# Scaling limits of planar maps under the Smith embedding

Federico Bertacco Imperial College London Ewain Gwynne University of Chicago
Scott Sheffield MIT

###### Abstract

The Smith embedding of a finite planar map with two marked vertices, possibly
with conductances on the edges, is a way of representing the map as a tiling
of a finite cylinder by rectangles. In this embedding, each edge of the planar
map corresponds to a rectangle, and each vertex corresponds to a horizontal
segment. Given a sequence of finite planar maps embedded in an infinite
cylinder, such that the random walk on both the map and its planar dual
converges to Brownian motion modulo time change, we prove that the a priori
embedding is close to an affine transformation of the Smith embedding at large
scales. By applying this result, we prove that the Smith embeddings of mated-
CRT maps with the sphere topology converge to $\gamma$-Liouville quantum
gravity ($\gamma$-LQG).

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Main result
    3. 1.3 Application to the mated-CRT map
    4. 1.4 Outline
  2. 2 Background and setup
    1. 2.1 Basic definitions
    2. 2.2 Universal cover
    3. 2.3 Random walks and electrical networks
    4. 2.4 Discrete harmonic conjugate
    5. 2.5 Construction of the Smith embedding
  3. 3 Some properties of the Smith embedding
    1. 3.1 Adding new vertices
    2. 3.2 Periodicity
    3. 3.3 Hitting distribution of a horizontal line
    4. 3.4 Expected horizontal winding
  4. 4 Proof of the main result
    1. 4.1 Height coordinate
    2. 4.2 Width coordinate
    3. 4.3 Proof of Theorem 1.3
  5. 5 Application to mated-CRT maps
    1. 5.1 SLE/LQG description of mated-CRT maps
    2. 5.2 Mated-CRT maps satisfy the assumptions
    3. 5.3 Convergence to LQG
  6. A Some standard estimates for planar Brownian motion

##  1 Introduction

###  1.1 Motivation

Over the past few decades, there has been a large amount of interest in the
study of random planar maps, i.e., graphs embedded in the plane viewed modulo
orientation-preserving homeomorphisms. Since the foundational work of Polyakov
in the context of bosonic string theory [Pol81], it has been believed that
various types of random planar maps converge, in various topologies, to
limiting random surfaces called _Liouville quantum gravity_ (LQG) surfaces.
The rigorous mathematical study of LQG has been explored, e.g., in works by
Duplantier and Sheffield [DS11] and Rhodes and Vargas [RV11]. Roughly
speaking, LQG surfaces can be thought of as random two-dimensional Riemannian
manifolds parameterized by a fixed underlying Riemann surface, indexed by a
parameter $\gamma\in(0,2]$. These surfaces are too rough to be Riemannian
manifolds in the literal sense, but one can still define, e.g., the associated
volume form (area measure) and distance function (metric) via regularization
procedures [Kah85, DS11, RV11, Ber17, DDDF20, DFG+20, GM20, GM21]. Many
properties of the $\gamma$-LQG area measure are well-known [RV14, BP21,
Ber23].

One way of formulating the convergence of random planar maps toward LQG
surfaces is to consider so-called _discrete conformal embeddings_ of the
random planar maps. Here, a discrete conformal embedding refers to a
particular way of “drawing” the map in the plane, which is in some sense a
discrete analog of the Riemann mapping. Suppose we have a random planar map
with $n$ vertices, along with a discrete conformal embedding of the map that
maps each vertex to a point in $\mathbb{C}$. This embedding creates a natural
measure on the plane, with each vertex given a mass of $1/n$. In many
settings, it is natural to conjecture that as $n$ tends to infinity, the
measure should converge weakly to the $\gamma$-LQG area measure, with the
parameter $\gamma$ depending on the particular planar map model under
consideration. Additionally, the random walk on the embedded map is expected
to converge in law to two-dimensional Brownian motion modulo time
parameterization (more precisely, the parameterized walk should converge to
the so-called _Liouville Brownian motion_ [Ber15, GRV16, BG22]). Several
precise scaling limit conjectures for random planar maps toward LQG surfaces
were formulated, e.g., in [DS11, DKRV16, She16, DMS21]. However, this very
general convergence ansatz has only been rigorously proven in a few specific
settings (see below).

One of the challenges in formulating a general scaling limit result for the
embedding of random planar maps is the existence of numerous discrete
conformal embeddings that could be regarded as natural in some sense. We
collect here some of the most commonly employed discrete conformal embeddings.

  * •

The _circle packing_ (see [Nac20] for a review), which represents the map as
the tangency graph of a collection of non-overlapping circles111We refer to
[RS87] for a proof of the fact that the circle packing for lattice
approximations of planar domains gives an approximation of the Riemann
mapping..

  * •

The _Smith embedding_ (a.k.a.  _rectangle packing_), which will be the focus
of the present paper, was introduced by Brooks, Smith, Stone, and Tutte in
[BSST40]. It is another popular method of embedding planar graphs, and it is
defined by means of a rectangle tiling of either a cylinder or a rectangle,
and in which vertices of the planar map correspond to horizontal segments in
the Smith embedding, and edges of the planar map correspond to rectangles in
the Smith embedding222We refer to [GP20] for a proof of the fact that the
Smith embedding for fine-mesh lattice graphs gives an approximation of the
Riemann mapping.. Several papers have studied properties of the Smith
embedding of planar maps [BS96, Geo16, HP17, CG20].

  * •

Other examples of discrete conformal embeddings include the _Tutte embedding_
[GMS21], the _Cardy–Smirnov embedding_ [HS21], and the _Riemann uniformization
embedding_ , obtained by viewing the planar map as a piecewise flat two-
dimensional Riemannian manifold where the faces are identified with unit side
length polygons.

Some cases of the aforementioned conjecture, that LQG describes the scaling
limit of random planar maps under discrete conformal embeddings, have been
proven. For example, in [GMS21], Gwynne, Miller, and Sheffield established the
convergence to $\gamma$-LQG under the Tutte embedding for a one-parameter
family of random planar maps defined using pairs of correlated Brownian
motions, known as the _mated-CRT maps_ (see below for a definition of this
family of random planar maps). Moreover, in [GMS20], the same authors proved
that the Tutte embedding of the Poisson Voronoi tessellation of the Brownian
disk converges to $\sqrt{8/3}$-LQG. In [HS21], Holden and Sun proved that the
scaling limit of uniformly sampled triangulations under the Cardy–Smirnov
embedding converges to $\sqrt{8/3}$-LQG. Finally, let us also mention that in
[GGJN19], the authors studied the circle-packing of the mated-CRT map and
showed that there are no macroscopic circles in the circle packing of this
random planar map. Roughly speaking, the main goal of this paper is to provide
a general convergence result for the Smith embedding of planar maps, which
works whenever random walk on both the map and its dual approximate Brownian
motion.

###  1.2 Main result

The main result of this paper concerns the scaling limit of general (random)
planar maps under the Smith embedding. More precisely, we will show that for a
sequence of finite planar maps satisfying an invariance principle assumption
both on the map and on its dual, the a priori embedding is close to an affine
transformation of the Smith embedding at large scales. We will then apply this
result to prove the convergence of the Smith embeddings of mated-CRT maps to
$\gamma$-LQG.

One advantage of the version of the Smith embedding considered in this paper
is that its definition is particularly natural for random planar maps without
boundary. This is in contrast to other embeddings under which random planar
maps have been shown to converge to LQG (such as the Tutte embedding [GMS21]
and the Cardy–Smirnov embedding [HS21]) which are most naturally defined for
planar maps with boundary.

Throughout the paper, we always write _weighted planar map_ for a planar map
with edge conductances. Moreover, throughout the article, we allow all of our
planar maps to have multiple edges and self-loops. In order to state our main
theorem, we need to introduce some notation. Given a planar graph
$\mathcal{G}$, we denote the sets of vertices, edges, and faces of
$\mathcal{G}$ by $\mathcal{V}\mathcal{G}$, $\mathcal{E}\mathcal{G}$, and
$\mathcal{F}\mathcal{G}$, respectively. We consider a doubly marked finite
weighted planar map $(\mathcal{G},c,v_{0},v_{1})$, where $v_{0}$,
$v_{1}\in\mathcal{V}\mathcal{G}$ are the two marked vertices, and
$c=\\{c_{e}\\}_{e\in\mathcal{E}\mathcal{G}}$ is a collection of positive
unoriented weights called conductances. We assume that we are given a _proper
embedding_ of the map in the infinite cylinder
$\mathcal{C}_{2\pi}:=\mathbb{R}/2\pi\mathbb{Z}\times\mathbb{R}$ in the sense
of the following definition.

######  Definition 1.1.

An embedding of the quadruplet $(\mathcal{G},c,v_{0},v_{1})$ in the infinite
cylinder $\mathcal{C}_{2\pi}$ is said to be _proper_ if:

  1. (a)

the edges in $\mathcal{E}\mathcal{G}$ are continuous and do not cross;

  2. (b)

the graph $\mathcal{G}$ is connected;

  3. (c)

the two marked vertices $v_{0}$ and $v_{1}$ are mapped to $-\infty$ and
$+\infty$, respectively, and they do not lie on the boundary of the same face.

We observe that, if $(\mathcal{G},c,v_{0},v_{1})$ is properly embedded in
$\mathcal{C}_{2\pi}$, then the set of vertices $\mathcal{V}\mathcal{G}$ is
contained in $\mathcal{C}_{2\pi}$; each edge in $\mathcal{E}\mathcal{G}$ is a
curve in $\mathcal{C}_{2\pi}$ that does not cross any other edge, except
possibly at its endpoints; and each face in $\mathcal{F}\mathcal{G}$ is a
connected component in $\mathcal{C}_{2\pi}$ of the complement of the embedded
graph $\mathcal{G}$. Since the two marked vertices are mapped to $\pm\infty$,
this implies that there is an infinite face at each end of the cylinder
$\mathcal{C}_{2\pi}$. In what follows, we use the convention to identify each
vertex in $\mathcal{V}\mathcal{G}$ with its a priori embedding, i.e., if
$x\in\mathcal{V}\mathcal{G}$ then we view $x$ as a point in
$\mathcal{C}_{2\pi}$. Furthermore, for a set $K\subset\mathcal{C}_{2\pi}$, we
write

$\mathcal{V}\mathcal{G}(K):=\bigl{\\{}x\in\mathcal{V}\mathcal{G}:x\in
K\bigr{\\}}.$

We denote by $\smash{(\widehat{\mathcal{G}},\widehat{c})}$ the weighted dual
planar graph associated to $\smash{(\mathcal{G},c)}$, where the conductance
$\smash{\widehat{c}_{\widehat{e}}}$ of a dual edge $\smash{\widehat{e}\
\in\mathcal{E}\widehat{\mathcal{G}}}$ is equal to the resistance of the
corresponding primal edge $e\in\mathcal{E}\mathcal{G}$, i.e., we set
$\smash{\widehat{c}_{\widehat{e}}:=1/c_{e}}$. We assume that
$\smash{(\widehat{\mathcal{G}},\widehat{c})}$ is _properly embedded_ in the
infinite cylinder $\mathcal{C}_{2\pi}$ in the sense of the following
definition.

######  Definition 1.2.

An embedding of the weighted dual planar graph
$(\widehat{\mathcal{G}},\widehat{c})$ associated to $(\mathcal{G},c)$ in the
infinite cylinder $\mathcal{C}_{2\pi}$ is said to be _proper_ if:

  1. (a)

every vertex of $\widehat{\mathcal{G}}$ is contained in a face of
$\mathcal{G}$;

  2. (b)

every edge $e$ of $\mathcal{G}$ is crossed by a single edge $\widehat{e}$ of
$\widehat{\mathcal{G}}$ which joins the two faces incident to $e$.

If an edge $e\in\mathcal{E}\mathcal{G}$ is oriented, the orientation of the
corresponding dual edge $\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}$ can
be obtained by rotating $e$ counter-clockwise. As for the primal graph, given
a set $K\subset\mathcal{C}_{2\pi}$, we write

$\mathcal{V}\widehat{\mathcal{G}}(K):=\bigl{\\{}\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}:\widehat{x}\in
K\bigr{\\}}.$

####  1.2.1 Smith embedding

We are now ready to provide a somewhat informal description of the Smith
embedding of a given doubly marked finite weighted planar map
$(\mathcal{G},c,v_{0},v_{1})$. The precise definition will be given in Section
2. As mentioned earlier, the Smith embedding of a planar map was first
introduced by Brooks, Smith, Stone, and Tutte in [BSST40], and later
generalized to infinite planar graphs by Benjamini and Schramm in [BS96].

The Smith embedding of the quadruplet $(\mathcal{G},c,v_{0},v_{1})$ is
constructed by means of a tiling by rectangles of a finite cylinder
$\mathcal{C}_{\upeta}:=\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$, where $\upeta$
is a positive number which depends on $(\mathcal{G},c)$ to be specified later.
Each vertex $x\in\mathcal{V}\mathcal{G}$ is represented by a horizontal line
segment $\mathrm{H}_{x}$, each edge $e\in\mathcal{E}\mathcal{G}$ by a
rectangle $\mathrm{R}_{e}$, and each dual vertex
$\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}$ by a vertical line segment
$\mathrm{V}_{\widehat{x}}$. In particular, since each edge
$e\in\mathcal{E}\mathcal{G}$ corresponds to a rectangle in the tiling, we need
to specify four coordinates for each edge. This is done by means of the
voltage function $\mathfrak{h}:\mathcal{V}\mathcal{G}\to[0,1]$ and its
discrete harmonic conjugate
$\mathfrak{w}:\mathcal{V}\widehat{\mathcal{G}}\to\mathbb{R}/\upeta\mathbb{Z}$.

The function $\mathfrak{h}$ is the unique function on $\mathcal{V}\mathcal{G}$
which is discrete harmonic on
$\mathcal{V}\mathcal{G}\setminus\\{v_{0},v_{1}\\}$ (with respect to the
conductances $c$) with boundary conditions given by $\mathfrak{h}(v_{0})=0$
and $\mathfrak{h}(v_{1})=1$, i.e.,

$\mathfrak{h}(x)=\mathbb{P}_{x}\bigl{(}\text{$X$ hits $v_{1}$ before
$v_{0}$}\bigr{)},\quad\forall x\in\mathcal{V}\mathcal{G},$

where $\mathbb{P}_{x}$ denotes the law of random walk $X^{x}$ on
$(\mathcal{G},c)$ started from $x$. We refer to Subsection 2.3 for more
details.

The function $\mathfrak{w}$ is the function on
$\mathcal{V}\widehat{\mathcal{G}}$ that satisfies the discrete Cauchy–Riemann
equation associated to $\mathfrak{h}$, i.e., it is the function on the set of
dual vertices $\mathcal{V}\widehat{\mathcal{G}}$ whose difference at the
endpoints of each edge of $\widehat{\mathcal{G}}$ is equal to the difference
of $\mathfrak{h}$ at the endpoints of the corresponding primal edge times its
conductance. As we will see in Subsection 2.4, the function $\mathfrak{w}$ is
only defined modulo $\upeta\mathbb{Z}$ and modulo an additive constant that
can be fixed by imposing that $\mathfrak{w}$ is equal to zero on a chosen dual
vertex. In particular, the choice of the additive constant of $\mathfrak{w}$
fixes the rotation of the cylinder in which the tiling takes place.

Now, we can specify the various objects involved in the definition of the
Smith embedding.

  * •

For each edge $e\in\mathcal{E}\mathcal{G}$, the rectangle $\mathrm{R}_{e}$
corresponds to the rectangle on $\mathcal{C}_{\upeta}$ such that the height
coordinates of the top and bottom sides are given by the values of
$\mathfrak{h}$ at the endpoints of $e$, and the width coordinates of the left
and right sides of $\mathrm{R}_{e}$ are given by the values of $\mathfrak{w}$
at the endpoints of the corresponding dual edge $\widehat{e}$.

  * •

For each vertex $x\in\mathcal{V}\mathcal{G}$, the horizontal segment
$\mathrm{H}_{x}$ corresponds to the maximal horizontal segment which lies on
the boundaries of all the rectangles corresponding to the edges incident to
$x$.

  * •

For each dual vertex $\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}$, the
vertical segment $\mathrm{V}_{\widehat{x}}$ corresponds to the maximal
vertical segment which is tangent with all rectangles corresponding to primal
edges surrounding $\widehat{x}$.

We call the map
$\SS:\mathcal{V}\mathcal{G}\cup\mathcal{E}\mathcal{G}\cup\mathcal{V}\widehat{\mathcal{G}}\to\mathcal{C}_{\upeta}$
such that $\SS(e):=\mathrm{R}_{e}$, $\SS(x):=\mathrm{H}_{x}$, and
$\SS(\widehat{x}):=\mathrm{V}_{\widehat{x}}$ the tiling map associated to the
quadruplet $(\mathcal{G},c,v_{0},v_{1})$. We refer to Figure 1 for a
diagrammatic illustration of the tiling map associated to a given quadruplet
$(\mathcal{G},c,v_{0},v_{1})$. We define the Smith embedding
$\dot{\mathcal{S}}$ associated to the quadruplet $(\mathcal{G},c,v_{0},v_{1})$
as the function from $\mathcal{V}\mathcal{G}$ to
$\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$ given by

$\dot{\mathcal{S}}(x):=\operatorname{mid}(\mathrm{H}_{x}),\quad\forall
x\in\mathcal{V}\mathcal{G},$ (1.1)

where $\operatorname{mid}(\mathrm{H}_{x})$ corresponds to the middle point of
the horizontal line segment $\mathrm{H}_{x}$333This definition of the Smith
embedding is somewhat arbitrarily. Indeed, for each
$x\in\mathcal{V}\mathcal{G}$, one can define $\dot{\mathcal{S}}(x)$ to be any
arbitrary point inside the horizontal segment $\mathrm{H}_{x}$.. We refer to
Subsection 2.5 for precise definitions.

Figure 1: Left: A doubly marked finite weighted planar graph
$(\mathcal{G},c,v_{0},v_{1})$, drawn in gray, properly embedded in the
infinite cylinder $\mathcal{C}_{2\pi}$ with the two marked vertices $v_{0}$
and $v_{1}$ drawn in black. Drawn in red is the corresponding weighted dual
planar graph $(\widehat{\mathcal{G}},\widehat{c})$, which is also properly
embedded in $\mathcal{C}_{2\pi}$. Right: The Smith diagram associated to
$\smash{(\mathcal{G},c,v_{0},v_{1})}$ constructed via the tiling map $\SS$.
The blue horizontal segment $\mathrm{H}_{x}$ corresponds to the vertex
$\smash{x\in\mathcal{V}\mathcal{G}}$. The blue vertical segment
$\smash{\mathrm{V}_{\widehat{x}}}$ corresponds to the dual vertex
$\smash{\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}}$. The blue rectangle
$\smash{\mathrm{R}_{e}}$ corresponds to the edge
$\smash{e\in\mathcal{E}\mathcal{G}}$. In both the left and right figure, the
two vertical lines with an arrow are identified with each other.

