# Convergence of the Probabilistic Interpretation of Modulus

Nathan Albin  Department of Mathematics, Kansas State University, Manhattan,
KS 66506, USA.  Joan Lind  Department of Mathematics, University of Tennessee,
Knoxville, TN 37996, USA.  Pietro Poggi-Corradini  Department of Mathematics,
Kansas State University, Manhattan, KS 66506, USA.

###### Abstract

Given a Jordan domain $\Omega\subset\mathbb{C}$ and two disjoint arcs $A,B$ on
$\partial\Omega$, the modulus $m$ of the curve family connecting $A$ and $B$
in $\Omega$ is equal to the modulus of the curve family connecting the
vertical sides in the rectangle $R=[0,1]\times[0,m]$. Also, $m>0$ is the
unique value such that there is a conformal map $\psi$ mapping $\Omega$ to
${\rm int}(R)$ so that $\psi$ extends continuously to a homeomorphism of
$\partial\Omega$ onto $\partial R$ and the arcs $A$ and $B$ are sent to the
vertical sides of $R$. Moreover, in the case of the rectangle the family of
horizontal segments connecting the two sides has the same modulus as the
entire connecting family. Pulling these segments back to $\Omega$ via $\psi$
yields a family of extremal curves (also known as horizontal trajectories)
connecting $A$ to $B$ in $\Omega$. In this paper, we show that these extremal
curves can be approximated by some discrete paths arising from an
orthodiagonal approximation of $\Omega$. Moreover, we show that there is a
natural probability mass function (pmf) on these paths, deriving from the
theory of discrete modulus, which converges to the transverse measure on the
set of extremal curves. The key ingredient is an algorithm that, for an
embedded planar graph, takes the current flow between two sets of nodes, and
produces a unique path decomposition with non-crossing paths. Moreover, some
care was taken to adapt recent results for harmonic convergence on
orthodiagonal maps, due to Gurel-Gurevich, Jerison, and Nachmias, to our
context. Finally, we generalize a result of N. Alrayes from the square grid
setting to the orthodiagonal setting, and prove that the discrete modulus of
the approximating non-crossing paths converges to the continuous modulus.

Keywords: continuous and discrete modulus, electric flows, rectangle-tilings,
harmonic convergence.

2010 Mathematics Subject Classification: 30C85 (Primary) 31C20; 05C85
(Secondary)

###### Contents

  1. 1 Introduction and results
  2. 2 Background and examples
    1. 2.1 Discrete modulus
    2. 2.2 The probabilistic interpretation of discrete modulus
    3. 2.3 Discrete harmonic functions
    4. 2.4 Examples
    5. 2.5 Orthodiagonal maps
    6. 2.6 Linearly connected and John domains
  3. 3 Non-crossing minimal subfamilies
    1. 3.1 Non-crossing minimal subfamily algorithm
    2. 3.2 Scaling limit of non-crossing minimal subfamilies
    3. 3.3 Further consequences of the minimal non-crossing subfamily
    4. 3.4 A numerical example
      1. 3.4.1 Approximating the modulus
      2. 3.4.2 Non-crossing paths
  4. 4 Convergence of discrete harmonic functions
    1. 4.1 Modified toolbox
    2. 4.2 Proof of Theorem 3
    3. 4.3 Proof of Theorem 4

##  1 Introduction and results

Many objects in complex analysis are approximated by discrete analogues. In
this paper, we establish the convergence of three discrete objects, discrete
modulus, discrete paths that are extremal for the modulus, and discrete
harmonic functions, to their continuous counterparts, the continuous modulus,
the extremal curves (or horizontal trajectories) for the modulus, and harmonic
functions.

Consider a Jordan domain $\Omega\subset\mathbb{C}$ and two disjoint arcs on
$\partial\Omega$. The modulus of the curve family connecting these arcs in
$\Omega$ is well understood. In particular, there is a subfamily of extremal
curves (or horizontal trajectories) that carry all the information for this
modulus. We wish to approximate $\Omega$ with a plane graph $G$ and
approximate the boundary arcs with sets of vertices $A$ and $B$. As previously
studied in [2, 3, 4], the discrete modulus of the family of paths between $A$
and $B$ in $G$ arises from putting a density on the edges of $G$. See Section
2.1 for the definitions.

Our first task, accomplished in the result below, is to identify a subfamily
of paths between $A$ and $B$ in $G$ that is the candidate approximating family
for the extremal curves in the classical setting. The non-crossing paths in
this subfamily are allowed to intersect (i.e. they may share vertices and
edges in common), but they may not cross. See Section 3 for the precise
definition.

######  Proposition 1.

Let $G=(V,E,\sigma)$ be a finite plane network with the set $V$ of vertices,
the set $E$ of edges, and edge-weights $\sigma\in\mathbb{R}^{E}_{>0}$. Assume
that there are nonempty sets of boundary vertices $A$ and $B$ defined as
follows: $A=S_{1}\cap V$ and $B=S_{2}\cap V$, where $S_{1},S_{2}$ are two
disjoint arcs from a Jordan curve $\mathcal{C}$ that encircles $G$ in the
closed outer face of $G$. Consider the family $\Gamma(A,B)$ of all paths in
$G$ from $A$ to $B$.

Then, there is a unique subfamily $\Gamma\subset\Gamma(A,B)$ such that

  * (i)

$\Gamma$ is a family of non-crossing paths (defined in Section 3);

  * (ii)

$\Gamma$ supports a pmf $\mu$ that is optimal for the probabilistic
interpretation of the $(2,\sigma)$-modulus of $\Gamma(A,B)$ (defined in
Section 2.2);

  * (iii)

and $\Gamma$ is minimal for the $(2,\sigma)$-modulus of $\Gamma(A,B)$ (defined
in Section 2.1).

Moreover, both $\Gamma$ and $\mu$ are output by the Non-crossing Minimal
Subfamily Algorithm described in Section 3.

Our next goal is to show that the candidate subfamily identified in
Proposition 1 does in fact approximate the family of extremal curves. For
this, we specialize to plane graphs that arise from finite orthodiagonal maps,
a class of graphs that generalize the square grid. Roughly, an othodiagonal
map is a plane graph where every interior face is a quadrilateral with
orthogonal diagonals. Associated with the othodiagonal map is the so-called
primal graph $G^{\bullet}$ that we utilize, and the edges of $G^{\bullet}$ are
equipped with canonical weights defined in Equation (9). See Section 2.5 for
definitions and notation.

Given a Jordan domain $\Omega\subset\mathbb{C}$ and two arcs $A,B$ on
$\partial\Omega$, let $\psi$ be the conformal map from $\Omega$ to a rectangle
$R=[0,1]\times[0,m]$ so that $A$ and $B$ are sent to the vertical sides
$\\{0\\}\times[0,m]$ and $\\{1\\}\times[0,m]$. Then the extremal curves for
modulus are the pullback under $\psi$ of the horizontal segments connecting
the two vertical sides of $R$. The transverse measure on the horizontal curves
in $R$ is given by the 1-dimensional Lebesgue measure on the height
($y$-coordinate in $R$) of these curves. We define the transverse measure on
the extremal curves from $A$ to $B$ in $\Omega$ to be the pullback via $\psi$
of the transverse measure on the horizontal curves in $R$.

