+1}$ for $1\leq i\leq n-1$. (It may be useful to keep in
mind the combinatorial setup of Section 2.1.) We assume that $\partial\Omega$
is locally connected and there exists a simply connected domain
$\tilde{\Omega}$ whose boundary is $C^{1}$ and simple such that
$\Omega\subset\tilde{\Omega}$ and $\partial\Omega\cap\partial\tilde{\Omega}$
equals $\cup_{i=1}^{n}(x_{2i-1}x_{2i})$. Suppose a sequence of discrete simply
connected domains
$(\Omega_{\delta};x_{1}^{\delta},\ldots,x^{\delta}_{2n};z^{\delta}_{1},\ldots,z^{\delta}_{n-1})$
converges to $(\Omega;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$ in the
following sense: there exists $C>0$ such that for $1\leq i\leq n$, we have

$d((x^{\delta}_{2i-1}x^{\delta}_{2i}),(x_{2i-1}x_{2i}))\leq C\delta,\quad
d((x^{\delta}_{2i}x^{\delta}_{2i+1}),(x_{2i}x_{2i+1}))\to 0;$ (4.1)

and for $1\leq i\leq n-1$, we have

$d(z_{i}^{\delta},z_{i})\to 0.$ (4.2)

Recall the spanning trees appearing in Temperley’s bijection are described by
a set $\mathcal{U}(\overline{\Omega}^{*}_{0})$, see Lemma 2.1.

It turns out dealing with $\mathcal{U}(\overline{\Omega}^{*}_{0})$ is hard
because, although it can still be seen as an asymptotically degenerate
conditioning of the uniform spanning tree with alternating boundary conditions
(which is well understood by [HLW20]), this conditioning is still to
complicated to be described directly. Instead, we will describe the complement
of the event which serves to condition the tree with alternating boundary
conditions, using an inclusion-exclusion description. First, let
$\tilde{\mathcal{T}}_{\delta}$ be the UST on $\Omega_{\delta}$ with
alternating boundary conditions (where $c_{i}=(x_{2i}x_{2i+1})$ is wired for
each $1\leq i\leq n$). Let

$\tilde{E}=\tilde{E}(z_{1}^{\delta}z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta}z_{n-1}^{\delta,+})$

be the event that the first step of the branch in
$\tilde{\mathcal{T}}_{\delta}$ connecting $(x_{2i}^{\delta}x_{2i+1}^{\delta})$
to $(x_{2n}^{\delta}x_{1}^{\delta})$ is through the edge
$z_{i}^{\delta}z_{i}^{\delta,+}$. Thus
$\mathcal{U}(\overline{\Omega}^{*}_{0})$ consists of the restriction of
alternating spanning trees $\tilde{\mathcal{T}}_{\delta}$ to this event
$\tilde{E}(z_{1}^{\delta}z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta}z_{n-1}^{\delta,+})$.

Now, we consider a graph $G^{*}$ with the boundary condition that
$\cup_{i=1}^{n}(x^{\delta}_{2i-1}x^{\delta}_{2i})$ is wired to be a single
vertex (thus not only $c_{i}=(x_{2i}x_{2i+1})$ is wired, but these arcs are
also wired together). Fix $1\leq s\leq n-1$, and fix a subset of indices
$1\leq i_{1}<\ldots<i_{s}\leq n-1$. Define the set of spanning trees

$\mathcal{A}_{i_{1},\ldots,i_{s}}:=\\{\mathcal{T}^{*}_{\delta}\text{ on
}G^{*},z_{i_{k}}^{\delta,+}\text{ connects to
}(x_{2i_{k+1}}^{\delta}x_{2i_{k+1}+1}^{\delta})\text{ in
}\Omega_{\delta}\cap\mathcal{T}^{*}_{\delta},\text{ for }1\leq k\leq s\\},$
(4.3)

where here we used the convention that $i_{s+1}=i_{1}$ in this description.
Thus an equivalent description of the event $\mathcal{A}_{i_{1},\ldots,i_{s}}$
is that the edges
$(z^{\delta}_{i_{1}}z_{i_{1}}^{\delta,+}),\ldots,(z^{\delta}_{i_{s}}z_{i_{s}}^{\delta,+})$
are all closed (in the sense that they are not part of the tree
$\mathcal{T}^{*}_{\delta}$), but adding them to the tree would create a loop
connecting $(c_{i_{1}},\ldots,c_{i_{s}})$ in $\Omega_{\delta}$.