####  1.2.2 Assumptions and statement of the main result

To state our main result, we need to consider a sequence of doubly marked
finite weighted planar maps

$\bigl{\\{}(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})\bigr{\\}}_{n\in\mathbb{N}},$

and the sequence of associated weighted dual planar graphs
$\\{(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})\\}_{n\in\mathbb{N}}$. We make
the following assumptions.

  1. (H1)

(Cylindrical embedding) For each $n\in\mathbb{N}$, the quadruplet
$(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})$ is properly embedded in the
infinite cylinder $\mathcal{C}_{2\pi}$ in the sense of Definition 1.1.
Furthermore, the associated weighted dual planar graph
$(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})$ is also properly embedded in
$\mathcal{C}_{2\pi}$ in the sense of Definition 1.2.

  2. (H2)

(Invariance principle on the primal graphs) For each $n\in\mathbb{N}$, view
the embedded random walk on $(\mathcal{G}^{n},c^{n})$, stopped when it hits
either $v_{0}^{n}$ or $v_{1}^{n}$, as a continuous curve in
$\mathcal{C}_{2\pi}$ obtained by piecewise linear interpolation at constant
speed. For each compact subset $K\subset\mathcal{C}_{2\pi}$ and for any $z\in
K$, the law of the random walk on $(\mathcal{G}^{n},c^{n})$ started from the
vertex $x_{z}^{n}\in\mathcal{V}\mathcal{G}^{n}$ nearest to $z$ weakly
converges as $n\to\infty$ to the law of the Brownian motion on
$\mathcal{C}_{2\pi}$ started from $z$ with respect to the local topology on
curves viewed modulo time parameterization specified in Subsection 2.1.2,
uniformly over all $z\in K$.

  3. (H3)

(Invariance principle on the dual graphs) For each $n\in\mathbb{N}$, view the
embedded random walk on $(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})$ as a
continuous curve in $\mathcal{C}_{2\pi}$ obtained by piecewise linear
interpolation at constant speed. For each compact subset
$K\subset\mathcal{C}_{2\pi}$ and for any $z\in K$, the law of the random walk
on $(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})$ started from the vertex
$\widehat{x}_{z}^{n}\in\mathcal{V}\widehat{\mathcal{G}}^{n}$ nearest to $z$
weakly converges as $n\to\infty$ to the law of the Brownian motion on
$\mathcal{C}_{2\pi}$ started from $z$ with respect to the local topology on
curves viewed modulo time parameterization specified in Subsection 2.1.2,
uniformly over all $z\in K$.

In what follows, given a point $x\in\mathcal{C}_{2\pi}$, we write
$\mathop{Re}\nolimits(x)\in[0,2\pi)$ for its horizontal coordinate and
$\mathop{Im}\nolimits(x)\in\mathbb{R}$ for its height coordinate. Similarly,
if $x\in\mathcal{C}_{\upeta}$, then $\mathop{Re}\nolimits(x)\in[0,\upeta)$
denotes its horizontal coordinate and $\mathop{Im}\nolimits(x)\in[0,1]$
denotes its height coordinate. We are now ready to state our main theorem.

######  Theorem 1.3 (Main theorem).

Consider a sequence
$\smash{\\{(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})\\}_{n\in\mathbb{N}}}$
of doubly marked finite weighted planar maps and let
$\smash{\\{(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})\\}_{n\in\mathbb{N}}}$
be the sequence of associated weighted dual planar graphs. Assume that
assumptions (H1), (H3), (H3) are satisfied. For each $n\in\mathbb{N}$, let
$\dot{\mathcal{S}}_{n}:\mathcal{V}\mathcal{G}^{n}\to\mathcal{C}_{\upeta_{n}}$
be the Smith embedding associated with the quadruplet
$(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})$ as specified in (1.1). There
exist sequences $\smash{\\{c^{\mathfrak{h}}_{n}\\}_{n\in\mathbb{N}}}$,
$\smash{\\{b^{\mathfrak{h}}_{n}\\}_{n\in\mathbb{N}}}$,
$\smash{\\{b^{\mathfrak{w}}_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}}$ such
that, if we let $\smash{T_{n}:\mathcal{C}_{\upeta_{n}}\to\mathcal{C}_{2\pi}}$
be the affine transformation of the form

$\mathop{Re}\nolimits(T_{n}x):=\left(\frac{2\pi}{\upeta_{n}}\mathop{Re}\nolimits(x)+b_{n}^{\mathfrak{w}}\right)\;\mathrm{mod}(2\pi)\quad\text{
and
}\quad\mathop{Im}\nolimits(T_{n}x):=c_{n}^{\mathfrak{h}}\mathop{Im}\nolimits(x)+b^{\mathfrak{h}}_{n},\qquad\forall
x\in\mathcal{C}_{\upeta_{n}},$

then, for all compact sets $K\subset\mathcal{C}_{2\pi}$, it holds that

$\lim_{n\to\infty}\sup_{x\in\mathcal{V}\mathcal{G}^{n}(K)}\mathrm{d}_{2\pi}\bigl{(}T_{n}\dot{\mathcal{S}}_{n}(x),x\bigr{)}=0,$

where $\mathrm{d}_{2\pi}$ denotes the Euclidean distance on the cylinder
$\mathcal{C}_{2\pi}$.

Figure 2: Consider a planar map $\mathcal{M}$ with $n$ edges and unit edge
conductances, and a distinguished spanning tree $\mathcal{T}$. Then
$(\mathcal{M},\mathcal{T})$ determines a quadrangulation $\mathcal{G}$ with
$n$ quadrilaterals obtained by replacing each edge of $\mathcal{M}$ by a
quadrilateral. Moreover, the path that “snakes” between $\mathcal{T}$ and its
dual crosses the edges of $\mathcal{G}$ in order. Shown here is the Smith
diagram of an instance of $\mathcal{G}$ corresponding to a uniformly random
$(\mathcal{M},\mathcal{T})$ pair with two random points chosen as roots. Left:
The squares in the figure are colored according to their place in the cyclic
ordering. Right: The squares in the figure are colored according to their
Euclidean size. It is interesting to compare the figure on the right with the
square subdivisions associated to the $\gamma$-LQG measure appearing, e.g., in
[DS11, Figures 1, 2, 3]. The interested reader can find the Mathematica code
used to generate the above simulations at the following link:
https://github.com/federico-bertacco/smith-embedding.git. Furthermore, at the
same link, there is also a PDF file available that provides further
explanation on how these simulations were produced.

Theorem 1.3 tells us that in order to say that the Smith embedding of
$(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})$ is close to a given a priori
embedding (up to translation and scaling), we only need to know a certain
invariance principle for random walk under the a priori embedding. This result
is in some ways not surprising, since it is natural to expect that if a simple
random walk (and its dual) approximate Brownian motion, then discrete harmonic
functions (and their conjugate duals) should approximate continuum harmonic
functions. However, showing that this is actually true in the limit, and that
the convergence is to the right continuum harmonic functions, will require
some new coupling tricks and some careful boundary behavior estimates, which
we hope may prove useful in other settings as well. More precisely, the
particular statement we obtain is far from obvious a priori, for two main
reasons.

  * •

Our hypotheses only concern the _macroscopic_ behavior of the random walk on
$(\mathcal{G}^{n},c^{n})$ in the _bulk_ of the cylinder. We do not need any
hypotheses about how the random walk behaves when it gets close to the marked
vertices $v_{0}^{n}$ and $v_{1}^{n}$. This may seem surprising at first glance
since one could worry, e.g., that the structure of $(\mathcal{G}^{n},c^{n})$
in small neighborhoods of $v_{0}^{n}$ and $v_{1}^{n}$ makes it much easier for
random walk to hit $v_{0}^{n}$ than for it to hit $v_{1}^{n}$, and so the
height coordinate function $\mathfrak{h}^{n}$ is close to zero on all of
$\mathcal{V}\mathcal{G}^{n}$. What allows us to get around this is the scaling
and translation sequences
$\smash{\\{c^{\mathfrak{h}}_{n}\\}_{n\in\mathbb{N}}}$ and
$\smash{\\{b^{\mathfrak{h}}_{n}\\}_{n\in\mathbb{N}}}$. We refer to Subsection
4.1 for more details.

  * •

The width coordinate $\mathfrak{w}^{n}$ is discrete harmonic but does not
admit a simple direct description in terms of the random walk on
$(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})$. For this reason, a fair amount
of work is required to get from the invariance principle for this random walk
to a convergence statement for $\mathfrak{w}^{n}$. We refer to Subsection 4.2
for more details.

We remark that the Smith embedding can be very far from the identity near the
ends of the cylinder: what is interesting, and perhaps surprising, is the
generality in which we can show that the “bad behavior” gets “smoothed out” in
the middle of the cylinder. This is apparent in the simulations presented in
Figure 2.

As we will discuss in the next section, one application of Theorem 1.3 is the
convergence of the mated-CRT map with the sphere topology to LQG under the
Smith embedding. More generally, Theorem 1.3 reduces the problem of proving
the convergence to LQG under the Smith embeddings for other types of random
planar maps to the problem of finding some a priori embeddings of the map and
its dual under which the counting measure on vertices converges to the LQG
measure and the random walk on the map converges to Brownian motion modulo
time parameterization.

###  1.3 Application to the mated-CRT map

Mated-CRT maps are a one-parameter family of random planar maps constructed
and studied, e.g., in [GHS19, GMS19, DMS21, GMS21]. The mated-CRT maps are
parameterized by a real parameter $\gamma\in(0,2)$ and are in the universality
class of $\gamma$-LQG. In this paper, we will be interested in mated-CRT maps
with the sphere topology. For each $n\in\mathbb{N}$ and $\gamma\in(0,2)$, the
the $n$-_mated-CRT map with the sphere topology_ is the random planar
triangulation $\mathcal{G}^{n}$ with vertex set given by

$\mathcal{V}\mathcal{G}^{n}:=\frac{1}{n}\mathbb{Z}\cap(0,1],$

and an edge set defined by means of a condition involving a pair of linear
Brownian motions. More precisely, consider a two dimensional Brownian motion
$(L,R)$ with covariance matrix given by

$\operatorname{\mathbb{V}ar}(L_{t})=\operatorname{\mathbb{V}ar}(R_{t})=|t|,\qquad\operatorname{\mathbb{C}ov}(L_{t},R_{t})=-\cos\left(\frac{\pi\gamma^{2}}{4}\right)|t|,$
(1.2)

and conditioned to stay in the first quadrant for one unit of time and end up
at $(0,0)$, i.e., $(L,R)$ is an excursion. Then, two vertices $x_{1}$,
$x_{2}\in\mathcal{V}\mathcal{G}^{n}$ are connected by an edge if and only if
either

$\displaystyle\max\left\\{\inf_{t\in[x_{1}-1/n,x_{1}]}L_{t},\inf_{t\in[x_{2}-1/n,x_{2}]}L_{t}\right\\}$
$\displaystyle\leq\inf_{t\in[x_{1},x_{2}-1/n]}L_{t},\quad\text{ or }$ (1.3)
$\displaystyle\max\left\\{\inf_{t\in[x_{1}-1/n,x_{1}]}R_{t},\inf_{t\in[x_{2}-1/n,x_{2}]}R_{t}\right\\}$
$\displaystyle\leq\inf_{t\in[x_{1},x_{2}-1/n]}R_{t}.$

The vertices in $\mathcal{V}\mathcal{G}^{n}$ are connected by two edges if
$|x_{1}-x_{2}|\neq 1/n$ and both the conditions in (1.3) hold. We observe that
the condition for $L$ in (1.3) is equivalent to the existence of a horizontal
line segment below the graph of $L$ whose end points are of the form
$(t_{1},L_{t_{1}})$ and $(t_{2},L_{t_{2}})$ for $t_{1}\in[x_{1}-1/n,x_{1}]$
and $t_{2}\in[x_{2}-1/n,x_{2}]$, and similarly for $R$. This allows us to give
an equivalent, more geometric, version of the definition of $\mathcal{G}^{n}$.
In particular, this procedure assigns a natural planar map structure to the
mated-CRT map $\mathcal{G}^{n}$, under which it is a triangulation. We refer
to Figure 3 for a diagrammatic explanation of this procedure.

Figure 3: Left: A diagram showing the construction of the mated-CRT map with
sphere topology and $n=8$ vertices. To geometrically construct the mated-CRT
map $\mathcal{G}^{n}$, we draw the graphs of $L$ and $C-R$ with a chosen large
constant $C>0$ to ensure that the graphs do not intersect. The region between
the graphs is then divided into vertical strips. Each strip corresponds to the
vertex $x\in\mathcal{V}\mathcal{G}^{n}$ which is the horizontal coordinate of
its rightmost point. Two vertices $x_{1}$,
$x_{2}\in\mathcal{V}\mathcal{G}^{n}$ are connected by an edge if and only if
their respective vertical strips are connected by a horizontal line segment
that is either below the graph of $L$ or above the graph of $C-R$. For each
pair of vertices for which the condition holds for $L$ (resp. $C-R$), we have
drawn the lowest (resp. highest) segment which joins the corresponding
vertical strips. We note that consecutive vertices are always connected by an
edge. Right: The graph $\mathcal{G}^{n}$ can be represented in the plane by
connecting two vertices $x_{1}$, $x_{2}\in\mathcal{V}\mathcal{G}^{n}$ with an
arc below (resp. above) the real line if their vertical strips are connected
by a horizontal segment below (resp. above) the graph of $L$ (resp. $C-R$).
Additionally, each pair of consecutive vertices in
$\mathcal{V}\mathcal{G}^{n}$ is connected by an edge. This representation
gives $\mathcal{G}^{n}$ a planar map structure under which it is a
triangulation. A similar illustration was shown in [GMS21].

In [GMS21], Gwynne, Miller, and Sheffield proved that the Tutte embeddings of
mated-CRT maps with the disk topology converge to $\gamma$-LQG. Thanks to our
main theorem, we can prove an analogous result for the Smith embeddings of
mated-CRT maps with the sphere topology. More precisely, for each
$n\in\mathbb{N}$, pick two marked vertices $v_{0}^{n}$,
$v_{1}^{n}\in\mathcal{V}\mathcal{G}^{n}$. Then, we can conformally map the
sphere into the infinite cylinder $\mathcal{C}_{2\pi}$ so that the marked
points are mapped to $\pm\infty$, and
$(\mathcal{G}^{n},v_{0}^{n},v_{1}^{n})$444Here, each edge in
$\mathcal{E}\mathcal{G}^{n}$ has unit conductance and so we do not specify the
sequence of weights $c$ as in the general case. is properly embedded in
$\mathcal{C}_{2\pi}$.

######  Theorem 1.4 (Convergence of mated-CRT map).

Fix $\gamma\in(0,2)$ and let
$\\{(\mathcal{G}^{n},v_{0}^{n},v_{1}^{n})\\}_{n\in\mathbb{N}}$ be the sequence
of doubly marked $n$-mated CRT map with the sphere topology embedded in
$\mathcal{C}_{2\pi}$ as specified above. There exists a sequence of random
affine transformations $\\{T_{n}\\}_{n\in\mathbb{N}}$ from
$\mathcal{C}_{\upeta_{n}}$ to $\mathcal{C}_{2\pi}$ of the form specified in
the statement of Theorem 1.3 such that, if we let $\mu_{n}$ be the push-
forward with respect to the mapping $z\mapsto T_{n}z$ of the counting measure
on the set $\dot{\mathcal{S}}_{n}(\mathcal{V}\mathcal{G}^{n})$ scaled by
$1/n$, then we have the following convergences in probability as $n\to\infty$.

  1. (a)

On each compact subset of $\mathcal{C}_{2\pi}$, the measure $\mu_{n}$ weakly
converges to the $\gamma$-LQG measure associated to a doubly marked unit area
quantum sphere parameterized by $\mathcal{C}_{2\pi}$ in such a way that its
marked points are at $\pm\infty$, as defined in [DMS21, Definition 4.21].

  2. (b)

On each compact subset of $\mathcal{C}_{2\pi}$, the image under the mapping
$z\mapsto T_{n}z$ of the space-filling path on the Smith embedded mated-CRT
map on $\mathcal{C}_{\upeta_{n}}$ obtained from the left-right ordering of the
vertices converges uniformly with respect to the two-point compactification
topology on the cylinder to space-filling SLEκ on $\mathcal{C}_{2\pi}$, with
$\kappa=16/\gamma^{2}$, parameterized by $\gamma$-LQG mass.

  3. (c)

For $z\in\mathcal{C}_{2\pi}$, let $x_{z}^{n}\in\mathcal{V}\mathcal{G}^{n}$ be
the vertex nearest to $z$. The conditional law given $\mathcal{G}^{n}$ of the
image under the mapping $z\mapsto T_{n}z$ of the simple random walk on the
Smith-embedded mated-CRT map started from $\dot{\mathcal{S}}^{n}(x_{z}^{n})$
and stopped when it hits one of the horizontal segments associated to the
marked vertices weakly converges to the law of Brownian motion on
$\mathcal{C}_{2\pi}$ started from $z$, modulo time parameterization and
uniformly over all $z$ in a compact subset of $\mathcal{C}_{2\pi}$.

To conclude, let us point out that item a of Theorem 1.4 solves [ABL16,
Question 1] for the case of mated-CRT maps.

###  1.4 Outline

Most of the paper is dedicated to proving Theorem 1.3, and it is organized as
follows. In the first part of Section 2, we provide some background material
on weighted planar graphs and the theory of electrical networks. We then move
on to the precise construction of the tiling map and the definition of the
Smith embedding in Subsection 2.5. As mentioned earlier, this is achieved by
introducing two harmonic maps: one on the planar map itself and one on the
associated dual planar map.

In Section 3, we state and prove several properties of the Smith embedding.
The most significant result of this section is Lemma 3.17, which heuristically
states that the conditional expected horizontal winding of the Smith-embedded
random walk given the vertical coordinate of the walk is equal to zero. This
property plays a key role in proving our main theorem. In order to prove this
intermediate result, we rely on Lemma 3.14, which essentially states that the
conditional probability, given the vertical coordinate of the walk, that
Smith-embedded random walk hits a certain horizontal line segment is
proportional to its width. We point out that a similar result, but without
conditioning on the vertical component of the walk, has been obtained by
Georgakopoulos [Geo16, Lemma 6.2] in the setting of infinite weighted planar
graphs.