######  Theorem 2.

Let $\Omega$ be a Jordan domain whose boundary can be decomposed into four
analytic arcs $A,\tau_{1},B,\tau_{2}$ that do not meet in cusps. Let
$\epsilon_{n}>0$ with $\epsilon_{n}\to 0$. Let $G_{n}$ be an orthodiagonal map
with edge length at most $\epsilon_{n}$ and with boundary arcs
$S^{n}_{1},T^{n}_{1},S^{n}_{2},T^{n}_{2}$, so that
$(G_{n};S^{n}_{1},T^{n}_{1},S^{n}_{2},T^{n}_{2})$ approximates
$(\Omega;A,\tau_{1},B,\tau_{2})$ within distance $\epsilon_{n}$ (in the sense
of Definition 17). Equip $G_{n}$ with canonical edge-weights $\sigma_{n}$ as
in Equation (9). For $A_{n}=V_{n}^{\bullet}\cap S^{n}_{1}$ and
$B_{n}=V_{n}^{\bullet}\cap S^{n}_{2}$, consider the family
$\Gamma(A_{n},B_{n})$ of all paths in the primal graph $G^{\bullet}_{n}$ from
$A_{n}$ to $B_{n}$. Let $\Gamma_{n}$ and $\mu_{n}$ be corresponding objects
for $\Gamma(A_{n},B_{n})$, as described in Proposition 1; namely,
$\Gamma_{n}\subset\Gamma(A_{n},B_{n})$ is the unique minimal subfamily with
non-crossing paths and $\mu_{n}$ is the pmf supported on $\Gamma_{n}$ that is
optimal for the probabilistic interpretation of
$\operatorname{Mod}_{2,\sigma_{n}}\Gamma(A_{n},B_{n})$ (defined in Section
2.2).

Then, as $n\to\infty$, $\mu_{n}$ converges to the transverse measure $\mu$ on
the extremal curves from $A$ to $B$ in $\Omega$ (under weak convergence using
the Hausdorff metric as the underlying metric).

The measures $\mu_{n},\mu$ in the above theorem are supported on different
sets of curves, but we view them as measures on the same space as follows. Let
$X=\left(\bigcup_{n}\Gamma_{n}\right)\cup\Gamma$ where $\Gamma$ is the set of
extremal curves from $A$ to $B$ in $\Omega$. Then we consider $\mu_{n},\mu$ as
elements of the set $\mathcal{P}(X)$ of probability measures on
$(X,\mathcal{B}(X))$, where $\mathcal{B}(X)$ is the Borel $\sigma$-algebra
induced by the Hausdorff metric.

Theorem 2 relies on the following result about discrete harmonic functions
approximating continuous harmonic functions, which is an adaptation of the
main theorem of [11].

######  Theorem 3.

Let $\Omega$ be a Jordan domain whose boundary is decomposed into four
analytic arcs $A,\tau_{1},B,\tau_{2}$ that do not meet in cusps. Let $\psi$ be
a conformal map of $\Omega$ onto the interior of the rectangle
$R=[0,1]\times[0,m]$ taking the arcs $A$ and $B$ to the vertical sides of the
rectangle, and let $h_{c}=\operatorname{Re}\psi$. Let
$\epsilon,\delta\in(0,\operatorname{diam}(\Omega))$. Let $G=(V^{\bullet}\sqcup
V^{\circ},E)$ be a finite orthodiagonal map with boundary arcs
$S_{1},T_{1},S_{2},T_{2}$ with maximal edge length at most $\epsilon$, so that
$(G;S_{1},T_{1},S_{2},T_{2})$ approximates $(\Omega;A,\tau_{1},B,\tau_{2})$
within distance $\delta$ (in the sense of Definition 17). Equip $G$ with
canonical edge-weights $\sigma$ as in Equation (9). Let
$h_{d}:V^{\bullet}\to\mathbb{R}$ be discrete harmonic on ${\rm
int}(V^{\bullet})$ as in (6), with the following boundary values: $h_{d}(z)=0$
for $z\in V^{\bullet}\cap S_{1}$ and $h_{d}(z)=1$ for $z\in V^{\bullet}\cap
S_{2}$. Then, for $z\in V^{\bullet}\cap\overline{\Omega}$,

$|h_{d}(z)-h_{c}(z)|\leq\frac{C}{\log^{1/2}\left(\operatorname{diam}(\Omega)/(\epsilon\vee\delta)\right)},$

where $C$ depends only on $\Omega$.

Theorem 2 shows that the discrete optimal measures for modulus weakly converge
to the continuous analog. However, we want to have a stronger notion of
convergence, namely, that the expected overlaps defined right after (3)
converge to the reciprocal of the continuous modulus. In view of Theorem 3, it
means proving that the discrete Dirichlet energy of the discrete harmonic
functions $h_{d}$ converge to the Dirichlet energy of the continuous harmonic
function $h_{c}$. In practice, this stronger claim confirms what we see
numerically in Figure 6, namely that the heights in those two pictures
converge to the continuous modulus. That is the content of our next main
result, where we show that the discrete modulus converges to the continuous
modulus. This generalizes a result by Norah Alrayes in [6] from the square
grid setting to the orthodiagonal maps setting. Finally, note that the
convergence in energy cannot be deduced from the results in [11] and instead
depends crucially on Fulkerson duality for discrete modulus, see Lemma 24.

######  Theorem 4.

In the setting of Theorem 2, as $\epsilon_{n}\to 0$ the discrete modulus of
the path family between $A_{n}$ and $B_{n}$ in $G^{\bullet}_{n}$ converges to
continuous modulus of the curve family between $A$ and $B$ in $\Omega$.

We end with a note about the organization of this paper. In Section 2, we
introduce the needed background, including discrete modulus, discrete harmonic
functions, and orthodiagonal maps, and we consider two illuminating examples.
We begin Section 3 by discussing the Non-crossing Minimal Subfamily Algorithm
and proving Proposition 1. Then we prove Theorem 2, using Theorem 3 as a tool.
We end by discussing an interpretation of Proposition 1 in terms of current
flow and a rectangle-packing result. Section 4 contains the proof of Theorem 3
and Theorem 4.

## Acknowledgments

The authors would like to thank the anonymous referee for a very careful
reading of an earlier version and for many comments that helped improve the
paper over all. Lind and Poggi-Corradini would like to thank MSRI for their
support while some of this research was completed. The work of Poggi-Corradini
and Albin is supported in part by NSF Grant n. 2154032.

Added in proof: After this paper was submitted, A. Nachmias sent us a personal
communication noting that he had a similar unpublished result as Theorem 4,
but with a different proof.

##  2 Background and examples

###  2.1 Discrete modulus

Let $G=(V,E,\sigma)$ be a finite weighted graph, which is also referred to as
a network, with vertex set $V$, edge set $E$, and edge weights
$\sigma:E\to(0,\infty)$. We will use the term vertex and node interchangeably.
We assume that $G$ has no self-loops, but we allow multiple edges. Let
$\Gamma$ be a collection of objects in $G$. In this paper, most of our
attention will be on the case when $\Gamma$ is a collection of paths, but we
will also need modulus to be defined for a collection of $AB$-cuts in Section
3. A path is a sequence of edges $(x_{k-1}x_{k})_{k=1}^{n}$ so that the
vertices $x_{0},\dots,x_{n}$ are all distinct. Given $A,B\subset V$, then
$S\subset V$ is an $AB$-cut if $A\subset S$ and $B\cap S=\emptyset$. The usage
matrix $\mathcal{N}$ is a $|\Gamma|\times|E|$ matrix; in the case of paths
$\mathcal{N}(\gamma,e)=\mathbbm{1}_{\\{e\in\gamma\\}}$, and in the case of
$AB$-cuts $\mathcal{N}(S,e)=\mathbbm{1}_{\\{e\in\delta S\\}}$, where $\delta
S=\\{e=xy\in E:|S\cap\\{x,y\\}|=1\\}$.