Finally, define the event

$E^{*}=E^{*}(\Omega_{\delta};z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta,+}):=\left(\cup_{s=1}^{n-1}\cup_{i_{1},\ldots,i_{s}}\mathcal{A}_{i_{1},\ldots,i_{s}}\right)^{c}.$
(4.4)

Note that there is a bijection from $\tilde{E}$ to $E^{*}$: for every
$\tilde{\mathcal{T}}_{\delta}\in\tilde{E}(z_{1}^{\delta}z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta}z_{n-1}^{\delta,+})$,
we delete the edges
$\\{z_{1}^{\delta}z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta}z_{n-1}^{\delta,+}\\}$
and connect $\cup_{i=1}^{n}(x_{2i}^{\delta}x_{2i+1}^{\delta})$ outside of
$\Omega_{\delta}$ through the wired vertex. The inclusion-exclusion principle
shows this is a bijection.

In this section, we will therefore focus on consider the convergence of the
uniform spanning tree $\mathcal{T}^{*}_{\delta}$ on $G^{*}$ after conditioning
it on the event $E^{*}$ above. We denote by $\gamma_{i}^{\delta}$ the path in
$\mathcal{T}^{*}_{\delta}$ starting from $z_{i}^{\delta,+}$ and ending at the
first time it hits $\cup_{j\neq i}(x^{\delta}_{2j}x^{\delta}_{2j+1})$. The
goal of this section is to show the following theorem.

######  Theorem 4.1.

The discrete tree $\mathcal{T}^{*}_{\delta}$ converges as $\delta\to 0$ under
the metric $d_{\mathcal{H}}$. Denote by $\mathcal{T}^{*}$ the limit. The
continuous tree $\mathcal{T}^{*}$ can be constructed as follows. Fix any dense
set $\\{z_{i}\\}_{i\geq 1}$ in $\Omega$. We first sample the coupling of
$\\{\gamma_{1},\ldots,\gamma_{n-1}\\}$ and the GFF
$h_{\Omega}+u_{\Omega}(\cdot;x_{1},\ldots,x_{2n};z_{1},\ldots,z_{n-1})$ as in
Theorem 2.4. Recall that we denote by $\Omega_{1},\ldots,\Omega_{n}$ the
connected components of $D\setminus\cup_{i=1}^{n-1}\gamma_{i}$. For each
$i\geq 1$, we denote by $j(i)$ the unique index $1\leq j(i)\leq n$ such that
$z_{i}\in\Omega_{j(i)}$. Then, we sample the flow line starting from interior
point $\gamma_{z_{i}}$ in $\Omega_{j(i)}$ similarly as in Theorem 3.1. Then
$\mathcal{T}^{*}$ is the closure of $\cup_{i\geq 1}\gamma_{z_{i}}$.

By the same arguments as in Section 3.2, it suffices to show the convergence
of the boundary branches:

######  Theorem 4.2.

The discrete curves $\\{\gamma_{1}^{\delta},\ldots,\gamma_{n-1}^{\delta}\\}$
converge as $\delta\to 0$ to the flow lines
$\\{\gamma_{1}^{I},\ldots,\gamma_{n-1}^{I}\\}$ defined in Theorem 2.4.