Section 4 is the core of this article and contains the proof of Theorem 1.3,
which can be divided into two main blocks: in Subsection 4.1 we study the
height coordinate function, and in Subsection 4.2 we study the width
coordinate function. Specifically, the main result of Subsection 4.1 is
Proposition 4.3, which roughly states that the height coordinate of the a
priori embedding is asymptotically close to an affine transformation of the
height coordinate of the Smith embedding. Similarly, the main result of
Subsection 4.2 is Proposition 4.8, which states the analogous fact for the
width coordinate. We refer to Subsections 4.1 and 4.2 for the proof outlines
of the height and width coordinate results, respectively. Finally, in
Subsection 4.3, we show how to combine the results for the height coordinate
and width coordinate to prove Theorem 1.3.

In Section 5, we provide a brief introduction to the relationship between
mated-CRT maps and LQG. We then demonstrate in Subsection 5.2 that this family
of random planar maps satisfies the assumptions of our main result.
Specifically, in Subsection 5.3, we apply our result to show that the scaling
limit of mated-CRT maps is $\gamma$-LQG, thereby proving Theorem 1.4.

######  Acknowledgements.

F.B. is grateful to the Royal Society for financial support through Prof. M.
Hairer’s research professorship grant RP\R1\191065. E.G. was partially
supported by a Clay research fellowship. Part of this work was carried out
during the Probability and Mathematical Physics ICM satellite conference at
Helsinki in Summer 2022. We thank the organizers of this conference for their
hospitality.

##  2 Background and setup

###  2.1 Basic definitions

####  2.1.1 Basic notation

We write $\mathbb{N}$ for the set of positive integers and $\mathbb{N}_{0}$
for the set of non-negative integers. Given $n\in\mathbb{N}$ and
$j\in\mathbb{N}_{0}$, we let $[n]:=\\{1,\ldots,n\\}$ and
$[n]_{j}:=\\{j,\ldots,n\\}$, Furthermore, for $n\in\mathbb{N}$ and
$j\in\mathbb{N}_{0}$, we write $[x_{n}]$ to denote the collection of objects
$\\{x_{1},\ldots,x_{n}\\}$ and $[x_{n}]_{j}$ to denote the collection of
objects $\\{x_{j},\ldots,x_{n}\\}$. If $a$ and $b$ are two real numbers, we
write $a\mathop{\preceq}\nolimits b$ if there is a constant $C>0$, independent
of the values of $a$ or $b$ and certain other parameters of interest, such
that $a\leq Cb$, and we highlight the dependence of the implicit constants
when necessary. If $a$ and $b$ are two real numbers depending on a variable
$x$, we write $a=o_{x}(b)$ if $a/b$ tends to $0$ as $x\to\infty$. We use the
convention to identify $\mathbb{R}^{2}$ with $\mathbb{C}$. In particular,
given a point $x\in\mathbb{R}^{2}$, we write $\mathop{Re}\nolimits(x)$ (resp.
$\mathop{Im}\nolimits(x)$) for its horizontal (resp. vertical) coordinate.

####  2.1.2 Metric on curves modulo time parameterization

For $T_{1}$, $T_{2}>0$, let $P_{1}:[0,T_{1}]\to\mathbb{R}^{2}$ and
$P_{2}:[0,T_{2}]\to\mathbb{R}^{2}$ be two continuous curves defined on
possibly different time intervals. We define

$\operatorname{\mathrm{d}^{\mathrm{CMP}}}\bigl{(}P_{1},P_{2}\bigr{)}:=\inf_{\phi}\sup_{t\in[0,T_{1}]}\bigl{|}P_{1}(t)-P_{2}(\phi(t))\bigr{|},$
(2.1)

where the infimum is taken over all increasing homeomorphisms
$\phi:[0,T_{1}]\to[0,T_{2}]$. It is known that
$\operatorname{\mathrm{d}^{\mathrm{CMP}}}$ induces a complete metric on the
set of curves viewed modulo time parameterization (see [AB99, Lemma 2.1]).

For curves defined for infinite time, it is convenient to have a local variant
of the metric $\operatorname{\mathrm{d}^{\mathrm{CMP}}}$. Assume that
$P_{1}:[0,\infty)\to\mathbb{R}^{2}$ and $P_{2}:[0,\infty)\to\mathbb{R}^{2}$
are two such curves. Then, for $r>0$, let $T_{1,r}$ (resp. $T_{2,r}$) be the
first exit time of $P_{1}$ (resp. $P_{2}$) from the ball $B(0,r)$ centred at
$0$ with radius $r$, or $0$ if the curve starts outside $B(0,r)$. We define

$\operatorname{\mathrm{d}_{\mathrm{loc}}^{\mathrm{CMP}}}\bigl{(}P_{1},P_{2}\bigr{)}:=\int_{1}^{\infty}e^{-r}\bigl{(}1\wedge\operatorname{\mathrm{d}^{\mathrm{CMP}}}\left(P_{1}|_{[0,T_{1,r}]},P_{2}|_{[0,T_{2,r}]}\bigr{)}\right)\mathrm{d}r.$
(2.2)

Moreover, we observe that given a sequence $\\{P_{n}\\}_{n\in\mathbb{N}}$ of
continuous curves defined for infinite time, then
$\lim_{n\to\infty}\operatorname{\mathrm{d}_{\mathrm{loc}}^{\mathrm{CMP}}}(P_{n},P)=0$,
if and only if, for Lebesgue almost every $r>0$, $P_{n}$ stopped at its first
exit time from $B(0,r)$ converges to $P$ stopped at its first exit time from
$B(0,r)$ with respect to the metric (2.1).

######  Remark 2.1.

In the remaining part of the article, we also need to consider curves taking
values in the infinite cylinder $\mathcal{C}_{2\pi}$. We equip the spaces
specified above, but with $\mathcal{C}_{2\pi}$ in place of $\mathbb{R}^{2}$,
with the same metrics. It will be clear from the context whether the metric
under consideration refers to curves in $\mathbb{R}^{2}$ or in
$\mathcal{C}_{2\pi}$.

####  2.1.3 Graph notation

Given a finite planar graph $\mathcal{G}$, besides the notation related to
$\mathcal{G}$ specified in the introduction, we need to introduce some further
nomenclature. In particular, in what follows, we use
$e\in\mathcal{E}\mathcal{G}$ to denote both oriented and unoriented edges. An
oriented edge $e\in\mathcal{E}\mathcal{G}$ is oriented from its tail $e^{-}$
to its head $e^{+}$. Furthermore, given a vertex $x\in\mathcal{V}\mathcal{G}$,
we write $\mathcal{V}\mathcal{G}(x)$ for the set of vertices $y$ adjacent to
$x$, i.e., such that there exists an edge connecting $x$ to $y$. For a vertex
$x\in\mathcal{V}\mathcal{G}$, we denote by $\mathcal{E}\mathcal{G}(x)$ the set
of edges in $\mathcal{E}\mathcal{G}$ incident to $x$. For a fixed orientation
of the edges in $\mathcal{E}\mathcal{G}(x)$, we let
$\mathcal{E}\mathcal{G}^{\downarrow}(x)$ (resp.
$\mathcal{E}\mathcal{G}^{\uparrow}(x)$) be the set of edges in
$\mathcal{E}\mathcal{G}(x)$ with heads (resp. tails) equal to $x$. Similar
notation will also be used for the dual planar graph $\widehat{\mathcal{G}}$.

##### Metric graph.

We will need to consider the metric space associated to a planar graph
$\mathcal{G}$ which can be canonically built as follows. For each edge
$e\in\mathcal{E}\mathcal{G}$, we choose an arbitrary orientation of $e$ and we
let $I_{e}$ be an isometric copy of the real unit interval $[0,1]$. We define
the metric space $\mathbb{G}$ associated with $\mathcal{G}$ to be the quotient
of $\cup_{e\in\mathcal{E}\mathcal{G}}I_{e}$ where we identify the endpoints of
$I_{e}$ with the vertices $e^{-}$ and $e^{+}$, and we equip it with the
natural path metric $\mathrm{d}^{\mathbb{G}}$. More precisely, for two points
$x$, $y$ lying on an edge of $\mathcal{G}$, we define
$\mathrm{d}^{\mathbb{G}}(x,y)$ to be the Euclidean distance between $x$ and
$y$. For points $x$, $y$ lying on different edges, we use the metric given by
the length of the shortest path between the two points, where distances are
measured along the edges using the Euclidean distance. We can also define the
dual metric graph $\smash{\widehat{\mathbb{G}}}$, and the associated metric
$\smash{\mathrm{d}^{\widehat{\mathbb{G}}}}$, in a similar way.

###  2.2 Universal cover

The concept of universal cover of a graph will play an important role in our
analysis. If $\mathcal{G}$ is a graph embedded in the infinite cylinder
$\mathcal{C}_{2\pi}$, then there is a canonical way to define its lift
$\mathcal{G}^{{\dagger}}$ to the universal covering space of
$\mathcal{C}_{2\pi}$. More precisely, consider the universal cover
$(\mathbb{R}^{2},\upsigma_{2\pi})$ of $\mathcal{C}_{2\pi}$, where the covering
map $\upsigma_{2\pi}:\mathbb{R}^{2}\to\mathcal{C}_{2\pi}$ is defined by

$\upsigma_{2\pi}(t,x):=\bigl{(}e^{it},x\bigr{)},\quad\forall(t,x)\in\mathbb{R}^{2}.$
(2.3)

Then, the lifted graph $\mathcal{G}^{{\dagger}}$ can be constructed by taking
every lift of every vertex and every edge of $\mathcal{G}$ in
$\mathcal{C}_{2\pi}$ to the covering space $\mathbb{R}^{2}$. We denote by
$\mathcal{V}\mathcal{G}^{{\dagger}}$ and $\mathcal{E}\mathcal{G}^{{\dagger}}$
the set of vertices and edges of the lifted graph $\mathcal{G}^{{\dagger}}$,
respectively. Moreover, we can also construct the lift of the dual graph
$\widehat{\mathcal{G}}$ to the universal covering space $\mathbb{R}^{2}$ in a
similar way, and we denote it by $\widehat{\mathcal{G}}^{{\dagger}}$. We adopt
the following notational convention: if $x\in\mathcal{V}\mathcal{G}$ is a
vertex, then we denote by $\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$ a
lift of $x$; if $e\in\mathcal{E}\mathcal{G}$ is an edge, then we denote by
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$ a lift of $e$; if
$\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}$ is a dual vertex, then we
denote by
$\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$ a lift
of $\widehat{x}$.

Figure 4: Left: A doubly marked finite weighted planar graph
$(\mathcal{G},c,v_{0},v_{1})$, drawn in gray, properly embedded in the
infinite cylinder $\mathcal{C}_{2\pi}$ with the two marked vertices $v_{0}$
and $v_{1}$ drawn in black. Middle: The same doubly marked finite weighted
planar graph as in the left figure together with its dual planar weighted
graph $\smash{(\widehat{\mathcal{G}},\widehat{c})}$, drawn in red, properly
embedded in $\mathcal{C}_{2\pi}$. Right: A portion of the lifted weighted
graph $\smash{(\mathcal{G}^{{\dagger}},c^{{\dagger}})}$, drawn in grey, and
the associated dual lifted graph
$\smash{(\widehat{\mathcal{G}}^{{\dagger}},\widehat{c}^{{\dagger}})}$, drawn
in red. In both the left and middle figure, the two vertical lines with an
arrow are identified with each other.

Moreover, if $(\mathcal{G},c)$ is a finite weighted planar graph embedded in
$\mathcal{C}_{2\pi}$, we can naturally assign to each lifted edge $\mathbf{e}$
the conductance $c^{{\dagger}}_{\mathbf{e}}:=c_{e}$, and we denote by
$(\mathcal{G}^{{\dagger}},c^{{\dagger}})$ the lifted weighted graph. By
definition, the lifted graph $\mathcal{G}^{{\dagger}}$ is periodic in the
sense that if $\mathbf{x}_{1}$,
$\mathbf{x}_{2}\in\mathcal{V}\mathcal{G}^{{\dagger}}$ are two points in
$\mathbb{R}^{2}$ such that
$\mathop{Im}\nolimits(\mathbf{x}_{1})=\mathop{Im}\nolimits(\mathbf{x}_{2})$
and
$|\mathop{Re}\nolimits(\mathbf{x}_{1})-\mathop{Re}\nolimits(\mathbf{x}_{2})|\in\mathbb{N}_{0}$,
then $\upsigma_{2\pi}(\mathbf{x}_{1})=\upsigma_{2\pi}(\mathbf{x}_{2})$.
Finally, for a set $K\subset\mathbb{R}^{2}$, we write

$\mathcal{V}\mathcal{G}^{{\dagger}}(K):=\bigl{\\{}\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}\,:\,\mathbf{x}\in
K\bigr{\\}},\qquad\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}(K):=\bigl{\\{}\widehat{\mathbf{x}}\in\mathcal{V}\mathcal{G}^{{\dagger}}\,:\,\widehat{\mathbf{x}}\in
K\bigr{\\}}.$

Before proceeding, we recall the following simple result. A oriented path in
$\mathcal{G}$ is a collection of oriented edges $e_{1}\cdots e_{n}$ in
$\mathcal{E}\mathcal{G}$ such that $\smash{e_{j}^{+}=e_{j+1}^{-}}$, for all
$\smash{j\in[n-1]}$. Furthermore, if also $\smash{e_{n}^{+}=e_{1}^{-}}$, then
$e_{1}\cdots e_{n}$ is called a oriented loop.

######  Lemma 2.2.

Let $e_{1}\cdots e_{n}$ be a oriented path in $\mathcal{G}$. Let
$\mathbf{e}_{1}$ be a lift of $e_{1}$ to the lifted graph
$\mathcal{G}^{{\dagger}}$, then there exists a unique path
$\mathbf{e}_{1}\cdots\mathbf{e}_{n}$ in $\mathcal{G}^{{\dagger}}$ such that
$\mathbf{e}_{j}$ is a lift of $e_{j}$, for all $j\in[n]_{2}$.

The main advantage of working in the universal cover of the cylinder is that
we can keep track of the winding of paths.

######  Definition 2.3.

Let $0\leq t_{1}<t_{2}$, consider a path
$P:[t_{1},t_{2}]\to\mathcal{C}_{2\pi}$, and let
$\mathbf{P}:[t_{1},t_{2}]\to\mathbb{R}^{2}$ be a lift of $P$ to the universal
cover. We define the winding of $P$ by letting

$\operatorname{wind}_{2\pi}(P):=\frac{\mathop{Re}\nolimits(\mathbf{P}(t_{2}))-\mathop{Re}\nolimits(\mathbf{P}(t_{1}))}{2\pi}.$

We say that $P$ winds around the cylinder if
$|\operatorname{wind}_{2\pi}(P)|\geq 1$. We say that $P$ does a
noncontractible loop around the cylinder if there exist times $t_{1}\leq
s_{1}<s_{2}\leq t_{2}$ such that $P|_{[s_{1},s_{2}]}$ winds around the
cylinder and $P(s_{1})=P(s_{2})$.

###  2.3 Random walks and electrical networks

In this subsection, we briefly recall the main concepts in the theory of
electrical networks and we refer to [LP16, Nac20] for a complete introduction.
Let $(\mathcal{G},c,v_{0},v_{1})$ be a doubly marked finite weighted planar
graph properly embedded in the infinite cylinder $\mathcal{C}_{2\pi}$ in the
sense of Definition 1.1 . The conductance of a vertex
$x\in\mathcal{V}\mathcal{G}$ is denoted by $\pi(x)$ and it is defined to be
the sum of the conductances of all the edges incident to $x$, i.e.,

$\pi(x):=\sum_{e\in\mathcal{E}\mathcal{G}(x)}c_{e},\quad\forall
x\in\mathcal{V}\mathcal{G}.$

##### Random walk.

The random walk on $(\mathcal{G},c)$ is the discrete time Markov chain
$X=\\{X_{n}\\}_{n\in\mathbb{N}_{0}}$ with state space $\mathcal{V}\mathcal{G}$
such that, for all $n\in\mathbb{N}_{0}$,

$\mathbb{P}\bigl{(}X_{n+1}=y\mid
X_{n}=x\bigr{)}=\begin{cases}c_{xy}/\pi(x),&y\in\mathcal{V}\mathcal{G}(x),\\\
0,&\text{otherwise}.\end{cases}$

Given a vertex $x\in\mathcal{V}\mathcal{G}$, we write $\mathbb{P}_{x}$ and
$\mathbb{E}_{x}$ for the law and expectation of $X$ started from $x$.
Moreover, we may write $X^{x}$ in order to emphasize that the random walk $X$
is started from the vertex $x\in\mathcal{V}\mathcal{G}$. With a slight abuse
of notation, we will also denote with $X=\\{X_{t}\\}_{t\geq 0}$ the continuous
time version of the random walk, where the continuous path is generated by
piecewise linear interpolation at constant speed. If the conductance on every
edge of the graph is equal to one, we call the random walk in this case simple
random walk.

We emphasize that, given a random walk $X$ on $(\mathcal{G},c)$, we can
canonically lift it to the lifted weighted planar graph
$(\mathcal{G}^{{\dagger}},c^{{\dagger}})$, and we denote the resulting walk by
$\mathbf{X}$. If $X^{x}$ is started from a point $x\in\mathcal{V}\mathcal{G}$,
then we need to specify the lift $\mathbf{x}\in\upsigma_{2\pi}^{-1}(x)$ of $x$
from which the lifted walk $\mathbf{X}^{\mathbf{x}}$ is started from. Similar
notation will be also adopted for the random walk on the dual graph.

##### Estimate on the total variation distance.

We now state and prove an elementary lemma for general weighted planar graphs
which allows to compare the total variation distance of the exit positions
from a set for two random walks started from two distinct points.

######  Lemma 2.4.

Let $(\mathcal{G},c)$ be a finite weighted planar graph and let
$W\subset\mathcal{V}\mathcal{G}$. For $x\in\mathcal{V}\mathcal{G}$, let
$X^{x}$ be the random walk on $(\mathcal{G},c)$ started from $x$ and let
$\uptau_{x}$ be the first time that $X^{x}$ hits $W$. Then, for $x$,
$y\in\mathcal{V}\mathcal{G}\setminus W$, it holds that

$\operatorname{\mathrm{d}^{\mathrm{TV}}}\bigl{(}X^{x}_{\uptau_{x}},X^{y}_{\uptau_{y}}\bigr{)}\leq\mathbb{P}\bigl{(}X^{x}|_{[0,\uptau_{x}]}\text{
does not disconnect }y\text{ from }W\bigr{)},$

where $\operatorname{\mathrm{d}^{\mathrm{TV}}}$ denotes the total variation
distance.