A function $\rho:E\to[0,\infty)$, referred to as a density, is called
admissible for $\Gamma$ if

$\ell_{\rho}(\gamma):=\sum_{e\in E}\mathcal{N}(\gamma,e)\rho(e)\geq
1\;\;\text{ for all }\gamma\in\Gamma.$

When $\gamma$ is a path, we consider $\ell_{\rho}(\gamma)$ to be the length of
$\gamma$ under $\rho$. This means that $\rho$ is admissible for a collection
of paths if every path has length at least 1. For $1\leq p<\infty$, the
(discrete) $(p,\sigma)$-modulus of $\Gamma$ is defined as

$\operatorname{Mod}_{p,\sigma}(\Gamma):=\inf\sum_{e\in
E}\sigma(e)\rho(e)^{p},$

where the infimum is taken over all admissible densities $\rho$. See [2, 3, 4]
for further background on the discrete modulus. In the rest of this paper, we
will specialize to the case when $p=2$.

The extremal density $\rho^{*}$ is an admissible density that satisfies

$\operatorname{Mod}_{2,\sigma}(\Gamma)=\sum_{e\in E}\sigma(e)\rho^{*}(e)^{2}.$

The extremal density is unique in this case and is characterized in the
following theorem, which is the $p=2$ case of a well-known result (see for
instance, [4, Theorem 2.1], [1, Theorem 4-4], and also [7]).

######  Theorem 5.

(Beurling’s Criterion) Let $G=(V,E,\sigma)$ be a network and $\Gamma$ a family
of paths on $G$. An admissible density $\rho$ is extremal for
$\operatorname{Mod}_{2,\sigma}(\Gamma)$ if there is a subfamily
$\tilde{\Gamma}\subset\Gamma$ that satisfies the following two conditions:

  1. (i)

$\ell_{\rho}(\gamma)=1$ for all $\gamma\in\tilde{\Gamma}$.

  2. (ii)

If $h:E\to\mathbb{R}$ satisfies $\ell_{h}(\gamma)=\sum_{e\in\gamma}h(e)\geq 0$
for all $\gamma\in\tilde{\Gamma}$, then

$\sum_{e\in E}h(e)\rho(e)\sigma(e)\geq 0.$

Furthermore, in this case
$\operatorname{Mod}_{2,\sigma}(\tilde{\Gamma})=\operatorname{Mod}_{2,\sigma}(\Gamma)$.

A subfamily $\tilde{\Gamma}$ that satisfies the two conditions of Beurling’s
Criterion for $\rho^{*}$ is called a Beurling subfamily of $\Gamma$. A minimal
subfamily of $\Gamma$ is a subfamily $\tilde{\Gamma}\subset\Gamma$ so that
$\operatorname{Mod}_{2,\sigma}(\tilde{\Gamma})=\operatorname{Mod}_{2,\sigma}(\Gamma)$
and such that removing any path from $\tilde{\Gamma}$ decreases the modulus,
i.e. for any $\gamma\in\tilde{\Gamma}$ we have
$\operatorname{Mod}_{2,\sigma}(\tilde{\Gamma}\setminus\\{\gamma\\})<\operatorname{Mod}_{2,\sigma}(\Gamma)$.
As shown in [4, Theorem 3.5], $\Gamma$ will always contain a minimal
subfamily, although it may not be unique, and any minimal subfamily is also a
Beurling subfamily for $\Gamma$.

###  2.2 The probabilistic interpretation of discrete modulus

Here we recall the fact that, following [4] and [3], Lagrangian duality yields
a probabilistic interpretation for discrete modulus.

Assume that $\Gamma$ is a finite family of objects on $G$ with usage matrix
$\mathcal{N}$. In order to formulate the probabilistic interpretation of
$p$-modulus, we let $\mathcal{P}(\Gamma)$ represent the set of probability
mass functions (pmfs) on the set $\Gamma$. In other words,
$\mathcal{P}(\Gamma)$ contains the set of vectors $\mu\in\mathbb{R}_{\geq
0}^{\Gamma}$ with the property that $\mu^{T}\mathbf{1}=1$. Given a pmf $\mu$,
we can define a $\Gamma$-valued random variable $\underline{\gamma}$ with
distribution:

$\mathbb{P}_{\mu}\left(\underline{\gamma}=\gamma\right)=\mu(\gamma).$

Then, for every edge $e\in E$, the usage $\mathcal{N}(\underline{\gamma},e)$
is also a random variable, and we represent the expected edge-usage (depending
on the pmf $\mu$) as
$\mathbb{E}_{\mu}\left[\mathcal{N}(\underline{\gamma},e)\right]$. With these
notations, Lagrangian duality yields the following probabilistic
interpretation

######  Theorem 6 (Theorem 2 of [3]).

Let $G=(V,E,\sigma)$ be a finite graph with edge weights
$\sigma\in\mathbb{R}^{E}_{>0}$, and let $\Gamma$ be a finite family of objects
on $G$ with usage matrix $\mathcal{N}$. Then, for any $1<p<\infty$, letting
$q=p/(p-1)$ be the conjugate exponent to $p$, we have

$\operatorname{Mod}_{p,\sigma}(\Gamma)^{-\frac{q}{p}}=\min_{\mu\in\mathcal{P}(\Gamma)}\sum_{e\in
E}\sigma(e)^{-\frac{q}{p}}\mathbb{E}_{\mu}\left[\mathcal{N}(\underline{\gamma},e)\right]^{q}.$
(1)

Moreover, any optimal measure $\mu^{*}$, must satisfy

$\mathbb{E}_{\mu^{*}}\left[\mathcal{N}(\underline{\gamma},e)\right]=\frac{\sigma(e)\rho^{*}(e)^{p-1}}{\operatorname{Mod}_{p,\sigma}(\Gamma)}\qquad\forall
e\in E,$ (2)

where $\rho^{*}$ is the unique extremal density for
$\operatorname{Mod}_{p,\sigma}(\Gamma)$.

In this paper, we only consider the case when $p=2$ and $\Gamma$ is a family
of paths. In this case, the probabilistic interpretation says that modulus can
be computed by finding a pmf on $\Gamma$ that minimizes the expected
(weighted) overlap of two independent paths in $\Gamma$. We describe this in
the following corollary.

######  Corollary 7.

Consider a graph $G=(V,E,\sigma)$ with edge-weights
$\sigma\in\mathbb{R}^{E}_{>0}$. Let $\Gamma$ be a finite family of paths with
usage matrix $\mathcal{N}\in\mathbb{R}^{\Gamma\times E}$, and let
$\Sigma\in\mathbb{R}^{E\times E}$ be the diagonal matrix of weights
$\Sigma(e,e):=\sigma(e)$, for all $e\in E$. Then

$\operatorname{Mod}_{2,\sigma}(\Gamma)^{-1}=\min_{\mu\in\mathcal{P}(\Gamma)}\mu^{T}\mathcal{N}\Sigma^{-1}\mathcal{N}^{T}\mu=\min_{\mu\in\mathcal{P}(\Gamma)}\mathbb{E}_{\mu}\left|\underline{\gamma}\cap\underline{\gamma}^{\prime}\right|_{\sigma},$
(3)

where $\underline{\gamma}$ and $\underline{\gamma}^{\prime}$ are two
independent random variables chosen according to $\mu$, and
$\left|\underline{\gamma}\cap\underline{\gamma}^{\prime}\right|_{\sigma}$ is
their weighted _overlap_ $\sum_{e\in
E}\sigma(e)^{-1}\mathcal{N}(\gamma,e)\mathcal{N}(\gamma^{\prime},e)$.