###  4.1 Tightness of the boundary branches

We start with the tightness of the boundary branches. First of all, we will
give an estimate about the probability of
$E^{*}(\Omega_{\delta};x_{1}^{\delta},\ldots,x_{2n}^{\delta};z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta,+})$.
Fix a small $r>0$ such that $\partial B(x_{i},2r)\cap\partial
B(x_{j},2r)=\emptyset$ for $i\neq j$. We denote by $\partial
B(x_{i}^{\delta},r)$ the discrete approximation of $\partial B(x_{i},r)$ on
$\Omega_{\delta}$. For every
$v_{\delta}\in\cup_{i=1}^{n-1}B(z_{i}^{\delta},r)$, we define the harmonic
function

$h^{\delta}_{r}(v_{\delta}):=\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
hits }\cup_{i=1}^{n-1}\partial B(z_{i}^{\delta},r)\text{ before
}\cup_{i=1}^{n}(x_{2i}^{\delta}x_{2i+1}^{\delta})],$

where $\mathcal{R}_{\delta}$ has the same law as the random walk killed at
$\cup_{i=1}^{n}(x_{2i}^{\delta}x_{2i+1}^{\delta})$, and with reflecting
boundaries $\cup_{i=1}^{n}(x_{2i-1}^{\delta}x_{2i}^{\delta})$.

######  Lemma 4.3.

There exists $C>0$ and $c>0$ such that

$c\leq\frac{\mathbb{P}[\mathcal{T}^{*}_{\delta}\in
E^{*}(\Omega_{\delta};x_{1}^{\delta},\ldots,x_{2n}^{\delta};z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta,+})]}{\Pi_{i=1}^{n-1}h^{\delta}_{r}(z_{i}^{\delta,+})}\leq
C.$

###### Proof.

For the lower bound, we consider $n-1$ disjoint open sets
$\\{U_{1},\ldots,U_{n}\\}$ such that $B(z_{i},r)\cap\Omega\subset U_{i}$ and
$\partial U_{i}\cap\partial
D\subset(x_{2n}x_{1})\cup(B(z_{i},r)\cap\partial\Omega)$. Denote
$U_{i}^{\delta}$ a discrete approximation of $U_{i}$ on $\Omega_{\delta}$ (see
Figure 4.1).

Figure 4.1: This is an illustration of the discrete neighbour $U_{1}^{\delta}$
and $U_{2}^{\delta}$ when $n=3$.

Note that

$\mathbb{P}[\mathcal{T}^{*}_{\delta}\in
E^{*}(\Omega_{\delta};x_{1}^{\delta},\ldots,x_{2n}^{\delta};z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta,+})]\geq\mathbb{P}[\cap_{i=1}^{n-1}\\{\gamma_{i}^{\delta}\subset
U_{i}^{\delta}\\}].$

For $v_{\delta}\in\Omega_{\delta}$, we define

$h_{i}^{\delta}(v_{\delta}):=\mathbb{P}_{v_{\delta}}[\mathcal{R}_{\delta}\text{
leaves }U_{i}^{\delta}\text{ in }(x_{2n}^{\delta}x_{1}^{\delta})],$

where $\mathcal{R}_{\delta}$, as usual, is a random walk killed at
$\cup_{i=1}^{n}(x_{2i}^{\delta}x_{2i+1}^{\delta})$, and with reflecting
boundaries on $\cup_{i=1}^{n}(x_{2i-1}^{\delta}x_{2i}^{\delta})$. Thus, by
Wilson’s algorithm, we have

$\mathbb{P}[\mathcal{T}^{*}_{\delta}\in
E^{*}(\Omega_{\delta};x_{1}^{\delta},\ldots,x_{2n}^{\delta};z_{1}^{\delta,+},\ldots,z_{n-1}^{\delta,+})]\geq\Pi_{i=1}^{n-1}h_{i}^{\delta}(z_{i}^{\delta,+})$

By Lemma 3.5, we have there exists a constant $c>0$, such that for every
$1\leq i\leq n$,

$\frac{h_{i}^{\delta}(z_{i}^{\delta,+})}{h^{\delta}_{r}(z_{i}^{\delta,+})}\geq
c.$

The lower bound follows.