###### Proof.

The proof is a variant of [GMS22, Lemma 3.12] with the difference that one
should consider a weighted spanning tree instead of a uniform spanning tree of
the finite weighted planar graph $(\mathcal{G},c)$. For the reader’s
convenience, we gather here a proof. The lemma is a consequence of Wilson’s
algorithm. Consider the weighted spanning tree $\mathcal{T}$ of the finite
weighted planar graph $(\mathcal{G},c)$, where all vertices of $W$ are wired
to a single point. We recall that the weighted spanning tree $\mathcal{T}$ is
chosen randomly from among all the spanning trees with probability
proportional to the product of the conductances along the edges of the tree.
For $x\in\mathcal{V}\mathcal{G}$, let $L^{x}$ be the unique path in
$\mathcal{T}$ from $x$ to $W$. For a path $P$ in $\mathcal{G}$, write
$\operatorname{LE}(P)$ for its chronological loop erasure. By Wilson’s
algorithm (see [LP16, Theorem 4.1]), we can generate the union $L^{x}\cup
L^{y}$ by using the following procedure.

  1. (a)

Run $X^{y}$ until time $\uptau_{y}$ and generate the loop erasure
$\operatorname{LE}(X^{y}|_{[0,\uptau^{y}]})$.

  2. (b)

Conditional on $X^{y}|_{[0,\uptau_{y}]}$, run $X^{x}$ until the first time
$\widetilde{\uptau}_{x}$ that it hits either
$\operatorname{LE}(X^{y}|_{[0,\uptau_{y}]})$ or $W$.

  3. (c)

Set $L^{x}\cup
L^{y}=\operatorname{LE}(X_{y}|_{[0,\uptau_{y}]})\cup\operatorname{LE}(X^{x}|_{[0,\widetilde{\uptau}_{x}]})$.

Note that $L_{y}=\operatorname{LE}(X^{y}|_{[0,\uptau_{y}]})$ in the above
procedure. Interchanging the roles of $x$ and $y$ in the above procedure shows
that $L^{x}$ and $\operatorname{LE}(X^{x}|_{[0,\uptau_{x}]})$ have the same
distribution. When constructing $L^{x}\cup L^{y}$ as described above, the
points at which $L^{x}$ and $L^{y}$ hit $W$ coincide if $X^{x}$ hits
$\operatorname{LE}(X^{y}|_{[0,\uptau_{y}]})$ before reaching $W$. In
particular, this occurs when $X^{x}|_{[0,\uptau_{x}]}$ disconnects $y$ from
$W$. Thus, there is a coupling between
$\operatorname{LE}(X^{x}|_{[0,\uptau_{x}]})$ and
$\operatorname{LE}(X^{y}|_{[0,\uptau_{y}]})$, where the probability that these
two loop erasures hit $W$ at the same point is at least
$\mathbb{P}_{x}(X|_{[0,\uptau_{x}]}\text{ disconnects }y\text{ from }W)$. Now,
by observing that $X^{x}_{\uptau_{x}}$ corresponds to the point at which
$\operatorname{LE}(X^{x}|_{[0,\uptau_{x}]})$ first hit $W$, and similarly for
$y$ in place of $x$, we obtain the desired result. ∎

##### Electrical network.

There is an extremely useful correspondence between random walks and
Kirchhoff’s theory of electric networks. Let $(\mathcal{G},c,v_{0},v_{1})$ be
as above and suppose that every edge $e\in\mathcal{E}\mathcal{G}$ is made of
conducting wires with conductance equals to $c_{e}$. Connect a battery between
$v_{1}$ and $v_{0}$ so that the voltage at $v_{1}$ is equal to one and the
voltage at $v_{0}$ is equal to zero. Then certain currents will flow along the
edges of the graph establishing the voltage at each vertex
$x\in\mathcal{V}\mathcal{G}\setminus\\{v_{0},v_{1}\\}$. An immediate
consequence of physical laws is that the voltage function is harmonic on
$\mathcal{V}\mathcal{G}$ except at $v_{0}$ and $v_{1}$. More formally, we have
the following definition.

######  Definition 2.5.

The voltage function associated to the quadruplet
$(\mathcal{G},c,v_{0},v_{1})$ is the unique function
$\mathfrak{h}:\mathcal{V}\mathcal{G}\to[0,1]$ such that
$\mathfrak{h}(v_{0})=0$, $\mathfrak{h}(v_{1})=1$, and

$\mathfrak{h}(x)=\frac{1}{\pi(x)}\sum_{y\in\mathcal{V}\mathcal{G}(x)}c_{xy}\mathfrak{h}(y),\quad\forall
x\in\mathcal{V}\mathcal{G}\setminus\\{v_{0},v_{1}\\}.$

In view of the role that $\mathfrak{h}$ will play in the construction of the
Smith embedding, we will also call $\mathfrak{h}$ the _height coordinate
function_.

Given an edge $e\in\mathcal{E}\mathcal{G}$, we say that $e$ is _harmonically
oriented_ if $\mathfrak{h}(e^{+})\geq\mathfrak{h}(e^{-})$. In what follows,
unless otherwise specified, we always consider the harmonic orientation of the
edges in $\mathcal{E}\mathcal{G}$. It is a remarkable fact that the voltage
function $\mathfrak{h}$ admits a representation in terms of a random walk $X$
on $(\mathcal{G},c)$. More precisely, if for all $v\in\mathcal{V}\mathcal{G}$,
we define $\uptau_{v}$ to be the first hitting time of $v$ from $X$, then one
can easily check that

$\mathfrak{h}(x)=\mathbb{P}_{x}\bigl{(}\uptau_{v_{1}}<\uptau_{v_{0}}\bigr{)},\quad\forall
x\in\mathcal{V}\mathcal{G}.$

Moreover, since the voltage function $\mathfrak{h}$ is harmonic on
$\mathcal{V}\mathcal{G}\setminus\\{v_{0},v_{1}\\}$, if $X^{x}$ is a random
walk on $(\mathcal{G},c)$ started from $x\in\mathcal{V}\mathcal{G}$ and killed
upon reaching the set $\\{v_{0},v_{1}\\}$, then the process
$\mathfrak{h}(X^{x})$ is a martingale with respect to the filtration generated
by $X^{x}$.

######  Remark 2.6.

We note that we can canonically lift the voltage function $\mathfrak{h}$ to
the lifted weighted graph $(\mathcal{G}^{{\dagger}},c^{{\dagger}})$ by setting
$\mathfrak{h}^{{\dagger}}:\mathcal{V}\mathcal{G}^{{\dagger}}\to[0,1]$ as
follows

$\mathfrak{h}^{{\dagger}}(\mathbf{x}):=\mathfrak{h}(\upsigma_{2\pi}(\mathbf{x})),\quad\forall\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}.$
(2.4)

######  Remark 2.7.

Note that we can naturally extend the definition of the voltage function to a
function on the metric graph $\mathbb{G}$ associated to $\mathcal{G}$, i.e.,
we can define the function $\mathfrak{h}:\mathbb{G}\to[0,1]$. This extension
can be done by linearly interpolating the values at the endpoints of every
edge $e\in\mathcal{E}\mathcal{G}$. More precisely, if $x\in\mathbb{G}$ is a
point lying on the harmonically oriented edge $e\in\mathcal{E}\mathcal{G}$,
then we set

$\mathfrak{h}(x):=(\mathfrak{h}(e^{+})-\mathfrak{h}(e^{-}))\mathrm{d}^{\mathbb{G}}(e^{-},x)+\mathfrak{h}(e^{-}).$

##### The flow induced by the voltage function.

We finish this subsection by introducing the flow across oriented edges
induced by the voltage function $\mathfrak{h}$. We denote such a flow by
$\nabla\mathfrak{h}:\mathcal{E}\mathcal{G}\to\mathbb{R}$ and we define it as
follows

$\nabla\mathfrak{h}(e):=c_{e}\bigl{(}\mathfrak{h}(e^{+})-\mathfrak{h}(e^{-})\bigr{)},\quad\forall
e\in\mathcal{E}\mathcal{G}.$

The flow $\nabla\mathfrak{h}$ satisfies the following well-known properties
(see [Nac20, Section 2.2]).

  1. (a)

(_Antisymmetry_) For every oriented edge $e\in\mathcal{E}\mathcal{G}$, it
holds that

$\nabla\mathfrak{h}(-e)=-\nabla\mathfrak{h}(e),$

where $-e$ stands for the edge $e$ endowed with opposite orientation.

  2. (b)

(_Kirchhoff’s node law_) For all
$x\in\mathcal{V}\mathcal{G}\setminus\\{v_{0},v_{1}\\}$, it holds that

$\sum_{e\in\mathcal{E}\mathcal{G}(x)}\nabla\mathfrak{h}(e)=0,$ (2.5)

where here the orientation of each $e\in\mathcal{E}\mathcal{G}(x)$ is fixed by
letting $e^{-}=x$.

  3. (c)

(_Kirchhoff’s cycle law_) For every directed cycle $e_{1}\cdots e_{n}$, it
holds that

$\sum_{i=1}^{n}\frac{1}{c_{e_{i}}}\nabla\mathfrak{h}(e_{i})=0.$ (2.6)

We denote the strength of the flow $\nabla\mathfrak{h}$ induced by
$\mathfrak{h}$ by setting

$\upeta:=\sum_{e\in\mathcal{E}\mathcal{G}^{\uparrow}(v_{0})}\nabla\mathfrak{h}(e),$
(2.7)

where $\smash{\mathcal{E}\mathcal{G}^{\uparrow}(v_{0})}$ denotes the set of
harmonically oriented edges in $\mathcal{E}\mathcal{G}$ with tails equal to
$v_{0}$. Furthermore, thanks to the harmonicity of $\mathfrak{h}$, a simple
computation yields that
$\smash{\upeta=\sum_{e\in\mathcal{E}\mathcal{G}^{\downarrow}(v_{1})}\nabla\mathfrak{h}(e)}$,
where $\smash{\mathcal{E}\mathcal{G}^{\downarrow}(v_{1})}$ denotes the set of
harmonically oriented edges in $\mathcal{E}\mathcal{G}$ with heads equal to
$v_{1}$.

###  2.4 Discrete harmonic conjugate

Let $(\mathcal{G},c,v_{0},v_{1})$ be a doubly marked finite weighted planar
graph properly embedded in the infinite cylinder $\mathcal{C}_{2\pi}$
according to Definition 1.1, and let
$\smash{(\widehat{\mathcal{G}},\widehat{c})}$ be the associated weighted dual
planar graph properly embedded in $\mathcal{C}_{2\pi}$ according to Definition
1.2. Moreover, let $\smash{\mathfrak{h}:\mathcal{V}\mathcal{G}\to[0,1]}$ be
the voltage function associated with $\smash{(\mathcal{G},c,v_{0},v_{1})}$ as
defined in Definition 2.5. We want to define the discrete harmonic conjugate
function of $\mathfrak{h}$, i.e., the function $\mathfrak{w}$ defined on the
set of dual vertices $\smash{\mathcal{V}\widehat{\mathcal{G}}}$ that satisfies
the discrete Cauchy–Riemann equation. More formally, for every directed edge
$e\in\mathcal{E}\mathcal{G}$ and its corresponding oriented dual edge
$\smash{\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}}$, the function
$\mathfrak{w}$ should satisfy the following identity

$\nabla\mathfrak{w}(\widehat{e}):=\widehat{c}_{\widehat{e}}\bigl{(}\mathfrak{w}(\widehat{e}^{+})-\mathfrak{w}(\widehat{e}^{-})\bigr{)}=\mathfrak{h}(e^{+})-\mathfrak{h}(e^{-}),$
(2.8)

where we recall that $\widehat{c}_{\widehat{e}}=1/c_{e}$.

To precisely define the function $\mathfrak{w}$ specified above, it will be
more convenient to work with the lifted weighted graph
$\smash{(\mathcal{G}^{{\dagger}},c^{{\dagger}})}$ and its dual
$\smash{(\widehat{\mathcal{G}}^{{\dagger}},\widehat{c}^{{\dagger}})}$. More
precisely, we consider the lifted voltage function
$\smash{\mathfrak{h}^{{\dagger}}:\mathcal{V}\mathcal{G}^{{\dagger}}\to[0,1]}$.
We fix an arbitrary vertex
$\smash{\widehat{\mathbf{x}}_{0}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}}$
on the lifted dual graph, and for every
$\smash{\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}}$
we consider a directed path of lifted dual edges
$\smash{\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}}$ connecting
$\smash{\widehat{\mathbf{x}}_{0}}$ to $\smash{\widehat{\mathbf{x}}}$.

######  Remark 2.8.

We recall that we are assuming that the two marked vertices $v_{0}$ and
$v_{1}$ do not lie on the boundary of the same face (see Definition 1.1). In
particular, this implies that there is a path of dual edges in
$\widehat{\mathcal{G}}$ that disconnects $v_{0}$ from $v_{1}$. Hence, the
lifted dual graph $\smash{\widehat{\mathcal{G}}^{{\dagger}}}$ is connected and
we can always find a path connecting $\smash{\widehat{\mathbf{x}}_{0}}$ to any
$\smash{\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}}$.

We define the function
$\mathfrak{w}^{{\dagger}}:\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}\to\mathbb{R}$
by setting

$\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}):=\sum_{j=1}^{n}\nabla\mathfrak{h}^{{\dagger}}(\mathbf{e}_{j}),\quad\forall\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}.$
(2.9)

where $\mathbf{e}_{j}\in\mathcal{E}\mathcal{G}^{{\dagger}}$ is the oriented
primal edge associated to $\widehat{\mathbf{e}}_{j}$. We call the function
$\mathfrak{w}^{{\dagger}}$ defined in this way the _lifted discrete harmonic
conjugate_ function associated to $\mathfrak{h}^{{\dagger}}$ with base vertex
$\widehat{\mathbf{x}}_{0}$. The following lemma guarantees that
$\mathfrak{w}^{{\dagger}}$ is actually well-defined.

######  Lemma 2.9.

For all $\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$,
the value $\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}})$ defined in (2.9)
does not depend on the choice of the directed path from
$\widehat{\mathbf{x}}_{0}$ to $\widehat{\mathbf{x}}$. Moreover, for any
$\widehat{\mathbf{x}}_{1}$,
$\widehat{\mathbf{x}}_{2}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$ such
that
$\upsigma_{2\pi}(\widehat{\mathbf{x}}_{1})=\upsigma_{2\pi}(\widehat{\mathbf{x}}_{2})$,
the following relation holds

$\frac{\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}_{1})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}_{2})}{\upeta}=\frac{\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{1})-\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{2})}{2\pi},$

where we recall that $\upeta$ denotes the strength of the flow induced by
$\mathfrak{h}$ as defined in (2.7).

###### Proof.

For the first part of the lemma, the proof is similar to that of [BS96, Lemma
3.2]. In particular, it is sufficient to prove that for any oriented loop
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ in
$\widehat{\mathcal{G}}^{{\dagger}}$, it holds that

$\sum_{j=1}^{n}\bigl{(}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+}_{j})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-}_{j})\bigr{)}=0.$
(2.10)

The key observation in [BS96, Lemma 3.2] is that every oriented loop
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ in
$\widehat{\mathcal{G}}^{{\dagger}}$ can be written as the disjoint union of
simple closed loops and of paths of length two consisting of a single dual
edge traversed in both directions. Here, by a simple closed path we mean that
$\widehat{\mathbf{e}}_{j}^{+}\neq\widehat{\mathbf{e}}_{k}^{+}$ for distinct
$j$, $k\in[n]$ when $n>2$, while when $n=2$, we mean that
$\widehat{\mathbf{e}}_{1}\neq-\widehat{\mathbf{e}}_{2}$. Therefore, since
(2.10) obviously holds if the path consists of a single dual edge traversed in
both directions, we can assume without loss of generality that
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ is a simple counter-
clockwise oriented closed loop. Let $K\subset\mathbb{R}^{2}$ be the bounded
connected component of
$\mathbb{R}^{2}\setminus\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$.
Then, thanks to (2.8), it holds that

$\sum_{j=1}^{n}\bigl{(}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+}_{j})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-}_{j})\bigr{)}=\sum_{j=1}^{n}\nabla\mathfrak{h}^{{\dagger}}(\mathbf{e}_{j})=\sum_{\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}(K)}\sum_{\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}(\mathbf{x})}\nabla\mathfrak{h}^{{\dagger}}(\mathbf{e}),$
(2.11)

where here the orientation of each edge
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}(\mathbf{x})$ is fixed by
letting $\mathbf{e}^{-}=\mathbf{x}$. The second equality in (2.11) follows
from the following argument. Fix
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}(K)$ and consider
$\mathbf{y}\in\mathcal{V}\mathcal{G}^{{\dagger}}(\mathbf{x})$. If
$\mathbf{y}\not\in K$, then $\mathbf{x}\mathbf{y}=\mathbf{e}_{j}$ for some
$j\in[n]$, while if $\mathbf{y}\in K$ then
$\nabla\mathfrak{h}^{{\dagger}}(\mathbf{x}\mathbf{y})$ cancels out with
$\nabla\mathfrak{h}^{{\dagger}}(\mathbf{y}\mathbf{x})$ thanks to the
antisymmetry of $\nabla\mathfrak{h}^{{\dagger}}$. The term on the right-hand
side of (2.11) is equal to zero thanks to the boundedness of $K$ and
Kirchhoff’s node law (2.5).