Moreover, any optimal pmf $\mu^{*}$ is related to the extremal density
$\rho^{*}$ for $\operatorname{Mod}_{2,\sigma}(\Gamma)$ as follows:

$\frac{\sigma(e)\rho^{*}(e)}{\operatorname{Mod}_{2,\sigma}(\Gamma)}=\left(\mathcal{N}^{T}\mu^{*}\right)(e)=\mathbb{E}_{\mu^{*}}\left[\mathcal{N}(\underline{\gamma},e)\right]=\mathbb{P}_{\mu^{*}}\left(e\in\underline{\gamma}\right),\qquad\forall
e\in E.$ (4)

###### Proof.

Note that when $p=2$, the quantity minimized in the right hand-side of (1)
becomes:

$\displaystyle\sum_{e\in
E}\sigma(e)^{-1}\left(\mathbb{E}_{\mu}\left[\mathcal{N}(\underline{\gamma},e)\right]\right)^{2}$
$\displaystyle=\sum_{e\in
E}\sigma(e)^{-1}\left[\left(\mathcal{N}^{T}\mu\right)(e)\right]^{2}$
$\displaystyle=\sum_{e\in
E}\sigma(e)^{-1}\left(\sum_{\gamma\in\Gamma}\mu(\gamma)\mathcal{N}(\gamma,e)\right)^{2}$
$\displaystyle=\sum_{\gamma,\gamma^{\prime}\in\Gamma}\mu(\gamma)\mu(\gamma^{\prime})\left(\sum_{e\in
E}\sigma(e)^{-1}\mathcal{N}(\gamma,e)\mathcal{N}(\gamma^{\prime},e)\right).$
$\displaystyle=\mu^{T}\mathcal{N}\Sigma^{-1}\mathcal{N}^{T}\mu=\mathbb{E}_{\mu}\left|\underline{\gamma}\cap\underline{\gamma}^{\prime}\right|_{\sigma}.$

∎

Note that in general the optimal pmf’s $\mu^{*}$ for (3) are not unique, even
though the extremal density $\rho^{*}$ is. When $\Gamma$ is minimal,
uniqueness of $\mu^{*}$ does hold. This fact can be deduced from [4, Theorem
3.5 and Section 5], however, it is not explicitly formulated there, so we
summarize it here, for clarity.

######  Proposition 8.

Let $G=(V,E,\sigma)$ be a finite graph with edge-weights
$\sigma\in\mathbb{R}^{E}_{>0}$, and let $\Gamma$ be a finite (non-empty)
family of paths in $G$. If $\tilde{\Gamma}\subset\Gamma$ is a minimal
subfamily for $\Gamma$, then there is a unique probability mass function (or
pmf) $\mu^{*}$ supported on $\tilde{\Gamma}$ so that for all $e\in E$

$\frac{\sigma(e)\rho^{*}(e)}{\operatorname{Mod}_{2,\sigma}(\Gamma)}=\mathbb{P}_{\mu^{*}}\left(e\in\underline{\gamma}\right)=\sum_{\gamma\in\Gamma}\mu^{*}(\gamma)\mathcal{N}(\gamma,e),$
(5)

where $\rho^{*}$ is the extremal density for
$\operatorname{Mod}_{2,\sigma}(\Gamma)$.

###### Proof.

Equation (5) holds for any optimal pmf, as we saw in (2). Therefore, it is
enough to show that there is a unique optimal pmf $\mu^{*}$ supported on
$\tilde{\Gamma}$. One way to see this is to use Theorem 3.5 (ii) of [4], which
shows that if $\tilde{\Gamma}$ is a minimal subfamily of $\Gamma$, then there
is a unique set of Lagrange multipliers $\lambda^{*}$ that are optimal for the
dual problem to modulus, so that
$\tilde{\Gamma}=\\{\gamma\in\Gamma:\lambda^{*}(\gamma)>0\\}$. The minimization
on the right hand-side of (1) is derived from the Lagrangian dual problem for
modulus by normalizing the Lagrangian multipliers $\lambda$ so as to have
total sum $\mathbf{1}^{T}\lambda$ equal to $1$. Therefore, the optimal pmf
$\mu^{*}$ for $\tilde{\Gamma}$ is obtained by dividing $\lambda^{*}$ by
$\mathbf{1}^{T}\lambda^{*}$, and is therefore also unique.

Alternatively, we can argue as with the Lagrange multipliers, but using the
pmf’s directly instead. We argue by contradiction. Suppose
$\mu_{0},\mu_{1}\in\mathcal{P}(\tilde{\Gamma})$ are two distinct optimal pmfs
for $\operatorname{Mod}_{2,\sigma}\tilde{\Gamma}$. Since
$\tilde{\Gamma}\subset\Gamma$ and
$\operatorname{Mod}_{2,\sigma}\tilde{\Gamma}=\operatorname{Mod}_{2,\sigma}\Gamma$,
we have that $\mu_{0}$ and $\mu_{1}$ are also optimal pmfs for
$\operatorname{Mod}_{2,\sigma}\Gamma$. In particular, they both satisfy
Equation (5). For $t\in\mathbb{R}$, set $\mu_{t}:=t\mu_{1}+(1-t)\mu_{0}$. Then
$\mathbf{1}^{T}\mu_{t}=1$. Moreover, we claim that there is a maximal interval
$(t_{0},t_{1})\supset[0,1]$ such that $\mu_{t}\geq 0$ for $t\in[t_{0},t_{1}]$.
The strict inclusion follows from the fact that $\tilde{\Gamma}$ is minimal,
which implies that $\min_{\gamma\in\tilde{\Gamma}}\mu_{j}(\gamma)>0$ for
$j=0,1$. In other words, $\mu_{t}\in\mathcal{P}(\Gamma)$ for
$t\in[t_{0},t_{1}]$. Furthermore, for $t$ in this range $\mu_{t}$ satisfies
Equation (5) and is therefore optimal for
$\operatorname{Mod}_{2,\sigma}\Gamma$. Finally, when $t=t_{0}$ or $t_{1}$ we
get an optimal pmf that must vanish on some $\gamma\in\tilde{\Gamma}$,
contradicting the minimality of $\tilde{\Gamma}$. ∎

###  2.3 Discrete harmonic functions

For a network $G=(V,E,\sigma)$, when $v,w\in V$ are adjacent, we will use the
notation $vw$ to refer to the edge incident to $v$ and $w$. A function
$f:V\to\mathbb{R}$ is discrete harmonic at $v\in V$ if

$f(v)\sum_{w:\,vw\in E}\sigma(vw)=\sum_{w:\,vw\in E}f(w)\sigma(vw),$ (6)

or in other words, the value of $f$ at $v$ is a weighted average of the values
of $f$ at the neighbors of $v$. The existence and uniqueness of discrete
harmonic functions with specified boundary values is well known; see for
instance Section 2.1 of [14].