The upper bound is much more delicate and uses a careful combinatorial
analysis together with choice of ordering in Wilson’s algorithm. Roughly, we
want to say that if we were to generate the spanning tree
$\mathcal{T}^{*}_{\delta}$ with Wilson’s algorithm, we would roughly have
$n-1$ independent walks which must at least verify the event defining
$h_{r}^{\delta}(z_{i}^{\delta,+})$. The trouble is that it is _a priori_
possible for a walk starting from $z_{i}^{\delta,+}$ to come very close to
$z_{j}^{\delta,+}$ (closer than distance $r$) for some $j\neq i$. This would
prevent us from comparing the probability of $E^{*}$ with that of $n-1$
independent events.

To deal with this, we will show up to a finite number of combinatorial
possibilities, it is always possible to reveal the curves one at a time, in in
a certain order such that, each time, the walk we choose to reveal must escape
the corresponding ball before touching the wired boundary (which includes the
paths already revealed so far, as dictated by Wilson’s algorithm).

More precisely, we must consider events of the following type. Call an index
$1\leq i\leq m-1$ _good_ if
$\gamma^{\delta}_{i}\cap(\cup_{j}B_{j})=\emptyset$, with
$B_{j}=B(z_{j}^{\delta,+},r)$, and call it bad otherwise. Let $I$ denote the
set of good indices. We then consider the events

$\mathbb{P}(\mathcal{T}^{*}_{\delta}\in E^{*},I=\\{i_{1},\ldots,i_{m}\\})$
(4.5)

where $1\leq m\leq n-1$, $1\leq i_{1}<\ldots<i_{m}\leq n-1$. For each such
event that we need to consider, we want to bound its probability by
$C\prod_{i=1}^{n-1}h_{r}^{\delta}(z_{i}^{\delta,+})$. To do so we use Wilson’s
algorithm with an order that depends specifically on the event we are
considering. Namely, in a first stage, we reveal the _good_ branches
$\gamma^{\delta}_{j}$ for $j\in I$, using independent random walks
$\mathcal{R}_{j}^{\delta}$ from the point $z_{j}^{\delta,+}$ for $j\in I$.
Clearly, for $E^{*}$ to be satisfied, it must be the case that during this
first stage, for each $j\in I$, $\mathcal{R}_{j}^{\delta}$ escapes $B_{j}$. By
independence, this has a probability $\prod_{j\in
I}h_{r}^{\delta}(z^{\delta,+}_{j})$. Now in a second stage, given the good
branches, let us wire together the target arc $(x_{2n},x_{1})$ and the good
branches; this forms a new wired boundary in Wilson’s algorithm; see Figure
4.2.

Figure 4.2:  An illustration of
$\gamma_{1}^{\delta},\gamma_{2}^{\delta},\gamma_{3}^{\delta}$ for $n=4$. In
this case, only the green curve $\gamma_{1}^{\delta}$ is good initially, and
so we start by revealing this in Wilson’s algorithm. Once it is revealed and
added to the wired boundary, we observe that the blue curve
$\gamma_{2}^{\delta}$ becomes good, and so we reveal it at this point. Only
once this is done, does the curve $\gamma_{3}^{\delta}$ become good and can be
revealed.

We can now consider, among the remaining bad branches, those which are _good_
with respect to this new target. By iterating this procedure (i.e., induction)
can reduce the number $n-1$ of branches that we need to consider; at each
stage of this induction we bound the conditional probability of $E^{*}$ by the
product over all the good indices of $h_{r}^{\delta}(z^{\delta,+}_{j})$. The
only exception is when all the branches are bad.

Thus let us assume that all the branches are bad, i.e., let us explain how to
bound the probability in (4.5) when $I=\emptyset$. Consider
$\gamma_{1}^{\delta}$, which is an oriented path from $z_{1}^{\delta,+}$ to
the (current) wired boundary. We let $J$ be the index $j$ of the last (with