Concerning the second part of the lemma, we can proceed as follows. Let
$\widehat{\mathbf{x}}_{1}$,
$\widehat{\mathbf{x}}_{2}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$ be
such that
$\upsigma_{2\pi}(\widehat{\mathbf{x}}_{1})=\upsigma_{2\pi}(\widehat{\mathbf{x}}_{2})$,
and let $k\in\mathbb{Z}$ be such that
$k=(\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{2})-\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{1}))/(2\pi)$.
Then we need to prove that
$\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}_{2})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}_{1})=\upeta
k$. In particular, thanks to the first part of the lemma, it is sufficient to
consider an arbitrary directed path
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ in
$\widehat{\mathcal{G}}^{{\dagger}}$ connecting $\widehat{\mathbf{x}}_{1}$ to
$\widehat{\mathbf{x}}_{2}$ and show that
$\sum_{j=1}^{n}(\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+}_{j})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-}_{j}))=\upeta
k$. We assume first that $k=1$. We can choose the lifted dual edges
$[\widehat{\mathbf{e}}_{n}]$ in such way that
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ is a simple path
oriented from left to right. Now, using an argument similar to the one used
above, it is not difficult to see that
$\sum_{j=1}^{n}\nabla\mathfrak{h}^{{\dagger}}(\mathbf{e}_{j})=\upeta$, and so
the conclusion follows in this case. Finally, the general case can be obtained
easily: we can just “glue” together, by eventually changing the orientation if
$k$ is negative, $k$ copies of the path used in the case $k=1$. ∎

From the definition (2.9) of the function $\mathfrak{w}^{{\dagger}}$, it
follows that for every oriented edge
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$ and for the associated dual
edge $\widehat{\mathbf{e}}\in\mathcal{E}\widehat{\mathcal{G}}^{{\dagger}}$, it
holds that

$\nabla\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}):=\widehat{c}^{{\dagger}}_{\widehat{\mathbf{e}}}(\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-}))=\mathfrak{h}^{{\dagger}}(\mathbf{e}^{+})-\mathfrak{h}^{{\dagger}}(\mathbf{e}^{-}),$

i.e., $\mathfrak{w}^{{\dagger}}$ satisfies the discrete Cauchy–Riemann
equation. Moreover, an immediate application of Kirchhoff’s cycle law (2.6)
implies that the function $\mathfrak{w}^{{\dagger}}$ is harmonic on
$\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$. Thanks to Lemma 2.9, we can
define the discrete harmonic conjugate function of $\mathfrak{h}$ as follows.

######  Definition 2.10.

The discrete harmonic conjugate function of $\mathfrak{h}$ with base vertex
$\widehat{x}_{0}\in\mathcal{V}\widehat{\mathcal{G}}$ is the unique function
$\mathfrak{w}:\mathcal{V}\widehat{\mathcal{G}}\to\mathbb{R}/\upeta\mathbb{Z}$
such that $\mathfrak{w}(\widehat{x}_{0})=0$ and

$\mathfrak{w}(\widehat{x})=\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}})\;\mathrm{mod}(\upeta),\quad\forall\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}},$

where
$\mathfrak{w}^{{\dagger}}:\mathcal{V}\mathcal{G}^{{\dagger}}\to\mathbb{R}$ is
the function defined in (2.9) with base vertex an arbitrary lift of
$\widehat{x}_{0}$. In view of the role that $\mathfrak{w}$ will play in the
construction of the Smith embedding, we will also call $\mathfrak{w}$ the
_width coordinate function_.

######  Remark 2.11.

As for the case of the voltage function $\mathfrak{h}$, we can naturally
extend the definition of $\mathfrak{w}$ to a function from the dual metric
graph $\widehat{\mathbb{G}}$, i.e.,
$\mathfrak{w}:\widehat{\mathbb{G}}\to\mathbb{R}/\upeta\mathbb{Z}$. To be
precise, if $\widehat{x}\in\widehat{\mathbb{G}}$ is a point on the edge
$\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}$ and
$\mathfrak{w}(\widehat{e}^{+})\geq\mathfrak{w}(\widehat{e}^{-})$, then we set

$\mathfrak{w}(\widehat{x}):=(\mathfrak{w}(\widehat{e}^{+})-\mathfrak{w}(\widehat{e}^{-}))\mathrm{d}^{\widehat{\mathbb{G}}}(\widehat{e}^{-},\widehat{x})+\mathfrak{w}(\widehat{e}^{-}).$

###  2.5 Construction of the Smith embedding

Let $(\mathcal{G},c,v_{0},v_{1})$ be a doubly marked finite weighted planar
graph properly embedded in the infinite cylinder $\mathcal{C}_{2\pi}$
according to Definition 1.1, and let
$\smash{(\widehat{\mathcal{G}},\widehat{c})}$ be the associated weighted dual
planar graph properly embedded in $\mathcal{C}_{2\pi}$ according to Definition
1.2. The aim of this subsection is to precisely define the _Smith embedding_
of $(\mathcal{G},c,v_{0},v_{1})$. As we have already explained in the
introduction, the Smith embedding is built in terms of a tiling of a finite
cylinder with rectangles in which every edge $e\in\mathcal{E}\mathcal{G}$
corresponds to a rectangle in the tiling, every vertex
$x\in\mathcal{V}\mathcal{G}$ corresponds to the maximal horizontal segment
tangent with all rectangles corresponding to the edges incident to $x$, and
every dual vertex $\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}$ corresponds
to the maxiaml vertical segment tangent with all rectangles corresponding to
primal edges surrounding $\widehat{x}$. The existence of such tiling was first
proven in [BSST40] and then successively extended in [BS96].

##### The main objects.

To precisely define the Smith embedding, it will be more convenient to work
with the lifted weighted graph
$\smash{(\mathcal{G}^{{\dagger}},c^{{\dagger}})}$ and its dual
$\smash{(\widehat{\mathcal{G}}^{{\dagger}},\widehat{c}^{{\dagger}})}$. More
precisely, we need to consider the lifted voltage function
$\mathfrak{h}^{{\dagger}}:\mathcal{V}\mathcal{G}^{{\dagger}}\to[0,1]$ and its
lifted discrete harmonic conjugate function
$\mathfrak{w}^{{\dagger}}:\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}\to\mathbb{R}$.
For every edge $\smash{\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}}$,
consider its harmonic orientation and let
$\smash{\widehat{\mathbf{e}}\in\mathcal{E}\widehat{\mathcal{G}}^{{\dagger}}}$
be the corresponding oriented dual edge. We define the intervals

$\mathrm{I}_{\mathbf{e}}:=\bigl{[}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-}),\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+})\bigr{]},\qquad\widehat{\mathrm{I}}_{\widehat{\mathbf{e}}}:=\bigl{[}\mathfrak{h}^{{\dagger}}(\mathbf{e}^{-}),\mathfrak{h}^{{\dagger}}(\mathbf{e}^{+})\bigr{]}.$

Then, we define the rectangle $\mathrm{R}_{\mathbf{e}}$ associated to the edge
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$ by letting

$\mathrm{R}_{\mathbf{e}}:=\mathrm{I}_{\mathbf{e}}\times\widehat{\mathrm{I}}_{\widehat{\mathbf{e}}}\subset\mathbb{R}\times[0,1],\quad\forall\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}.$
(2.12)

Recalling the definition (2.9) of the lifted discrete harmonic conjugate
function $\mathfrak{w}^{{\dagger}}$, it holds that

$\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{+})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{-})=c_{\mathbf{e}}(\mathfrak{h}^{{\dagger}}(\mathbf{e}^{+})-\mathfrak{h}^{{\dagger}}(\mathbf{e}^{-})).$

Therefore, the aspect ratio of the rectangle $\mathrm{R}_{\mathbf{e}}$ is
equal to the conductance $c_{\mathbf{e}}$ of the edge
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$. In particular, this implies
that if an edge $\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$ has unit
conductance, then $\mathrm{R}_{\mathbf{e}}$ is a square. For a vertex
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$, we define the closed
horizontal line segment $\mathrm{H}_{\mathbf{x}}$ by setting

$\mathrm{H}_{\mathbf{x}}:=\bigcup_{\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger},\downarrow}(\mathbf{x})}\mathrm{I}_{\mathbf{e}}\times\\{\mathfrak{h}^{{\dagger}}(\mathbf{x})\\}\subset\mathbb{R}\times[0,1],\quad\forall\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}},$
(2.13)

where $\mathcal{E}\mathcal{G}^{{\dagger},\downarrow}(\mathbf{x})$ denotes the
set of harmonically oriented lifted edges with heads equal to $\mathbf{x}$.
Finally, for a dual vertex
$\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$, we
define the closed vertical line segment $\mathrm{V}_{\widehat{\mathbf{x}}}$ by
letting

$\mathrm{V}_{\widehat{\mathbf{x}}}:=\bigcup_{\widehat{\mathbf{e}}\in\mathcal{E}\widehat{\mathcal{G}}^{{\dagger},\downarrow}(\widehat{\mathbf{x}})}\\{\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}})\\}\times\widehat{\mathrm{I}}_{\widehat{\mathbf{e}}}\subset\mathbb{R}\times[0,1],\quad\forall\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}},$
(2.14)

where
$\mathcal{E}\widehat{\mathcal{G}}^{{\dagger},\downarrow}(\widehat{\mathbf{x}})$
denotes the set of harmonically oriented lifted dual edges with heads equal to
$\widehat{\mathbf{x}}$. Thanks to the harmonicity of the lifted height
coordinate function, we observe that in the definition of
$\mathrm{H}_{\mathbf{x}}$, one can replace
$\mathcal{E}\mathcal{G}^{{\dagger},\downarrow}(\mathbf{x})$ with
$\mathcal{E}\mathcal{G}^{{\dagger},\uparrow}(x)$, and similarly for
$\mathrm{V}_{\widehat{\mathbf{x}}}$.

##### Construction of the tiling.

We recall that $\upeta$ denotes the strength of the flow induced by
$\mathfrak{h}$ as defined in (2.7). We consider the cylinder

$\mathcal{C}_{\upeta}:=\mathbb{R}/\upeta\mathbb{Z}\times[0,1],$

where $\mathbb{R}/\upeta\mathbb{Z}$ denotes the circle of length $\upeta$. We
let $(\mathbb{R}\times[0,1],\upsigma_{\upeta})$ be the universal cover of
$\mathcal{C}_{\upeta}$, where the covering map
$\upsigma_{\upeta}:\mathbb{R}\times[0,1]\to\mathcal{C}_{\upeta}$ is defined by

$\upsigma_{\upeta}(t,x):=\bigl{(}e^{i2\pi
t/\upeta},x\bigr{)},\quad\forall(t,x)\in\mathbb{R}\times[0,1].$ (2.15)

We are now ready to define the tiling map. For each
$e\in\mathcal{E}\mathcal{G}$, $x\in\mathcal{V}\mathcal{G}$, and
$\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}$, we define the following
objects

$\mathrm{R}_{e}:=\upsigma_{\upeta}(\mathrm{R}_{\mathbf{e}}),\qquad\mathrm{H}_{x}:=\upsigma_{\upeta}(\mathrm{H}_{\mathbf{x}}),\qquad\mathrm{V}_{\widehat{x}}:=\upsigma_{\upeta}(\mathrm{V}_{\widehat{\mathbf{x}}}),$
(2.16)

where $\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$,
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$, and
$\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$ are
lifts of $e$, $x$, and $\widehat{x}$, respectively. An immediate consequence
of Lemma 2.9 is that $\mathrm{R}_{e}$, $\mathrm{H}_{x}$, and
$\mathrm{V}_{\widehat{x}}$ are well-defined, i.e., they do not depend on the
particular choice of the lifts $\mathbf{e}$, $\mathbf{x}$ and
$\widehat{\mathbf{x}}$.

The following properties are well-known (see [BS96, Theorem 3.1]).

  1. (a)

The collection of rectangles
$\\{\mathrm{R}_{e}\\}_{e\in\mathcal{E}\mathcal{G}}$ constitutes a tiling of
$\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$, i.e., for each pair of distinct
edges $e_{1}$, $e_{2}\in\mathcal{E}\mathcal{G}$, the interiors of the
rectangles $\mathrm{R}_{e_{1}}$ and $\mathrm{R}_{e_{2}}$ are disjoint and
$\cup_{e\in\mathcal{E}\mathcal{G}}\mathrm{R}_{e}=\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$.

  2. (b)

For each two distinct edges $e_{1}$, $e_{2}\in\mathcal{E}\mathcal{G}$, the
interiors of the vertical sides of the rectangles $\mathrm{R}_{e_{1}}$ and
$\mathrm{R}_{e_{2}}$ have a non-trivial intersection only if $e_{1}$ and
$e_{2}$ both lie in the boundary of some common face of $\mathcal{G}$.

  3. (c)

Two rectangles intersect along their horizontal (resp. vertical) boundaries if
and only if the corresponding primal (resp. dual) edges share an endpoint.

We note that if $e\in\mathcal{E}\mathcal{G}$ is such that no current flows
through it, i.e., $\mathfrak{h}(e^{-})=\mathfrak{h}(e^{+})$, then the
corresponding rectangle $\mathrm{R}_{e}$ is degenerate and consists only of a
single point. We also remark that the existence of the aforementioned tiling
was proven by Benjamini and Schramm in [BS96]. Originally, their proof was
stated specifically for the case of edges with unit conductance, however, it
can be readily extended to our setting.

######  Definition 2.12 (Tiling map).

The tiling map associated to the quadruplet $(\mathcal{G},c,v_{0},v_{1})$ is
the map

$\SS:\mathcal{E}\mathcal{G}\cup\mathcal{V}\mathcal{G}\cup\mathcal{V}\widehat{\mathcal{G}}\to\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$

such that

$\SS(e):=\mathrm{R}_{e},\quad\forall
e\in\mathcal{E}\mathcal{G};\qquad\SS(x):=\mathrm{H}_{x},\quad\forall
x\in\mathcal{V}\mathcal{G};\qquad\SS(\widehat{x}):=\mathrm{V}_{\widehat{x}},\quad\forall\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}},$

where $\mathrm{R}_{e}$, $\mathrm{H}_{x}$, and $\mathrm{V}_{\widehat{x}}$ are
as defined in (2.16). The image of the tiling map $\SS$ is called the Smith
diagram associated to $(\mathcal{G},c,v_{0},v_{1})$.

We refer to Figure 1 for an illustration of the Smith diagram associated to a
given quadruplet $(\mathcal{G},c,v_{0},v_{1})$.

######  Remark 2.13.

Since the height coordinate function $\mathfrak{h}$ can be extended to the
metric graph $\mathbb{G}$, we can view each rectangle $\smash{\mathrm{R}_{e}}$
of the tiling as being foliated into horizontal segments, one for each inner
point of the corresponding edge $\smash{e\in\mathcal{E}\mathcal{G}}$.
Similarly, since the width coordinate function $\mathfrak{w}$ can be extended
to the dual metric graph $\smash{\widehat{\mathbb{G}}}$, we can also view each
rectangle $\mathrm{R}_{e}$ of the tiling as being foliated into vertical
segments, one for each inner point of the corresponding dual edge
$\smash{\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}}$.

It will be also convenient to introduce the lifted tiling map associated to
$(\mathcal{G},c,v_{0},v_{1})$ which is the map

$\SS^{{\dagger}}:\mathcal{E}\mathcal{G}^{{\dagger}}\cup\mathcal{V}\mathcal{G}^{{\dagger}}\cup\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}\to\mathbb{R}\times[0,1]$

such that $\SS^{{\dagger}}(\mathbf{x}):=\mathrm{H}_{\mathbf{x}}$ for each
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$,
$\SS^{{\dagger}}(\mathbf{e}):=\mathrm{R}_{\mathbf{e}}$ for each
$\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}$, and
$\SS^{{\dagger}}(\widehat{\mathbf{x}}):=\mathrm{V}_{\widehat{\mathbf{x}}}$ for
each $\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}$. We
emphasize that, since the collection
$\\{R_{e}\\}_{e\in\mathcal{E}\mathcal{G}}$ forms a tiling of the cylinder
$\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$, the collection of rectangles
$\\{R_{\mathbf{e}}\\}_{\mathbf{e}\in\mathcal{E}\mathcal{G}^{{\dagger}}}$ forms
a periodic tiling of $\mathbb{R}\times[0,1]$ of period $\upeta$. We are now
ready to precisely define the Smith embedding.

######  Definition 2.14 (Smith embedding).

The Smith embedding associated to the quadruplet $(\mathcal{G},c,v_{0},v_{1})$
is the function
$\dot{\mathcal{S}}:\mathcal{V}\mathcal{G}\to\mathbb{R}/\upeta\mathbb{Z}\times[0,1]$
such that

$\dot{\mathcal{S}}(x)=\operatorname{mid}(\mathrm{H}_{x}),\quad\forall
x\in\mathcal{V}\mathcal{G},$

where $\operatorname{mid}(\mathrm{H}_{x})$ denotes the middle point of the
horizontal line segment $\SS(x)$. Moreover, we define the lifted Smith
embedding
$\dot{\mathcal{S}}^{{\dagger}}:\mathcal{V}\mathcal{G}^{{\dagger}}\to\mathbb{R}\times[0,1]$
as the map that assigns to each
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$ the middle point of the
horizontal line segment $\SS^{{\dagger}}(\mathbf{x})$.

We emphasize once again that the choice to define the Smith embedding by
picking the middle point of each horizontal line segment is rather arbitrary.
Indeed, the main result of this paper holds also if one chose an arbitrary
point inside each horizontal segment. Finally, for technical reasons, we also
need to introduce the following map.

######  Definition 2.15.

We define the map $\dot{\mathcal{S}}^{{\dagger},\rm{rand}}$ that assigns to
each vertex $\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$ the random
variable $\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x})$ which is
uniformly distributed on the horizontal line segment
$\SS^{{\dagger}}(\mathbf{x})$.

##  3 Some properties of the Smith embedding

In this section, we collect some results that follow directly from the
construction of the Smith embedding. We fix throughout this section a doubly
marked finite weighted planar graph $(\mathcal{G},c,v_{0},v_{1})$ properly
embedded in the infinite cylinder $\mathcal{C}_{2\pi}$ according to Definition
1.1, and we also consider the associated weighted dual planar graph
$\smash{(\widehat{\mathcal{G}},\widehat{c})}$ properly embedded in
$\mathcal{C}_{2\pi}$ according to Definition 1.2. In what follows, we consider
the metric graph $\mathbb{G}$ associated to $\mathcal{G}$, and we let
$\mathfrak{h}:\mathbb{G}\to[0,1]$ be the extended height coordinate function
as specified in Remark 2.7. Furthermore, we also consider the dual metric
graph $\widehat{\mathbb{G}}$, and we let
$\mathfrak{w}:\widehat{\mathbb{G}}\to\mathbb{R}/\upeta\mathbb{Z}$ be the
extended width coordinate function as specified in Remark 2.11.

###  3.1 Adding new vertices

In this subsection, we see how declaring a finite amount of interior points of
some edges in the graph to be vertices affects the height coordinate function
and the random walk on the graph. More precisely, we start with the following
definition.

######  Definition 3.1.

Let $W\subset\mathbb{G}$ be a finite subset of the metric graph. We define the
weighted planar graph $(\mathcal{G}^{\prime},c^{\prime})$ associated to
$(\mathcal{G},c)$ and $W$ as follows:

  1. (a)

The set of vertices $\mathcal{V}\mathcal{G}^{\prime}$ is given by
$\mathcal{V}\mathcal{G}\cup W$;

  2. (b)

If the interior of an edge $e\in\mathcal{E}\mathcal{G}$ contains
$n\in\mathbb{N}$ points of $W$, then $e$ is split into $n+1$ new edges
$\smash{[e^{\prime}_{n+1}]}$ according to the points in the interior of $e$.
The edge $e$ remains unchanged otherwise.