######  Proposition 9.

(Existence and uniqueness of discrete harmonic functions) Let $G=(V,E,\sigma)$
be a finite connected network. For $U\subsetneq V$ and $g:V\setminus
U\to\mathbb{R}$, there is a unique function $h:V\to\mathbb{R}$ such that $h=g$
on $V\setminus U$ and $h$ is discrete harmonic on $U$.

The function $h$ given by Proposition 9 is called the solution to the discrete
Dirichlet problem on $U$ with boundary data $g$. Viewed from the electric
network perspective, $h$ is called the voltage function. From its discrete
gradient, we obtain the current flow $f=\sigma dh$ defined by

$f(vw)=\sigma(vw)\left[h(w)-h(v)\right]\;\;\;\text{ for all }vw\in\vv{E}.$

In the above, we use $\vv{E}$ for the set of directed edges, which contains
both possible orientations for all the edges of $E$. In this setting, $vw$
represents the directed edge from $v$ to $w$.

More generally, a flow on $U\subset V$ is a function $\theta$ on $\vv{E}$ so
that $\theta(vw)=-\theta(wv)$ for all $vw\in\vv{E}$ and each $v\in U$
satisfies Kirchhoff’s node law, i.e.

$\sum_{w:\,vw\in\vv{E}}\theta(vw)=0.$

Furthermore, if $A$ and $B$ are nonempty and disjoint subsets in $V$ and
$U:=V\setminus(A\cup B)$, then we say that $\theta$ is a flow from $A$ to $B$,
if $\theta$ is a flow on $U$. In this case, the strength (or value) of
$\theta$ is given by the formula:

${\rm strength}(\theta):=\sum_{w:\,vw\in\vv{E},v\in
A}\theta(vw)=\sum_{v:\,vw\in\vv{E},w\in B}\theta(vw).$

Moreover, such a flow is said to be a unit flow if it has unit strength.

The energy of a symmetric or anti-symmetric function
$\theta:\vv{E}\to\mathbb{R}$ is given (with a slight abuse of notation) by

$\mathcal{E}(\theta)=\sum_{e\in E}\frac{1}{\sigma(e)}\theta^{2}(e).$

The energy of $h:V\to\mathbb{R}$ is defined to be the energy of its discrete
gradient, i.e.

$\mathcal{E}(h)=\mathcal{E}(\sigma dh)=\sum_{vw\in
E}\sigma(vw)\left[h(w)-h(v)\right]^{2}.$

We are now able to explain the close connection between the discrete 2-modulus
and discrete harmonic functions. Suppose $G$ is network with disjoint nonempty
sets $A,B\subset V$, and edge-weights $\sigma\in\mathbb{R}^{E}_{>0}$. Let
$\Gamma$ be the family of paths in $G$ from $A$ to $B$, and let $h$ be the
solution to the discrete Dirichlet problem on $V\setminus(A\cup B)$ with
$h|_{A}=0$ and $h|_{B}=1$. Then,

$\operatorname{Mod}_{2,\sigma}(\Gamma)=\mathcal{E}(h),$ (7)

and the extremal density is related to the solution of the Dirichlet problem
as follows

$\rho^{*}(vw)=|h(w)-h(v)|$ (8)

for all $vw\in E$. See Theorem 4.2 in [2].

###  2.4 Examples

We wish to illustrate the concepts from the previous sections and our main
results by taking a look at two simple examples.

######  Example 10.

For the first example, let $R$ be the rectangle $[0,L]\times[0,1]$, where
$L\in\mathbb{N}$. Let $G$ be the square grid in $R$ with lattice size $1/n$
and weights $\sigma\equiv 1$. Let $\Gamma$ be the family of all paths in $G$
from $A=\\{v\in V\,:\,\text{Re}(v)=0\\}$ to $B=\\{v\in
V\,:\,\text{Re}(v)=L\\}$.

For this example, we claim the following:

  1. 1.

The extremal density is $\rho^{*}(e)=\begin{cases}\frac{1}{Ln}&\quad\text{if
}e\text{ is a horizontal edge}\\\ 0&\quad\text{if }e\text{ is a vertical
edge}\\\ \end{cases}.$

  2. 2.

The subfamily of horizontal paths $\tilde{\Gamma}$ is a Beurling subfamily.

  3. 3.

$\operatorname{Mod}_{2}(\Gamma)=\operatorname{Mod}_{2}(\tilde{\Gamma})=\frac{1}{L}\left(1+\frac{1}{n}\right)$.

  4. 4.

$\tilde{\Gamma}$ is the unique minimal subfamily.

  5. 5.

The optimal pmf $\mu^{*}$ is unique and is uniform on $\tilde{\Gamma}$.

We will use Beurling’s criterion to establish the first two claims. Note that
each horizontal path $\gamma\in\tilde{\Gamma}$ has $Ln$ horizontal edges, and
so $\ell_{\rho^{*}}(\gamma)=1$. To establish the second condition of
Beurling’s criterion, assume that $h:E\to\mathbb{R}$ satisfies
$\sum_{e\in\gamma}h(e)\geq 0$ for all $\gamma\in\tilde{\Gamma}$. Then, since
$\rho^{*}(e)=0$ when $e$ is vertical,

$\sum_{e\in E}h(e)\rho^{*}(e)=\frac{1}{Ln}\sum_{e\text{
horizontal}}h(e)=\frac{1}{Ln}\sum_{\gamma\in\tilde{\Gamma}}\sum_{e\in\gamma}h(e)\geq
0.$

Thus Beurling’s criterion (Theorem 5) shows that $\rho^{*}$ is the extremal
density and
$\operatorname{Mod}_{2}(\Gamma)=\operatorname{Mod}_{2}(\tilde{\Gamma})$. We
compute

$\operatorname{Mod}_{2}(\Gamma)=\sum_{e\in
E}\rho^{*}(e)^{2}=\left(\frac{1}{Ln}\right)^{2}(\text{\\# of horizonal
edges})=\frac{1}{L}\left(1+\frac{1}{n}\right),$

since there are $Ln(n+1)$ horizontal edges. Notice that as $n\to\infty$, the
discrete modulus converges to $\frac{1}{L}$, which is the corresponding
continuous modulus.

An alternate approach to finding $\rho^{*}$ would be to check that
$h(v)=\frac{1}{L}\text{Re}(v)$ is discrete harmonic on $V\setminus(A\cup B)$.
Thus $h$ is the solution to the discrete Dirichlet problem with boundary
values $h|_{A}=0$ and $h|_{B}=1$, and we could obtain $\rho^{*}$ from (8).

We observe that $\tilde{\Gamma}$ is a minimal subfamily, since removing any
curve $\gamma\in\tilde{\Gamma}$ will decrease the modulus. To see this, change
$\rho^{*}$ to be zero on the removed path $\gamma$ and use Beurling’s
criterion to show that the new density is extremal for
$\tilde{\Gamma}\setminus\\{\gamma\\}$. To show that $\tilde{\Gamma}$ is the
unique minimal subfamily, assume to the contrary that $\hat{\Gamma}$ is
another minimal subfamily. Then the optimal pmf $\hat{\mu}$ on $\hat{\Gamma}$
satisfies that the expected edge usage of $e$ under $\hat{\mu}$ is given by
$\rho^{*}(e)/\operatorname{Mod}_{2}(\Gamma)$ by (5). This implies that the
expected edge usage for any vertical edge must be zero, and the paths in
$\hat{\Gamma}$ cannot use any vertical edges. Therefore
$\hat{\Gamma}\subset\tilde{\Gamma}$. However, since
$\operatorname{Mod}_{2}(\hat{\Gamma})=\operatorname{Mod}_{2}(\tilde{\Gamma})$,
we must have $\hat{\Gamma}=\tilde{\Gamma}$.

Let $\mu^{*}$ be the optimal pmf on $\tilde{\Gamma}$. From (5), we see that if
$e$ is a horizontal edge and $\gamma$ is the corresponding horizontal path,
then

$\mu^{*}(\gamma)=\mathbb{P}_{\mu^{*}}\left[\gamma\text{ contains
}e\right]=\frac{\rho^{*}(e)}{\operatorname{Mod}_{2}(\Gamma)}=\frac{1}{n+1}.$

This implies that $\mu^{*}$ is uniform on $\tilde{\Gamma}$.

We wish to vary $n$, and so we update our notation to show the dependence on
$n$, i.e. let $\mu^{*}_{n}=\mu^{*}$ which is supported on
$\tilde{\Gamma}_{n}=\tilde{\Gamma}$. For this example, the result of Theorem 2
is fairly intuitive: as $n\to\infty$, $\mu^{*}_{n}$ converges to the
transverse measure on all horizontal curves in $R$, which are the extremal
curves in the continuous setting.

######  Example 11.

In our second example, we will look at a rotated rectangle. Let $R$ be the
rectangle $[0,\sqrt{2}L]\times[0,\sqrt{2}]$ rotated by $\pi/4$ (i.e. the
corners of $R$ are $0,L+iL,L-1+i(L+1),-1+i$.) In R, take the (non-rotated)
square grid with lattice size $1/n$ and weights $\sigma\equiv 1$. Let $\Gamma$
be the family of all paths in $R$ from the left side
$A=\\{v\,:\,\text{Im}(v)=-\text{Re}(v)\\}$ to the right side
$B=\\{v\,:\,\text{Im}(v)=-\text{Re}(v)+2L\\}$. See Figure 1.

Figure 1: The graph for Example 11 with $L=2$, $n=3$, the vertices of $A$
shown in red, and the vertices of $B$ shown in blue.

In contrast to Example 10, there is not a unique minimal subfamily in this
case, and we will describe two such subfamilies. Consider first the subfamily
$\Gamma^{z}$ of “zig-zag” paths, which alternate between increasing the real
component and increasing the imaginary component (or vice-versa). These paths
each have $2Ln$ steps, and each edge is used in exactly 1 path. By arguing as
before, Beurling’s criterion can be used to show that the extremal density is
$\frac{1}{2Ln}$ on all edges and $\Gamma^{z}$ is a Beurling subfamily. Further
we see that $\Gamma^{z}$ is also a minimal subfamily, since removing any path
will decrease the modulus. From the extremal density and the fact that there
are $4n^{2}L$ total edges, we see that
$\operatorname{Mod}_{2}(\Gamma)=\frac{1}{L}$, which is equal to the
corresponding continuous modulus.

Another minimal subfamily $\Gamma^{r}$ consists of the “reflected” paths which
keep increasing the same component (whether real or imaginary) until they hit
the boundary of $R$ and then they switch to increasing the other component.
The paths in this family will also have $2Ln$ steps each, and each edge is
used in exactly 1 path. One can check that this will also be a Beurling
subfamily and a minimal subfamily.

Each subfamily will have an associated optimal pmf: let $\mu_{n}^{z}$ be the
pmf on the zig-zag subfamily $\Gamma^{z}$, and let $\mu_{n}^{r}$ be the pmf on
the reflected subfamily $\Gamma^{r}$. Both measures will be uniform, giving
the $2n$ paths in the relevant subfamily weight $\frac{1}{2n}$. Further both
$\mu_{n}^{z}$ and $\mu_{n}^{r}$ will have a limit as $n\to\infty$, but the
limiting objects will be different. The limit for $\mu_{n}^{z}$ is the
transverse measure on the straight line paths of slope 1 in $R$ (which are the
extremal curves in the continuous setting). However, the limit for
$\mu_{n}^{r}$ will be supported on paths that move in a straight line either
horizontally or vertically until reflecting off the boundary of the rectangle.

To illustrate the point that not all Beurling subfamilies are minimal, we wish
to examine one further subfamily. Let

$\hat{\Gamma}=\\{\gamma\in\Gamma\,:\,\gamma\text{ has }2Ln\text{ steps}\\}.$

In other words, this is the collection of all curves that have length 1 under
the extremal density. This family contains both $\Gamma^{z}$ and $\Gamma^{r}$.
In a moment, we will show that this is a Beurling subfamily, and hence it will
be the largest Beurling subfamily by construction.

Let $\hat{\mu}$ be the pmf that arises from viewing the paths in
$\hat{\Gamma}$ as 1-dim simple random walk (SRW) trajectories started on the
left side of $R$ and reflected to stay in the rectangle. In other words, we
can think of creating the paths dynamically as follows: Each of the $2n$
initial edges has equal probability $\frac{1}{2n}$. Once the first edge is
chosen, we chose the next edge with probability $1/2$ of going up or going to
the right. Then we repeat. If we happen to be on the boundary, then we take
the only available edge with probability 1. One can show for any fixed edge
$e$ that $\mathbb{P}_{\hat{\mu}}[e\in\gamma]=\frac{1}{2n}$, see Claim 12
below.

With this understanding of $\hat{\mu}$, we will be able to show that
$\hat{\Gamma}$ is a Beurling subfamily by verifying Beurling’s Criterion. Let
$h\in\mathbb{R}^{E}$ with $\sum_{e\in\gamma}h(e)\geq 0$ for all
$\gamma\in\hat{\Gamma}$. Then

$\sum_{e\in E}h(e)\cdot\frac{1}{2nL}=\sum_{e\in
E}h(e)\cdot\frac{1}{L}\sum_{\gamma\in\hat{\Gamma}}\hat{\mu}(\gamma)\mathbbm{1}_{\\{e\in\gamma\\}}=\frac{1}{L}\sum_{\gamma\in\hat{\Gamma}}\hat{\mu}(\gamma)\sum_{e\in\gamma}h(e)\geq
0.$

A short computation will show that $\hat{\mu}$ is a pmf on $\hat{\Gamma}$ that
minimizes the expected overlap of two independent randomly chosen paths in
$\hat{\Gamma}$:

$\displaystyle\mathbb{E}_{\hat{\mu}}|\gamma\cap\gamma^{\prime}|$
$\displaystyle=\mathbb{E}_{\hat{\mu}}\left[\sum_{e\in
E}\mathbbm{1}_{\\{e\in\gamma\cap\gamma^{\prime}\\}}\right]$
$\displaystyle=\sum_{e\in
E}\mathbb{P}_{\hat{\mu}}\left[e\in\gamma\cap\gamma^{\prime}\right]$
$\displaystyle=\left(\frac{1}{2n}\right)^{2}\cdot 4n^{2}L$ $\displaystyle=L.$

By (3), we have

$L=\operatorname{Mod}_{2}(\Gamma)^{-1}=\min_{\mu\in\mathcal{P}(\Gamma)}\mathbb{E}_{\mu}|\gamma\cap\gamma^{\prime}|.$

Therefore, $\hat{\mu}$ is a pmf on $\hat{\Gamma}$ that minimizes expected
overlap. Hence, $\hat{\mu}$ is an optimal pmf on $\hat{\Gamma}$. Note that it
may not be unique, since we are only guaranteed a unique optimal pmf for a
minimal subfamily, as we saw in Proposition 8

To highlight the fact that the limit in Theorem 2 depends on the correct
choice of subfamily, we mention that the limit of the pmf $\hat{\mu}$ as
$n\to\infty$ yields 1-dim reflected Brownian motion trajectories started at a
uniform point on the left side of $R$.