  3. (c)

If the interior of an edge $e\in\mathcal{E}\mathcal{G}$ is split into
$\smash{[e^{\prime}_{n+1}]}$ new edges, for some $n\in\mathbb{N}$, then we set

$c^{\prime}_{e^{\prime}_{i}}:=\frac{c_{e}}{\mathrm{d}^{\mathbb{G}}\bigl{(}e_{i}^{\prime,-},e_{i}^{\prime,+}\bigr{)}},\quad\forall
i\in[n+1].$

The conductance of $e$ remain unchanged otherwise.

The weighted dual graph
$(\widehat{\mathcal{G}}^{\prime},\widehat{c}^{\prime})$ can be naturally
constructed from $(\mathcal{G}^{\prime},c^{\prime})$.

######  Remark 3.2.

At the level of the Smith diagram, adding new vertices to the interior of some
edges of the graph according to the procedure described above corresponds to
horizontally dissecting the rectangles associated to such edges. More
precisely, let us assume for simplicity that only one point is added to the
the interior of an edge $e$, and let $e_{1}^{\prime}$ and $e_{2}^{\prime}$ be
the new edges in which $e$ is split into. Suppose that
$\smash{e_{1}^{\prime,-}=e^{-}}$ and $\smash{e_{2}^{\prime,+}=e^{+}}$. Let
$\SS^{\prime}$ be the tiling map associated to new weighted graph. Then it is
immediate to check that
$\SS(e)=\SS^{\prime}(e_{1}^{\prime})\cup\SS^{\prime}(e_{2}^{\prime})$. In
particular the rectangles $\SS^{\prime}(e_{1}^{\prime})$ and
$\SS^{\prime}(e_{2}^{\prime})$ have the same width of the rectangle $\SS(e)$,
and the height of $\SS^{\prime}(e_{1}^{\prime})$ is proportional to
$\smash{\mathrm{d}^{\mathbb{G}}(e_{1}^{\prime,-},e_{1}^{\prime,+})}$, while
that of $\SS^{\prime}(e_{2}^{\prime})$ is proportional to
$\smash{\mathrm{d}^{\mathbb{G}}(e_{2}^{\prime,-},e_{2}^{\prime,+})}$.

Let $\mathfrak{h}^{\prime}:\mathcal{V}\mathcal{G}^{\prime}\to[0,1]$ be the
voltage function associated to the quadruplet
$(\mathcal{G}^{\prime},c^{\prime},v_{0},v_{1})$. The following lemma relates
the function $\mathfrak{h}$ with $\mathfrak{h}^{\prime}$.

######  Lemma 3.3.

For every $x\in\mathcal{V}\mathcal{G}$, it holds that
$\mathfrak{h}(x)=\mathfrak{h}^{\prime}(x)$.

###### Proof.

We denote by
$K:=\\#\\{\mathcal{V}\mathcal{G}^{\prime}\setminus\mathcal{V}\mathcal{G}\\}$
the number of new vertices added to the graph $\mathcal{G}^{\prime}$. If
$K=0$, then the result follows immediately. Let us now assume that $K=1$ and
suppose that the edge $e\in\mathcal{E}\mathcal{G}$ is split into two new edges
$e_{1}^{\prime}$ and $e_{2}^{\prime}$ with conductances
$\smash{c^{\prime}_{e_{1}^{\prime}}}$ and
$\smash{c^{\prime}_{e_{2}^{\prime}}}$ as specified in Definition 3.1. Then,
the desired result follows immediately from the series law of electrical
network (cf. [LP16, Section 2.3.I]). The general case follows from a simple
induction argument on $K$. ∎

######  Lemma 3.4.

For $x\in\mathcal{V}\mathcal{G}$, let $X^{\prime,x}$ be the random walk on
$(\mathcal{G}^{\prime},c^{\prime})$ started from $x$. Let $\uptau_{0}:=0$ and,
for every $k\in\mathbb{N}_{0}$, we define inductively
$\uptau_{k+1}:=\inf\\{j>\uptau_{k}\,:\,X^{\prime,x}_{j}\in\mathcal{V}\mathcal{G}\\}$.
Then $\\{X_{\uptau_{k}}^{\prime,x}\\}_{k\in\mathbb{N}}$ has the same
distribution as the random walk on $(\mathcal{G},c)$ started from $x$.

###### Proof.

As in the proof of Lemma 3.3, let
$K:=\\#\\{\mathcal{V}\mathcal{G}^{\prime}\setminus\mathcal{V}\mathcal{G}\\}$.
If $K=0$, then the result is obvious. Let us now assume that $K=1$ and let
$y\in W$ be the vertex added to $\mathcal{G}^{\prime}$. We claim that the
distribution of $X^{\prime,x}_{\uptau_{1}}$ is equal to that of $X^{x}_{1}$.
If all the edges in $\mathcal{E}\mathcal{G}(x)$ do not contain $y$, then the
claim is obvious. Let us assume that there exists an edge in
$\mathcal{E}\mathcal{G}(x)$ which contains $y$. Then, thanks to an easy
computation, one can verify that

$\mathbb{P}_{x}(X^{\prime}_{\uptau_{1}}=v)=c_{xv}/\pi(x),\quad\forall
v\in\mathcal{V}\mathcal{G}(x).$

Therefore, thanks to the strong Markov property of $X^{\prime,x}$, we get that
for all $k\geq 2$, it holds that

$\mathbb{P}_{x}(X^{\prime}_{\uptau_{k}}=v\mid
X^{\prime}_{\uptau_{k-1}}=w)=\begin{cases}c_{vw}/\pi(w),&v\in\mathcal{V}\mathcal{G}(w)\\\
0,&\text{otherwise},\end{cases}\quad\forall k\geq 2,$

which proves the result if $K=1$. The general case follows from a simple
induction argument on $K$. ∎

######  Remark 3.5.

Given a finite set $\widehat{W}\subset\widehat{\mathbb{G}}$, following a
similar procedure to the one described above, one can also construct the
weighted dual graph $(\widehat{\mathcal{G}}^{\prime},\widehat{c}^{\prime})$
associated to $(\widehat{\mathcal{G}},c)$ and $\widehat{W}$. In particular,
results similar to the one stated above hold also for the respective dual
counterparts. For example, if
$\mathfrak{w}^{\prime}:\mathcal{V}\widehat{\mathcal{G}}^{\prime}\to\mathbb{R}/\upeta\mathbb{Z}$
is the width coordinate function associated to the new weighted graph, then
$\mathfrak{w}^{\prime}$ restricted to $\mathcal{V}\widehat{\mathcal{G}}$
coincides with the original width coordinate function. Moreover, adding new
dual vertices to the interior of some dual edges corresponds to vertically
dissecting the associated rectangles in the Smith diagram.

###  3.2 Periodicity

We collect here some properties of the lifted Smith diagram that are due to
its periodicity. We recall that the map
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}$ is defined in Definition 2.15.

######  Lemma 3.6.

Let $\mathbf{x}_{1}$, $\mathbf{x}_{2}\in\mathcal{V}\mathcal{G}^{{\dagger}}$ be
such that $\upsigma_{2\pi}(\mathbf{x}_{1})=\upsigma_{2\pi}(\mathbf{x}_{2})$.
Then, uniformly over all possible realizations of
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{1})$ and of
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{2})$, it holds that

$\left|\frac{\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{2}))-\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{1}))}{\upeta}-\frac{\mathop{Re}\nolimits(\mathbf{x}_{2})-\mathop{Re}\nolimits(\mathbf{x}_{1})}{2\pi}\right|\leq
1.$

###### Proof.

We observe that the proof of this result is an easy consequence of Lemma 2.9.
However, it does not follow immediately from Lemma 2.9 since the points
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{1})$ and
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{2})$ are points chosen
uniformly at random from the horizontal segments
$\SS^{{\dagger}}(\mathbf{x}_{1})$ and $\SS^{{\dagger}}(\mathbf{x}_{2})$,
respectively. We give here the details for completeness.

Fix a realization of $\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{1})$
and of $\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{2})$. Let
$\mathcal{E}\mathcal{G}^{{\dagger},\uparrow}(\mathbf{x}_{1})=[\mathbf{e}^{1}_{n}]$
be the set of lifted harmonically oriented edges with tails equals to
$\mathbf{x}_{1}$ ordered in such a way that
$\widehat{\mathbf{e}}^{1}_{1}\cdots\widehat{\mathbf{e}}^{1}_{n}$ forms a path
in $\widehat{\mathcal{G}}^{{\dagger}}$ oriented from left to right. Define
$\mathcal{E}\mathcal{G}^{{\dagger},\uparrow}(\mathbf{x}_{2})=[\mathbf{e}^{2}_{n}]$
in a similar way. Then, by construction of the map
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}$, it holds that

$\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{1}))\in\bigl{[}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,-}_{1}),\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,+}_{n})\bigr{]},\qquad\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}_{2}))\in\bigl{[}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,-}_{1}),\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,+}_{n})\bigr{]}.$

Moreover, by Lemma 2.9, we have that

$\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,-}_{1})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,-}_{1})=\frac{\upeta}{2\pi}\left(\mathop{Re}\nolimits(\mathbf{x}_{1})-\mathop{Re}\nolimits(\mathbf{x}_{2})\right),\qquad\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,+}_{n})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,+}_{n})=\frac{\upeta}{2\pi}\left(\mathop{Re}\nolimits(\mathbf{x}_{1})-\mathop{Re}\nolimits(\mathbf{x}_{2})\right),$

and also

$\bigl{|}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,+}_{n})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{1,-}_{1})\bigr{|}=\bigl{|}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,+}_{n})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{2,-}_{1})\bigr{|}\leq\upeta,$

where the first equality is again due to Lemma 2.9, and the last inequality is
due to the fact that, by construction of the Smith diagram, every horizontal
line segment has width at most $\upeta$. Therefore, putting together all these
facts, one can readily obtain the desired result. ∎

In order to state and prove the next lemma of this subsection, we need to
introduce some notation. We assume it exists $a\in(0,1)$ such that all the
points in the set $\mathfrak{h}^{-1}(a)$ are vertices in
$\mathcal{V}\mathcal{G}$. We define the set of harmonically oriented edges
$\mathcal{E}\mathcal{G}_{a}$ and the corresponding set of dual edges
$\mathcal{E}\widehat{\mathcal{G}}_{a}$ as follows

$\mathcal{E}\mathcal{G}_{a}:=\bigl{\\{}e\in\mathcal{E}\mathcal{G}\,:\,\mathfrak{h}(e^{-})\leq
a<\mathfrak{h}(e^{+})\bigr{\\}},\qquad\mathcal{E}\widehat{\mathcal{G}}_{a}:=\bigl{\\{}\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}\,:\,\widehat{e}\text{
is the dual of some }e\in\mathcal{E}\mathcal{G}_{a}\bigr{\\}},$

and we define the following sets of vertices

$\mathcal{V}\mathcal{G}_{a}:=\bigl{\\{}x\in\mathcal{V}\mathcal{G}\,:\,x=e^{-}\text{
for some
}e\in\mathcal{E}\mathcal{G}_{a}\bigr{\\}},\qquad\mathcal{V}\widehat{\mathcal{G}}_{a}:=\bigl{\\{}\widehat{x}\in\mathcal{V}\widehat{\mathcal{G}}\,:\,\widehat{x}=\widehat{e}^{-}\text{
for some }\widehat{e}\in\mathcal{E}\widehat{\mathcal{G}}_{a}\bigr{\\}}.$

Furthermore, we denote by $\smash{\mathcal{E}\mathcal{G}_{a}^{{\dagger}}}$,
$\smash{\mathcal{E}\widehat{\mathcal{G}}^{{\dagger}}_{a}}$,
$\smash{\mathcal{V}\mathcal{G}_{a}^{{\dagger}}}$,
$\smash{\mathcal{V}\widehat{\mathcal{G}}_{a}^{{\dagger}}}$ the lifts to the
universal cover of the corresponding objects. Now, we let
$\smash{\widehat{\mathbf{x}}^{a}_{0}}$ be the vertex in
$\smash{\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}_{a}}$ whose real
coordinate is nearest to $0$. We note that the set
$\smash{\mathcal{E}\mathcal{G}_{a}}$ is a cutset and that the associated set
of oriented dual edges $\smash{\mathcal{E}\widehat{\mathcal{G}}_{a}}$ admits
an enumeration $\smash{[\widehat{e}_{n}]}$ such that
$\smash{\widehat{e}_{1}\cdots\widehat{e}_{n}}$ forms a counter-clockwise
oriented noncontractible simple closed loop in the dual graph
$\smash{\widehat{\mathcal{G}}}$. In particular, every edge
$\smash{\widehat{e}_{i}}$ admits a lift $\smash{\widehat{\mathbf{e}}_{i}}$
such that $\smash{\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}}$
forms a simple path oriented from left to right joining
$\smash{\widehat{\mathbf{x}}^{a}_{0}}$ to the shifted vertex
$\smash{\widehat{\mathbf{x}}^{a}_{0}+(0,2\pi)}$ (see Lemma 2.2).

######  Definition 3.7.

Letting $[\widehat{\mathbf{e}}_{n}]$ be as specified above, we define the set
of lifted dual vertices

$\widehat{\mathcal{P}}_{a}^{\dagger}:=\bigl{\\{}\widehat{\mathbf{x}}\in\mathcal{V}\widehat{\mathcal{G}}^{{\dagger}}_{a}\,:\,\widehat{\mathbf{x}}=\widehat{\mathbf{e}}_{i}^{-}\text{
for some }i\in[n]\bigr{\\}},$ (3.1)

and we also define the set of lifted vertices

$\mathcal{P}_{a}^{\dagger}:=\bigl{\\{}\mathbf{x}\in(\mathfrak{h}^{{\dagger}})^{-1}(a)\,:\,\mathbf{x}=\mathbf{e}_{i}^{-}\text{
for some }i\in[n]\bigr{\\}},$ (3.2)

where $[\mathbf{e}_{n}]$ is the set of lifted edges associated to
$[\widehat{\mathbf{e}}_{n}]$.

######  Remark 3.8.

Consider the infinite strip
$\smash{\mathrm{S}^{a}_{2\pi}:=\bigl{[}\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{0}^{a}),\mathop{Re}\nolimits(\widehat{\mathbf{x}}_{0}^{a})+2\pi\bigr{)}\times\mathbb{R}}$.
We remark that in general the set
$\smash{\widehat{\mathcal{P}}_{a}^{\dagger}}$ is not fully contained in
$\smash{\mathrm{S}^{a}_{2\pi}}$. We also observe that, thanks to the fact that
$\mathcal{E}\mathcal{G}_{a}$ is a cutset, it holds that

$\smash{\upsigma_{2\pi}(\mathcal{P}_{a}^{\dagger})=\mathfrak{h}^{-1}(a)}\quad\text{
and
}\quad\smash{\\#\upsigma_{2\pi}(\mathcal{P}_{a}^{\dagger})=\\#\mathfrak{h}^{-1}(a)}.$

We are now ready to state and prove the next lemma of this subsection. We
refer to Figure 5 for a diagram illustrating the various objects involved in
the proof of Lemma 3.9.

Figure 5: A diagram illustrating the proof of Lemma 3.9. Left: A portion of
the lifted graph $\smash{\mathcal{G}^{{\dagger}}}$ embedded in
$\mathbb{R}^{2}$. The blue/green vertices are the vertices in the set
$\smash{\mathcal{P}_{a}^{\dagger}}$. The green vertex $\mathbf{x}$ is a fixed
vertex in $\smash{\mathcal{P}_{a}^{\dagger}}$. The red dual vertices are the
vertices in $\smash{\widehat{\mathcal{P}}_{a}^{\dagger}}$. The blue edges
belong to the set $\smash{\mathcal{E}\mathcal{G}_{a}^{{\dagger}}}$, and the
red dual edges belong to the set
$\mathcal{E}\widehat{\mathcal{G}}^{{\dagger}}_{a}$. The leftmost red dual
vertex corresponds to the vertex $\widehat{\mathbf{x}}^{a}_{0}$. The two red
dual edges with an arrow are the edges
$\smash{\widehat{\mathbf{e}}_{1}^{\mathbf{x}}}$ and
$\smash{\widehat{\mathbf{e}}_{2}^{\mathbf{x}}}$. Right: A portion of the
tiling of $\mathbb{R}\times[0,1]$ constructed via the lifted tiling map
$\SS^{{\dagger}}$. The blue horizontal line corresponds to
$\smash{\cup_{\mathbf{x}\in\mathcal{P}_{a}^{\dagger}}\SS^{{\dagger}}(\mathbf{x})}$.
The vertical red segments correspond to
$\smash{\SS^{{\dagger}}(\widehat{\mathbf{e}}_{1}^{\mathbf{x},-})}$ and
$\smash{\SS^{{\dagger}}(\widehat{\mathbf{e}}_{2}^{\mathbf{x},+})}$. The green
point corresponds to a possible realization of
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x})$.