######  Claim 12.

$\mathbb{P}_{\hat{\mu}}[e\in\gamma]=\frac{1}{2n}$ for any edge $e$.

###### Proof.

This claim can be shown by induction. In particular, we can subdivide the
edges into $2nL$ distinct sets of $2n$ edges

$E_{k}=\\{\text{ all the edges that could be step $k$ in a SRW trajectory
}\\},$

and we do induction on $k$ to show that $\mathbb{P}_{\hat{\mu}}[e\in\gamma]$
is the same for all edges $e$ in $E_{k}$. The base case follows by definition,
since each of the initial edges has equal probability. We will show the basic
idea behind the induction proof by explaining the reasoning for $E_{2}$ and
$E_{3}$. Consider the vertices that edges from $E_{1}$ lead into (we consider
the edges oriented so that the SRW trajectories flow from the left side to the
right side of $R$). Each of these vertices has exactly two edges leading into
it, meaning that our SRW induces a uniform distribution on these vertices.
Since each of these vertices has two edges leading out, and we will choose
between them with equal probability, we have uniform probability for all edges
in $E_{2}$. Now consider the vertices that the edges from $E_{2}$ lead into.
There are two special vertices, one on the top boundary and one on the bottom
boundary, that only have one edge leading into them and one edge leading out.
All the rest have two edges leading in and two edges leading out. While we no
longer have a uniform distribution induced on these vertices, we still obtain
the uniform distribution on $E_{3}$. (Reason: We have the same chance of
arriving at any internal vertex, and we have exactly half this probability of
arriving at a boundary vertex. From the internal vertex, we have two edges to
choose from with equal probability, while at the boundary vertex we must take
the only available edge.) ∎

Note that the paths in $\Gamma^{z}$ are non-crossing, while in $\Gamma^{r}$
and $\hat{\Gamma}$ they do cross. In Section 3 we will give an algorithm that,
in the case of plane graphs, always produces the unique non-crossing minimal
subfamily.

###  2.5 Orthodiagonal maps

A finite orthodiagonal map is a finite connected plane graph that satisfies
the following:

  * •

Each edge is a straight line segment.

  * •

Each inner face is a quadrilateral with orthogonal diagonals.

  * •

The boundary of the outer face is a simple closed curve.

See the top left image in Figure 2 for an illustration.

We will briefly describe some of the structure associated with an
orthodiagonal map $G=(V,E)$. Since $G$ is a bipartite graph, we obtain a
bipartition of the vertices $V=V^{\bullet}\sqcup V^{\circ}$. Each inner face
$Q$ is determined by the four vertices on its boundary, which must alternate
between $V^{\bullet}$ and $V^{\circ}$. We will write
$Q=[v_{1},w_{1},v_{2},w_{2}]$ with the convention that $v_{1},v_{2}\in
V^{\bullet}$, $w_{1},w_{2}\in V^{\circ}$, and the vertices are ordered
counterclockwise around $Q$. We can create two additional graphs associated
with $G$, the primal graph $G^{\bullet}=(V^{\bullet},E^{\bullet})$ and the
dual graph $G^{\circ}=(V^{\circ},E^{\circ})$. There is one primal edge and one
dual edge for each inner face of $G$. In particular, $e_{Q}^{\bullet}\in
E^{\bullet}$ is an edge between $v_{1}$ and $v_{2}$ in $Q$; it is defined to
be the line segment $v_{1}v_{2}$, if this segment is contained in $Q$, and
otherwise it is the union of the line segments $v_{1}p$ and $pv_{2}$, where
$p=(w_{1}+w_{2})/2$ is the midpoint of $w_{1}w_{2}$. The edge
$e_{Q}^{\circ}\in E^{\circ}$ between $w_{1}$ and $w_{2}$ in $Q$ is defined
similarly. For an edge $e\in E^{\bullet}$, the dual edge to $e$ is the unique
edge $e^{\circ}\in E^{\circ}$ so that there is an inner face $Q$ with
$e=e_{Q}^{\bullet}$ and $e^{\circ}=e_{Q}^{\circ}$. The primal network
$(G^{\bullet},\sigma)$ and the dual network $(G^{\circ},\sigma)$ have the
following canonical edge-weights:

$\sigma(e_{Q}^{\bullet})=\frac{|w_{1}w_{2}|}{|v_{1}v_{2}|}\;\;\;\text{ and
}\;\;\;\sigma(e_{Q}^{\circ})=\frac{|v_{1}v_{2}|}{|w_{1}w_{2}|},$ (9)

where the notation $|ab|$ is the Euclidean distance between $a$ and $b$. Note
that this distance is not necessarily the length of the edge.

One way to interpret these weights is through a finite-volume approximation of
the Laplace equation. Let $v_{1}$ be an interior vertex in the primal graph,
and consider the collection of quadrilaterals containing this vertex. The
bottom right picture in Figure 2 shows an example of such a collection of
quadrilaterals (light edges in the figure). The edges of the dual graph
surround a polygonal region $D$ (thick dashed lines in the figure). If $h$ is
the harmonic function for the continuum problem, then the divergence theorem
implies that

$0=\int_{D}\Delta h\;dA=\int_{\partial D}\frac{\partial h}{\partial n}\;dS.$

The integral on the right is a sum of integrals along the edges of $D$. Let
$w_{1}w_{2}$ be some such edge, as indicated in the figure. There is a
corresponding edge $v_{1}v_{2}$ in the primal graph. Since these edges are
orthogonal, $(h(v_{2})-h(v_{1}))/|v_{1}v_{2}|$ provides an approximation of
the normal derivative along $w_{1}w_{2}$. Thus, the contribution of
$w_{1}w_{2}$ to the boundary integral can be approximated as

$\int_{w_{1}w_{2}}\frac{\partial h}{\partial
n}\;dS\approx\frac{|w_{1}w_{2}|}{|v_{1}v_{2}|}(h(v_{2})-h(v_{1})).$

Performing a similar approximation on the remaining edges of $D$ yields the
approximation

$0\approx\sum_{u:v_{1}u\in E}\frac{|w_{1}w_{2}|}{|v_{1}u|}(h(u)-h(v_{1})).$

Thus, when the continuous harmonic function is restricted to the graph, it
approximately satisfies the discrete harmonic equation (6) at each interior
vertex, with the edge conductances defined in (9). Moreover, this
approximation improves as the lengths of the graph edges converge to zero.