######  Lemma 3.9.

Fix $a\in(0,1)$ and assume that all the points in $\mathfrak{h}^{-1}(a)$ are
vertices in $\mathcal{V}\mathcal{G}$. Let
$\widehat{\mathbf{x}}^{a}_{0}\in\mathcal{V}\widehat{\mathcal{G}}_{a}^{{\dagger}}$
be as specified above, then it holds that

$0\leq\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}^{a}_{0})\leq\upeta,\quad\forall\widehat{\mathbf{x}}\in\widehat{\mathcal{P}}_{a}^{\dagger},$

and also, uniformly over all possible realizations of
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\cdot)$, it holds that

$0\leq\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}))-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}^{a}_{0})\leq\upeta,\quad\forall\mathbf{x}\in\mathcal{P}_{a}^{\dagger}.$

###### Proof.

We start by proving the first part of the lemma. We consider the set of dual
edges $[\widehat{\mathbf{e}}_{n}]$ such that
$\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ forms a simple path
oriented from left to right joining $\widehat{\mathbf{x}}^{a}_{0}$ to the
shifted vertex $\widehat{\mathbf{x}}^{a}_{0}+(0,2\pi)$, as specified before
the lemma statement. Thanks to Lemma 2.9, we know that

$\sum_{i=1}^{n}\nabla\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}_{i})=\upeta,$

and each summand in the sum is non-negative. Now, let
$\widehat{\mathbf{x}}\in\widehat{\mathcal{P}}_{a}^{\dagger}$, then there
exists a subpath
$\widehat{\mathbf{e}}^{\widehat{\mathbf{x}}}_{1}\cdots\widehat{\mathbf{e}}^{\widehat{\mathbf{x}}}_{k}$
of $\widehat{\mathbf{e}}_{1}\cdots\widehat{\mathbf{e}}_{n}$ which connects
$\widehat{\mathbf{x}}^{a}_{0}$ to $\widehat{\mathbf{x}}$. Therefore, we have
that

$\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}})-\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{x}}^{a}_{0})=\sum_{i=1}^{k}\nabla\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{\widehat{\mathbf{x}}}_{i}),$

and the conclusion then follows thanks to the fact that

$0\leq\sum_{i=1}^{k}\nabla\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{\widehat{\mathbf{x}}}_{i})\leq\sum_{i=1}^{n}\nabla\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}_{i})=\upeta.$

We now prove the second part of the lemma. To this end, we fix
$\mathbf{x}\in\mathcal{P}_{a}^{\dagger}$ and a possible realization of
$\smash{\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x})}$. We let
$\smash{\mathcal{E}\mathcal{G}^{{\dagger},\uparrow}(\mathbf{x})=[\mathbf{e}^{\mathbf{x}}_{k}]}$
be the set of lifted harmonically oriented edges with tails equal to
$\mathbf{x}$ ordered in such a way that the corresponding lifted dual edges
$\smash{\widehat{\mathbf{e}}^{\mathbf{x}}_{1}\cdots\widehat{\mathbf{e}}^{\mathbf{x}}_{k}}$
forms a path in $\smash{\widehat{\mathcal{G}}^{{\dagger}}}$ oriented from left
to right (see Figure 5). By construction of the map
$\smash{\dot{\mathcal{S}}^{{\dagger},\rm{rand}}}$, it holds that

$\mathop{Re}\nolimits(\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{x}))\in\bigl{[}\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{\mathbf{x},-}_{1}),\mathfrak{w}^{{\dagger}}(\widehat{\mathbf{e}}^{\mathbf{x},+}_{k})\bigr{]}.$

Finally, since both $\widehat{\mathbf{e}}^{\mathbf{x},-}_{1}$ and
$\widehat{\mathbf{e}}^{\mathbf{x},+}_{k}$ belong to
$\widehat{\mathcal{P}}_{a}^{\dagger}$, the conclusion follows thanks to the
first part of the lemma. ∎

###  3.3 Hitting distribution of a horizontal line

Roughly speaking, the main goal of this subsection is to characterize the
hitting distribution of a horizontal line on the Smith diagram for the Smith
embedded random walk. Before precisely stating the result, we need to
introduce some notation.

######  Definition 3.10.

We define $\operatorname{length}:\mathcal{V}\mathcal{G}\to[0,\upeta)$ as the
function that assigns to each vertex $x\in\mathcal{V}\mathcal{G}$ the length
of the horizontal segment $\SS(x)$ associated to $x$ by the Smith embedding.
More precisely, we define

$\operatorname{length}(x):=\sum_{\mathbf{e}\in\mathcal{E}\mathcal{G}^{\downarrow}(x)}\nabla\mathfrak{h}(e),\quad\forall
x\in\mathcal{V}\mathcal{G},$

where $\mathcal{E}\mathcal{G}^{\downarrow}(x)$ denotes the set of harmonically
oriented edges in $\mathcal{E}\mathcal{G}$ with heads equal to $x$.

We recall that the difference of the width coordinate function between the
endpoints of a dual edge is equal to the gradient of the height coordinate of
the corresponding primal edge. Moreover, thanks to the fact that the height
coordinate function is harmonic, it is plain to see that

$\operatorname{length}(x)=\sum_{\mathbf{e}\in\mathcal{E}\mathcal{G}^{\uparrow}(x)}\nabla\mathfrak{h}(e),\quad\forall
x\in\mathcal{V}\mathcal{G},$

where $\mathcal{E}\mathcal{G}^{\uparrow}(x)$ denotes the set of harmonically
oriented edges in $\mathcal{E}\mathcal{G}$ with tails equal to $x$. We can
also naturally extend the definition of the length function to the metric
graph $\mathbb{G}$. More precisely, given a point
$x\in\mathbb{G}\setminus\mathcal{V}\mathcal{G}$, if $x$ lies in the interior
of the edge $e\in\mathcal{E}\mathcal{G}$, then we set
$\operatorname{length}(x):=\nabla\mathfrak{h}(e)$.

######  Definition 3.11.

Given a value $a\in(0,1)$, we define on the set
$\mathfrak{h}^{-1}(a)\subset\mathbb{G}$ the measure $\mu_{a}$ by letting

$\mu_{a}(x):=\frac{\operatorname{length}(x)}{\upeta},\quad\forall
x\in\mathfrak{h}^{-1}(a).$

Since, by construction of the Smith diagram, it holds that
$\smash{\sum_{x\in\mathfrak{h}^{-1}(a)}\operatorname{length}(x)=\upeta}$, the
measure $\mu_{a}$ is a probability measure on the set
$\smash{\mathfrak{h}^{-1}(a)}$.

######  Remark 3.12.

We emphasize that, thanks to Remark 3.8, one can also view $\mu_{a}$ as a
probability measure on the set $\mathcal{P}_{a}^{\dagger}$, where we recall
that $\mathcal{P}_{a}^{\dagger}$ is the set defined in (3.2).

From now on, we assume throughout the whole subsection that all the points in
the set $\cup_{x\in\mathcal{V}\mathcal{G}}\mathfrak{h}^{-1}(\mathfrak{h}(x))$
are vertices in $\mathcal{V}\mathcal{G}$. At the level of the Smith diagram,
this means that for all $x\in\mathcal{V}\mathcal{G}$, it holds that the set
$\smash{\cup_{y\in\mathfrak{h}^{-1}(\mathfrak{h}(x))}\SS(y)}$ is equal to the
noncontractible closed loop
$\smash{\mathbb{R}/\upeta\mathbb{Z}\times\\{\mathfrak{h}(x)\\}\subset\mathcal{C}_{\upeta}}$.
We observe that, for any given finite weighted planar graph, we can always
canonically construct from it another finite weighted planar graph that
satisfies this assumption. Indeed, suppose that it exists a vertex
$x\in\mathcal{V}\mathcal{G}$ such that
$\smash{\mathfrak{h}^{-1}(\mathfrak{h}(x))\not\subset\mathcal{V}\mathcal{G}}$,
then, using the procedure described in Subsection 3.1, we can declare all the
points in the finite set
$\smash{\mathfrak{h}^{-1}(\mathfrak{h}(x))\setminus\mathcal{V}\mathcal{G}}$ to
be vertices of the graph.

We fix a value $a\in(0,1)$ such that all the points in the set
$\mathfrak{h}^{-1}(a)$ are vertices in $\mathcal{V}\mathcal{G}$, and we let
$X^{\mu_{a}}$ be the random walk on $(\mathcal{G},c)$ started from a point in
$\mathfrak{h}^{-1}(a)$ sampled according to the probability measure $\mu_{a}$.

######  Definition 3.13.

Let $X^{\mu_{a}}$ be as specified above. For $N\in\mathbb{N}$, we say that a
finite sequence of height coordinates $[a_{N}]_{0}\subset(0,1)$ is
_admissible_ for the random walk $X^{\mu_{a}}$ if $a_{0}=a$ and

$\mathbb{P}_{\mu_{a}}\bigl{(}\mathfrak{h}(X_{n+1})=a_{n+1}\mid\mathfrak{h}(X_{n})=a_{n}\bigr{)}>0,\quad\forall
n\in[N-1]_{0}.$

We can now state the main result of this subsection.

######  Lemma 3.14.

Let $X^{\mu_{a}}$ be as specified above. For $N\in\mathbb{N}$, let
$[a_{N}]_{0}\subset(0,1)$ be an admissible sequence of height coordinates for
the random walk $X^{\mu_{a}}$. Then, for all $i\in[N]_{0}$, it holds that

$\mathbb{P}_{\mu_{a}}\bigl{(}X_{i}=x_{i}\mid\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=1}^{N}\bigr{)}=\mu_{a_{i}}(x),\quad\forall
x_{i}\in\mathfrak{h}^{-1}(a_{i}).$

Intuitively, the proof of the above lemma goes as follows. If $i=1$ and we
only had the conditioning on the event $\mathfrak{h}(X_{1})=a_{1}$, then the
result would follow from a simple computation. However, since, in general,
$i\neq 1$, and since we are also conditioning on the events
$\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=1}^{i-1}$ and
$\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=i+1}^{N}$, which represent the height
coordinates of the random walk for past and future times respectively, the
proof of the result is not immediate. However, the proof follows by a simple
induction argument in which we show that we can “forget” about the
conditioning on past and future times. Roughly speaking, the reason why we can
forget about past times is due to the fact that the height component of the
random walk is itself a Markov process, while the reason why we can forget
about future times is due to the harmonicity of the height coordinate
function. We now proceed with the proof of the lemma.

###### Proof of Lemma 3.14.

The proof involves three main steps.

Step 1: We start by proving that
$\smash{\mathbb{P}_{\mu_{a}}\bigl{(}\mathfrak{h}(X_{1})=a_{1}\mid
X_{0}=x_{0})}$ depends only on $\mathfrak{h}(X_{0})=a_{0}$, for all
$\smash{x_{0}\in\mathfrak{h}^{-1}(a)}$. Since we are assuming that
$\smash{\mathcal{V}\mathcal{G}=\cup_{x\in\mathcal{V}\mathcal{G}}\mathfrak{h}^{-1}(\mathfrak{h}(x))}$,
at its first step, the random walk can only reach two heights: height $a_{1}$
or height $\widetilde{a}_{1}$ say. If $a_{1}>a$, since $\mathfrak{h}$ is
harmonic at $x_{0}$, it holds that

$\mathfrak{h}(x_{0})=\frac{\sum_{e\in\mathcal{E}\mathcal{G}^{\downarrow}(x_{0})}c_{e}\mathfrak{h}(e^{-})+\sum_{e\in\mathcal{E}\mathcal{G}^{\uparrow}(x_{0})}c_{e}\mathfrak{h}(e^{+})}{\pi(x_{0})}=\widetilde{a}_{1}\frac{\sum_{e\in\mathcal{E}\mathcal{G}^{\downarrow}(x_{0})}c_{e}}{\pi(x_{0})}+a_{1}\frac{\sum_{e\in\mathcal{E}\mathcal{G}^{\uparrow}(x_{0})}c_{e}}{\pi(x_{0})},$
(3.3)

where, as usual, $\mathcal{E}\mathcal{G}^{\downarrow}(x_{0})$ (resp.
$\mathcal{E}\mathcal{G}^{\uparrow}(x_{0})$) denotes the set of harmonically
oriented edges with heads (resp. tails) equal to $x_{0}$. Now, we observe that

$\mathbb{P}_{\mu_{a}}\bigl{(}\mathfrak{h}(X_{1})=a_{1}\mid
X_{0}=x_{0}\bigr{)}=\frac{\sum_{e\in\mathcal{E}\mathcal{G}^{\uparrow}(x_{0})}c_{e}}{\pi(x_{0})},\qquad\mathbb{P}_{\mu_{a}}\bigl{(}\mathfrak{h}(X_{1})=\widetilde{a}_{1}\mid
X_{0}=x_{0}\bigr{)}=\frac{\sum_{e\in\mathcal{E}\mathcal{G}^{\downarrow}(x_{0})}c_{e}}{\pi(x_{0})}.$

In particular, plugging these expressions into (3.3) and rearranging, we
obtain that

$\mathbb{P}_{\mu_{a}}\bigl{(}\mathfrak{h}(X_{1})=a_{1}\mid
X_{0}=x_{0}\bigr{)}=\frac{|\widetilde{a}_{1}-a_{0}|}{|\widetilde{a}_{1}-a_{1}|},\quad\forall
x_{0}\in\mathfrak{h}^{-1}(a),$

from which the desired result follows since the right-hand side of the above
expression only depends on $\mathfrak{h}(X_{0})=a_{0}$. A similar conclusion
also holds if $a_{1}<a$. Now, proceeding by induction, one can prove that for
any $i\in[N]_{0}$ and for all
$x_{0}\in\mathfrak{h}^{-1}(a),\ldots,x_{i}\in\mathfrak{h}^{-1}(a_{i})$, it
holds that
$\smash{\mathbb{P}_{\mu_{a}}(\\{\mathfrak{h}(X_{i+n})=a_{i+n}\\}_{n=1}^{N-i}\mid\\{X_{j}=x_{j}\\}_{j=0}^{i})}$
depends only on $\mathfrak{h}(X_{i})=a_{i}$.

Step 2: Thanks to previous step and Bayes’ rule, we have that
$\smash{\mathbb{P}_{\mu_{a}}\bigl{(}X_{0}=x_{0}\mid\mathfrak{h}(X_{1})=a_{1}\bigr{)}=\mu_{a}(x_{0})}$,
for all $x_{0}\in\mathfrak{h}^{-1}(a)$. In particular, using this fact, we
will now prove that

$\mathbb{P}_{\mu_{a}}\bigl{(}X_{1}=x_{1}\mid\mathfrak{h}(X_{1})=a_{1}\bigr{)}=\mu_{a_{1}}(x_{1}),\quad\forall
x_{1}\in\mathfrak{h}^{-1}(a_{1}).$

To this end, fix $x_{1}\in\mathfrak{h}^{-1}(a_{1})$ and suppose that
$a_{1}>a$. Then we can proceed as follows

$\displaystyle\mathbb{P}_{\mu_{a}}\bigl{(}X_{1}=x_{1}\mid\mathfrak{h}(X_{1})=a_{1}\bigr{)}$
$\displaystyle\qquad=\sum_{x_{0}\in\mathfrak{h}^{-1}(a)\cap\mathcal{V}\mathcal{G}(x_{1})}\mathbb{P}_{x_{0}}\bigl{(}X_{1}=x_{1}\mid\mathfrak{h}(X_{1})=a_{1}\bigr{)}\mathbb{P}_{\mu_{a}}\bigl{(}X_{0}=x_{0}\mid\mathfrak{h}(X_{1})=a_{1}\bigr{)}$
$\displaystyle\qquad=\sum_{x_{0}\in\mathfrak{h}^{-1}(a)\cap\mathcal{V}\mathcal{G}(x_{1})}\frac{c_{x_{0}x_{1}}}{\sum_{v\in\mathfrak{h}^{-1}(a_{1})\cap\mathcal{V}\mathcal{G}(x_{0})}c_{x_{0}v}}\mu_{a}(x_{0})$
$\displaystyle\qquad=\sum_{x_{0}\in\mathfrak{h}^{-1}(a)\cap\mathcal{V}\mathcal{G}(x_{1})}\frac{c_{x_{0}x_{1}}}{\sum_{v\in\mathfrak{h}^{-1}(a_{1})\cap\mathcal{V}\mathcal{G}(x_{0})}c_{x_{0}v}}\frac{|a_{1}-a|\sum_{v\in\mathfrak{h}^{-1}(a_{1})\cap\mathcal{V}\mathcal{G}(x_{0})}c_{x_{0}v}}{\upeta}$
$\displaystyle\qquad=\frac{1}{\upeta}\sum_{e\in\mathcal{E}\mathcal{G}^{\downarrow}(x)}\nabla\mathfrak{h}(e)$
$\displaystyle\qquad=\mu_{a_{1}}(x),$

where $\mathcal{E}\mathcal{G}^{\downarrow}(x)$ denotes the set of harmonically
oriented edges with heads equal to $x$. In order to pass from the second line
to the third line of the above display, we used the definition of the
probability measure $\mu_{a}$, and the fact that, for all
$e\in\mathcal{E}\mathcal{G}^{\downarrow}(x)$, it holds that
$\mathfrak{h}(e^{-})=a$ and $\mathfrak{h}(e^{+})=a_{1}$. One can proceed
similarly if $a_{1}<a$. Now, proceeding by induction, one can easily prove
that, for all $i\in[N]$, it holds that

$\mathbb{P}_{\mu_{a}}\bigl{(}X_{i}=x_{i}\mid\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=1}^{i}\bigr{)}=\mu_{a_{i}}(x_{i}),\quad\forall
x_{i}\in\mathfrak{h}^{-1}(a_{i}).$

Step 3: For $i\in[N]_{0}$, we observe that a consequence of Step 1 is that the
sequence of random variables $\\{\mathfrak{h}(X_{i+n})\\}_{n=1}^{N-i}$ is
conditionally independent from $\\{X_{j}\\}_{j=0}^{i}$ given
$\mathfrak{h}(X_{i})=a_{i}$. Therefore, the conditional law of
$\\{X_{j}\\}_{j=0}^{i}$ given $\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=1}^{i}$ is
the same as the conditional law given
$\\{\mathfrak{h}(X_{n})=a_{n}\\}_{n=1}^{N}$. Hence, this implies that

$\mathbb{P}_{\mu_{a}}\bigl{(}X_{i}=x_{i}\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{)}=\mathbb{P}_{\mu_{a}}\bigl{(}X_{i}=x_{i}\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{i}\bigr{)},\quad\forall
x_{i}\in\mathfrak{h}^{-1}(a_{i}),$

and so the conclusion follows from Step 2. ∎

###  3.4 Expected horizontal winding

Roughly speaking, the main goal of this subsection is to establish that the
average winding that the Smith-embedded random walk does before hitting a
given horizontal line on the Smith diagram is zero. Before precisely stating
the result, we need to introduce some notation.

######  Definition 3.15.

Given the random walk $\mathbf{X}^{\mathbf{x}}$ on the lifted weighted graph
$(\mathcal{G}^{{\dagger}},c^{{\dagger}})$ started from
$\mathbf{x}\in\mathcal{V}\mathcal{G}^{{\dagger}}$, we let

$\mathbb{N}_{0}\ni n\mapsto\dot{\mathbf{X}}^{\mathbf{x}}_{n}$

be the discrete time process taking values in $\mathbb{R}\times[0,1]$ such
that, for each $n\in\mathbb{N}_{0}$, the conditional law of
$\dot{\mathbf{X}}^{\mathbf{x}}_{n}$, given $\mathbf{X}^{\mathbf{x}}_{n}$, is
equal to the law of
$\dot{\mathcal{S}}^{{\dagger},\rm{rand}}(\mathbf{X}_{n}^{\mathbf{x}})$, where
we recall that $\dot{\mathcal{S}}^{{\dagger},\rm{rand}}$ is defined in
Definition 2.15. We call the process $\dot{\mathbf{X}}^{\mathbf{x}}$ the
_lifted Smith-embedded random walk_ associated to $\mathbf{X}^{\mathbf{x}}$.

It follows from the definition of $\dot{\mathbf{X}}^{\mathbf{x}}$ that, at
each time step $n\in\mathbb{N}_{0}$, the location of the point
$\dot{\mathbf{X}}^{\mathbf{x}}_{n}$ is sampled uniformly at random and
independently of everything else from the horizontal line segment
$\smash{\SS^{{\dagger}}(\mathbf{X}^{\mathbf{x}}_{n})}$. With a slight abuse of
notation, we also denote by $\smash{\dot{\mathbf{X}}^{\mathbf{x}}}$ the
continuous time version $\smash{\\{\dot{\mathbf{X}}^{\mathbf{x}}_{t}\\}_{t\geq
0}}$, where the continuous path is generated by piecewise linear interpolation
at constant speed.

We assume also in this subsection that all the points in the set
$\cup_{x\in\mathcal{V}\mathcal{G}}\mathfrak{h}^{-1}(\mathfrak{h}(x))$ are
vertices in $\mathcal{V}\mathcal{G}$. Furthermore, we fix a value $a\in(0,1)$
such that all the points in the set $\mathfrak{h}^{-1}(a)$ are vertices in
$\mathcal{V}\mathcal{G}$, and we let $X^{\mu_{a}}$ be the random walk on
$(\mathcal{G},c)$ started from a point in $\mathfrak{h}^{-1}(a)$ sampled
according to the probability measure $\mu_{a}$ defined in Definition 3.11. We
also adopt the convention to denote by

$\mathbb{N}_{0}\ni n\mapsto\mathbf{X}^{\mu_{a}}_{n}$

the lift of the random walk $X^{\mu_{a}}$ to the lifted weighted graph
$(\mathcal{G}^{{\dagger}},c^{{\dagger}})$ started from a point in
$\mathcal{P}_{a}^{\dagger}$ sampled according to the the probability measure
$\mu_{a}$ (see Remark 3.12). Moreover, we denote by
$\dot{\mathbf{X}}^{\mu_{a}}$ the lifted Smith-embedded random walk associated
to $\mathbf{X}^{\mu_{a}}$ as specified above.

In complete analogy with the definition of winding of a path in the infinite
cylinder $\mathcal{C}_{2\pi}$, we have the following definition of winding on
the cylinder $\mathcal{C}_{\upeta}$.

######  Definition 3.16.

Let $0\leq t_{1}<t_{2}$, consider a path
$P:[t_{1},t_{2}]\to\mathcal{C}_{\upeta}$, and let
$\mathbf{P}:[t_{1},t_{2}]\to\mathbb{R}\times[0,1]$ be its lift to the covering
space $\mathbb{R}\times[0,1]$. Then, we define the winding of $P$ as follows

$\operatorname{wind}_{\upeta}(P)=\frac{\mathop{Re}\nolimits(\mathbf{P}(t_{2}))-\mathop{Re}\nolimits(\mathbf{P}(t_{1}))}{\upeta}.$
(3.4)

We are now ready to state the main result of this subsection.

######  Lemma 3.17.

Let $X^{\mu_{a}}$ and $\dot{\mathbf{X}}^{\mu_{a}}$ be as specified above. For
$N\in\mathbb{N}$, let $[a_{N}]_{0}\subset(0,1)$ be an admissible sequence of
height coordinates for the random walk $X^{\mu_{a}}$ as specified in
Definition 3.13. Then it holds that

$\mathbb{E}_{\mu_{a}}\bigl{[}\operatorname{wind}_{\upeta}\bigl{(}\dot{\mathbf{X}}|_{[0,N]}\bigr{)}\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}=0.$

Heuristically speaking, the proof of the above lemma goes as follows. First,
we observe that, by definition of winding, it holds that

$\operatorname{wind}_{\upeta}(\dot{\mathbf{X}}|_{[0,N]})=\sum_{j=0}^{N-1}\operatorname{wind}_{\upeta}(\dot{\mathbf{X}}|_{[j,j+1]}).$

In particular, this implies that we can reduce the problem to studying the
expected winding at each time step of the random walk. Suppose for a moment
that $N=1$ in the lemma statement. Then we only need to prove that the random
walk started uniformly at random from height $a$ and conditioned to hit height
$a_{1}$ at its first time step has zero expected winding. The reason why this
is true is due to the fact that the re-randomization procedure that takes
place inside each small segment ensures that the conditional hitting
distribution of the segment at height $a_{1}$ is also uniformly distributed.
To extend the argument to the general case $N\in\mathbb{N}$, we just need to
use Lemma 3.14 in a suitable way.

###### Proof of Lemma 3.17.

We divide the proof in two steps.

Step 1: The main goal of this first step is to find an equivalent condition
for the relation stated in the lemma to be true. To this end, we start by
observing that, thanks to the Definition 3.16 of winding, it is holds that

$\mathbb{E}_{\mu_{a}}\bigl{[}\operatorname{wind}_{\upeta}\bigl{(}\dot{\mathbf{X}}|_{[0,N]}\bigr{)}\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}=\frac{1}{\upeta}\sum_{j=0}^{N-1}\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})-\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}.$
(3.5)

Now, we claim that

$\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}=\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=j}^{j+1}\bigr{]},\quad\forall
j\in[N-1]_{0}.$ (3.6)

To show that this relation holds true, we first notice that the left-hand side
of (3.6) can be written as follows

$\displaystyle\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}$
$\displaystyle\qquad=\sum_{x\in\mathfrak{h}^{-1}(a_{j+1})}\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid
X_{j+1}=x\bigr{]}\mathbb{P}_{\mu_{a}}\bigl{(}X_{j+1}=x\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{)}$
$\displaystyle\qquad=\sum_{x\in\mathfrak{h}^{-1}(a_{j+1})}\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid
X_{j+1}=x\bigr{]}\mu_{a_{j+1}}(x),$

where in the last equality we used Lemma 3.14. Similarly, one can also obtain
the same expression for the right-hand side of (3.6). To do so, one can simply
condition on all possible height trajectories that join height $a$ to height
$a_{j}$ in $j$ steps, and then proceed in the same exact way as above. In a
similar way, one can also prove that

$\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=1}^{N}\bigr{]}=\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j})\mid\mathfrak{h}(X_{j})=a_{j}\bigr{]},\quad\forall
j\in[N-1]_{0}.$ (3.7)

Therefore, combining (3.6) and (3.7) and going back to (3.5), we showed that
the proof of the lemma follows if we prove that the following equality holds

$\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j+1})\mid\bigl{\\{}\mathfrak{h}(X_{n})=a_{n}\bigr{\\}}_{n=j}^{j+1}\bigr{]}=\mathbb{E}_{\mu_{a}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}_{j})\mid\mathfrak{h}(X_{j})=a_{j}\bigr{]},\quad\forall
j\in[N-1]_{0}.$ (3.8)

Now, for every $j\in[N-1]_{0}$, we let $X^{j}$ be the random walk on the
weighted graph $(\mathcal{G},c)$ started from a point in
$\mathfrak{h}^{-1}(a_{j})$ sampled according to the probability measure
$\mu_{a_{j}}$. As one can easily verify, a consequence of Lemma 3.14 is that
the conditional law of $X^{\mu_{a}}$, given
$\mathfrak{h}(X_{j}^{\mu_{a}})=a_{j}$, is equal to the law of $X^{j}$. In
particular, this implies that (3.8) is equivalent to the fact that

$\mathbb{E}_{\mu_{a_{j}}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}^{j}_{1})\mid\mathfrak{h}(X^{j}_{1})=a_{j+1}\bigr{]}-\mathbb{E}_{\mu_{a_{j}}}\bigl{[}\mathop{Re}\nolimits(\dot{\mathbf{X}}^{j}_{0})\bigr{]}=0,\quad\forall
j\in[N-1]_{0},$ (3.9)

where $\smash{\mathbf{X}^{j}}$ denotes the lift of $\smash{X^{j}}$ started
from a point in $\smash{\mathcal{P}^{{\dagger}}_{a_{j}}}$, and
$\smash{\dot{\mathbf{X}}^{j}}$ is the lifted Smith-embedded random walk
associated to $\mathbf{X}^{j}$. Let us also emphasize that here we are relying
on the fact that the definition of winding does not depend on the particular
choice of the lift.

Step 2: The main goal of this second step is to prove that the equality in
(3.9) holds. To this end, we observe that, since every horizontal segment in
the Smith diagram has length at most $\upeta$,
$\smash{|\mathop{Re}\nolimits(\dot{\mathbf{X}}^{j}_{1})-\mathop{Re}\nolimits(\dot{\mathbf{X}}^{j}_{0})|\leq\upeta}$.
In particular, this implies that (3.9) can be rewritten as follows

$\mathbb{E}_{\mu_{a_{j}}}\bigl{[}\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1}))\mid\mathfrak{h}(X^{j}_{1})=a_{j+1}\bigr{]}-\mathbb{E}_{\mu_{a_{j}}}\bigl{[}\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{0}))\bigr{]}=0,\quad\forall
j\in[N-1]_{0},$ (3.10)

where we recall that $\upsigma_{\upeta}$ is defined in (2.15) and denotes the
covering map of the cylinder $\mathcal{C}_{\upeta}$. By construction, the
random variable
$\smash{\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{0}))}$ is
uniformly distributed on the interval $[0,\upeta)$, and so the result follows
if we prove that also the conditional law of
$\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1}))$, given
$\mathfrak{h}(X^{j}_{1})=a_{j+1}$, is uniformly distributed on the interval
$[0,\upeta)$. To this end, let $\mathrm{I}\subset[0,\upeta)$ be a fixed
measurable set, then we have that

$\displaystyle\mathbb{P}_{\mu_{a_{j}}}\bigl{(}\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1}))\in\mathrm{I}\mid\mathfrak{h}(X^{j}_{1})=a_{j+1}\bigr{)}$
$\displaystyle\qquad=\sum_{x\in\mathfrak{h}^{-1}(a_{j+1})}\mathbb{P}_{\mu_{a_{j}}}\bigl{(}\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1}))\in\mathrm{I}\mid
X^{j}_{1}=x\bigr{)}\mathbb{P}_{\mu_{a_{j}}}\bigl{(}X^{j}_{1}=x\mid\mathfrak{h}(X^{j}_{1})=a_{j+1}\bigr{)}$
$\displaystyle\qquad=\sum_{x\in\mathfrak{h}^{-1}(a_{j+1})}\mathbb{P}_{\mu_{a_{j}}}\bigl{(}\mathop{Re}\nolimits(\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1}))\in\mathrm{I}\mid
X^{j}_{1}=x\bigr{)}\mu_{a_{j+1}}(x)$
$\displaystyle\qquad=\sum_{x\in\mathfrak{h}^{-1}(a_{j+1})}\frac{\operatorname{length}\bigl{(}\mathrm{I}\cap\mathop{Re}\nolimits(\SS(x))\bigr{)}|}{\operatorname{length}(x)}\frac{\operatorname{length}(x)}{\upeta}$
$\displaystyle\qquad=\frac{\operatorname{length}\bigl{(}\mathrm{I}\bigr{)}}{\upeta},$

where in the second equality we used Lemma 3.14, while in the third equality
we used the fact that the conditional law of
$\upsigma_{\upeta}(\dot{\mathbf{X}}^{j}_{1})$, given $X_{1}^{j}=x$, is
uniformly distributed on the interval $\SS(x)$ and the definition of the
probability measure $\mu_{a_{j+1}}$. Since this prove (3.10), which is
equivalent to (3.9), the proof of the lemma is concluded. ∎

##  4 Proof of the main result

The main goal of this section is to prove Theorem 1.3. To this end, we fix a
sequence $\\{(\mathcal{G}^{n},c^{n},v_{0}^{n},v_{1}^{n})\\}_{n\in\mathbb{N}}$
of doubly marked finite weighted planar maps and we let
$\\{(\widehat{\mathcal{G}}^{n},\widehat{c}^{n})\\}_{n\in\mathbb{N}}$ be the
sequence of associated weighted dual planar graphs. We assume throughout this
section that such sequences satisfy hypotheses (H1), (H3), (H3).

Before moving to the proof of the main theorem, we prove a couple of simple
results which are immediate consequences of our assumptions and which will be
useful later on. In particular, the next lemma is basically an analogue of
assumptions (H2) and (H3) in the setting of the universal cover.

######  Lemma 4.1.

For each $n\in\mathbb{N}$, view the embedded random walk on the lifted
weighted graph $\smash{(\mathcal{G}^{{\dagger},n},c^{{\dagger},n})}$, stopped
when it traverses an unbounded edge, as a continuous curve in $\mathbb{R}^{2}$
obtained by piecewise linear interpolation at constant speed. For each $R>0$
and for any $z\in\mathbb{R}\times[-R,R]$, the law of the random walk on
$\smash{(\mathcal{G}^{{\dagger},n},c^{{\dagger},n})}$ started from the vertex
$\smash{\mathbf{x}_{z}^{n}\in\mathcal{V}\mathcal{G}^{{\dagger},n}}$ nearest to
$z$ weakly converges as $n\to\infty$ to the law of the Brownian motion in
$\mathbb{R}^{2}$ started from $z$ with respect to the local topology on curves
viewed modulo time parameterization specified in Subsection 2.1.2, uniformly
over all $z\in\mathbb{R}\times[-R,R]$. Furthermore, the same result holds for
the random walk on the sequence of lifted weighted dual graphs
$\smash{\\{(\widehat{\mathcal{G}}^{{\dagger},n},\widehat{c}^{{\dagger},n})\\}_{n\in
N}}$.

###### Proof.

Obviously, we can just prove the result for the random walk on the sequence of
primal graphs as the result for the random walk on the sequence of dual graphs
can be proved in the same exact way. Fix $R>0$ and
$z\in\mathbb{R}\times[-R,R]$, and let $\mathbf{X}^{n,z}$ be the continuous
time random walk on $(\mathcal{G}^{{\dagger},n},c^{{\dagger},n})$ started from
$\mathbf{x}_{z}^{n}\in\mathcal{V}\mathcal{G}^{{\dagger},n}$ as specified in
the lemma statement. We let $\uptau_{0}:=0$ and for $k\in\mathbb{N}_{0}$ we
define inductively

$\uptau_{k+1}:=\inf\bigl{\\{}t\geq\uptau_{k}\,:\,\mathop{Re}\nolimits(X_{t})\not\in[\mathop{Re}\nolimits(X_{\uptau_{k}})-\pi,\mathop{Re}\nolimits(X_{\uptau_{k}})+\pi)\bigr{\\}}.$

For each $k\in\mathbb{N}_{0}$, we observe that the universal covering map
$\upsigma_{2\pi}:\mathbb{R}^{2}\to\mathcal{C}_{2\pi}$ is a biholomorphism when
restricted to
$[\mathop{Re}\nolimits(X_{\uptau_{k}})-\pi,\mathop{Re}\nolimits(X_{\uptau_{k}})+\pi)$.
Moreover, by assumption (H2), we know that the law of
$\upsigma_{2\pi}(\mathbf{X}^{n,z})$ weakly converges as $n\to\infty$ to the
law of the Brownian motion in $\mathcal{C}_{2\pi}$ with respect to the metric
$\operatorname{\mathrm{d}_{\mathrm{loc}}^{\mathrm{CMP}}}$ specified in
Subsection 2.1.2. Therefore, since Brownian motion is conformally invariant
and putting together the previous facts, we obtain that the law of the random
walk $\mathbf{X}^{n,z}$ weakly converges as $n\to\infty$ to the law of the
Brownian motion in $\mathbb{R}^{2}$ with respect to the metric
$\operatorname{\mathrm{d}^{\mathrm{CMP}}}$ on the time interval
$[\uptau_{k},\uptau_{k+1}]$, for each $k\in\mathbb{N}_{0}$. Hence, the desired
result follows. ∎

For each $n\in\mathbb{N}$, we recall that $\mathcal{F}\mathcal{G}^{n}$ denotes
the set of faces of the graph $\mathcal{G}^{n}$. The next lemma states that,
thanks to the invariance principle assumption on the sequence of weighted
graphs, the maximal diameter of the faces in $\mathcal{F}\mathcal{G}^{n}$
which intersect a compact subset of $\mathcal{C}_{2\pi}$ is of order
$o_{n}(1)$.

######  Lemma 4.2.

Let $R>0$, and, for any $n\in\mathbb{N}$, consider the set
$\mathcal{F}\mathcal{G}^{n}(R)$ of faces in $\mathcal{F}\mathcal{G}^{n}$ that
intersect $\mathbb{R}/2\pi\mathbb{Z}\times[-R,R]$. Then, for any
$\upvarepsilon>0$ and for any $n>n(R,\upvarepsilon)$ large enough, it holds
that

$\operatorname{diam}(f)\leq\upvarepsilon,\quad\forall
f\in\mathcal{F}\mathcal{G}^{n}(R).$

The same result holds also for the sequence of dual graphs, i.e., with
$\mathcal{F}\widehat{\mathcal{G}}^{n}$ in place of
$\mathcal{F}\mathcal{G}^{n}$.

###### Proof.

Obviously, we can just prove the result for the sequence of primal graphs as
the result for the sequence of dual graphs can be proved in the same exact
way. We proceed by contradiction by assuming that it exists $\upvarepsilon>0$
such that, for any $n\in\mathbb{N}$, it exists a face
$f_{n}\in\mathcal{F}\mathcal{G}^{n}(R)$ for which
$\operatorname{diam}(f_{n})>\upvarepsilon$. We notice that each set
$f_{n}\cap\mathbb{R}/2\pi\mathbb{Z}\times[-2R,2R]$ is compact and contains a
path $P_{n}$ with $\operatorname{diam}(P_{n})>\upvarepsilon$. By compactness
of the Hausdorff distance $\operatorname{\mathrm{d}^{\mathrm{H}}}$, we can
assume, by possibly taking a subsequential limit, that
$\lim_{n\to\infty}\operatorname{\mathrm{d}^{\mathrm{H}}}(P_{n},P)=0$ for some
compact and connected set $P\subset\mathbb{R}/2\pi\mathbb{Z}\times[-2R,2R]$.
Now, choose a rectangle $\mathrm{Q}$ such that $P$ disconnects the left and
right sides of $\mathrm{Q}$, or the top and bottom sides of $\mathrm{Q}$. For
any $n>n(R,\upvarepsilon,\mathrm{Q})$ large enough, also the path $P_{n}$
disconnects the rectangle $\mathrm{Q}$ in the same way as $P$. Therefore, for
any $x\in\mathbb{R}/2\pi\mathbb{Z}\times[-R,R]$, the random walk $X^{n,x}$ on
the weighted graph $(\mathcal{G}^{n},c^{n})$ started from $x$ cannot cross the
rectangle $\mathrm{Q}$ from left to right, or from top to bottom. However, the
Brownian motion $B^{x}$ on $\mathcal{C}_{2\pi}$ started from $x$ has positive
probability to do so. This is in contradiction with assumption (H2), and so