We list some further notation relating to an orthodiagonal map $G$:

$\displaystyle\partial G$ $\displaystyle=\text{ the topological boundary of
the outer face of }G$ $\displaystyle{\rm int}(G)$ $\displaystyle=\text{ the
open subset enclosed by }\partial G$ $\displaystyle\hat{G}$
$\displaystyle={\rm int}(G)\cup\partial G$ $\displaystyle\tilde{G}$
$\displaystyle=\text{ the union of the convex hulls of the closures of the
inner faces of }G$

In [11], Gurel-Gurevich, Jerison, and Nachmias show that one can approximate
continuous harmonic functions by discrete harmonic functions on an
orthodiagonal map, generalizing the work of [20] and [22]. In particular, they
are concerned with the solution $h_{c}:\overline{\Omega}\to\mathbb{R}$ to the
continuous Dirichlet problem on $\Omega$ with boundary data
$g|_{\partial\Omega}$; this terminology means that $h_{c}=g$ on
$\partial\Omega$, $h_{c}$ is harmonic on $\Omega$, and $h_{c}$ is continuous
on $\overline{\Omega}$.

######  Theorem 13 (Theorem 1.1 of [11]).

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected domain, and let
$g:\mathbb{C}\to\mathbb{R}$ be a $C^{2}$-smooth function. Given
$\epsilon,\delta\in(0,\operatorname{diam}(\Omega))$, let $G=(V^{\bullet}\sqcup
V^{\circ},E)$ be a finite orthodiagonal map with maximal edge length at most
$\epsilon$ such that the Hausdorff distance between $\partial G$ and
$\partial\Omega$ is at most $\delta$. Let
$h_{c}:\overline{\Omega}\to\mathbb{R}$ be the solution to the continuous
Dirichlet problem on $\Omega$ with boundary data $g|_{\partial\Omega}$, and
let $h_{d}:V^{\bullet}\to\mathbb{R}$ be the solution to the discrete Dirichlet
problem on $V^{\bullet}\setminus\partial G$ with boundary data $g|_{\partial
G\cap V^{\bullet}}$. Set

$C_{1}=\sup_{z\in\tilde{\Omega}}|\nabla g(z)|\;\;\;\text{ and
}\;\;\;C_{2}=\sup_{z\in\tilde{\Omega}}||Hg(z)||_{2},$

where $\tilde{\Omega}=\operatorname{conv}(\overline{\Omega}\cup\hat{G})$. Then
there is a universal constant $C<\infty$ such that for all $z\in
V^{\bullet}\cap\overline{\Omega}$,

$|h_{d}(z)-h_{c}(z)|\leq\frac{C\operatorname{diam}(\Omega)(C_{1}+C_{2}\epsilon)}{\log^{1/2}(\operatorname{diam}(\Omega)/(\delta\vee\epsilon)}.$

In the above result, the boundary condition involves the vertices of the
primal graph that are in the topological boundary of $G$. In our work, we need
a similar result with different boundary conditions. This leads us to making
the following definition.

######  Definition 14.

The graph $G=(V^{\bullet}\sqcup V^{\circ},E)$ is a finite orthodiagonal map
with boundary arcs $S_{1},T_{1},S_{2},T_{2}$, if $G$ is a finite orthodiagonal
map with the following boundary condition:

  * •

(Boundary condition) The topological boundary of the outer face of $G$ is
subdivided into four nonempty paths $S_{1},T_{1},S_{2},T_{2}$, listed
counterclockwise, with $S_{j}\cap T_{k}$ equal to a single vertex, for
$j,k=1,2$.

This boundary condition allows us to think of $\hat{G}$ as a conformal
rectangle with “sides” $S_{1},S_{2}$ and “top/bottom” $T_{1},T_{2}$. For the
associated primal and dual graphs, we define the (graph) boundaries as
follows:

$\partial V^{\bullet}:=V^{\bullet}\cap\left(S_{1}\cup S_{2}\right)\;\;\text{
and }\;\;\partial V^{\circ}:=V^{\circ}\cap\left(T_{1}\cup T_{2}\right).$

Note that $\partial V^{\bullet}$ and $\partial V^{\circ}$ do not include all
vertices in the topological boundary of $G$. Further, we define

${\rm int}(V^{\bullet}):=V^{\bullet}\setminus\partial V^{\bullet}\;\text{ and
}\;{\rm int}(V^{\circ}):=V^{\circ}\setminus\partial V^{\circ},$

and we point out that these are different from the collections of vertices in
the topological interior of $G^{\bullet}$ and $G^{\circ}$. See Figure 2.

$S_{1}$$T_{1}$$S_{2}$$T_{2}$

Figure 2: Top left: An example of an orthodiagonal map with the bipartation of
the vertices. Top right: The topological boundary of the map decomposed into
four paths: $S_{1},S_{2}$ (solid) and $T_{1},T_{2}$ (dashed). Bottom left: The
primal graph $G^{\bullet}$ (black/red vertices and solid edges) with $\partial
V^{\bullet}$ shown in red and the dual graph $G^{\circ}$ (white vertices and
dashed edges). Bottom right: Finite volume interpretation of the edge weights
assigned to an orthodiagonal map.

Let $\Omega$ be a Jordan domain, and let $F:\mathbb{D}\to\Omega$ be a
conformal map, which therefore extends as a homeomorphism
$F:\overline{\mathbb{D}}\to\overline{\Omega}$ by Carathéodory’s Therorem. We
say $\partial\Omega$ has a cusp at $F(e^{it})$, if

$\lim_{s\to t^{+}}\arg[F(e^{is})-F(e^{it})]-\lim_{s\to
t^{-}}\arg[F(e^{is})-F(e^{it})]=0\text{ mod}(2\pi).$

######  Definition 15.

A (closed) analytic arc is the image of the interval $[0,1]\times\\{0\\}$
under a conformal map defined in an open set containing $[0,1]\times\\{0\\}$.

######  Definition 16.

Let $\Omega$ be a simply connected Jordan domain in the plane such
$\partial\Omega$ has no cusps and can be decomposed into four analytic arcs
$A,\tau_{1},B,\tau_{2}$, listed in counter-clockwise order. We will call
$\Omega$ an analytic quadrilateral.

We wish to approximate an analytic quadrilateral with an orthodiagonal map. To
define this, we must specify how we will measure distances. The notation
dist${}_{\hat{G}}(x,y)$ means the minimum length of all paths between $x$ and
$y$ that remain in $\hat{G}$. So the notation
dist${}_{\hat{G}}(S_{1},A)\leq\delta$ means that for all $x\in S_{1}$, there
is a point $y\in A$ with dist${}_{\hat{G}}(x,y)\leq\delta$ and vice versa.

######  Definition 17.

Let $G$ be a finite orthodiagonal map with boundary arcs
$S_{1},T_{1},S_{2},T_{2}$, and let $\Omega$ be an analytic quadrilateral with
boundary arcs $A,\tau_{1},B,\tau_{2}$. We say that
$(G;S_{1},T_{1},S_{2},T_{2})$ approximates $(\Omega;A,\tau_{1},B,\tau_{2})$
within distance $\delta$, when $\overline{\Omega}\subset\hat{G}$ and
$\mathrm{dist}_{\hat{G}}(S_{1},A)$, $\mathrm{dist}_{\hat{G}}(S_{2},B)$, and
$\mathrm{dist}_{\hat{G}}(T_{k},\tau_{k})$, for $k=1,2$, are bounded above by
$\delta$.

###  2.6 Linearly connected and John domains

A simply connected domain $\Omega$ is linearly connected if there exists $K>0$
so that any two points $z,w\in\Omega$ can be connected by a curve
$\gamma\subset\Omega$ with diam$(\gamma)\leq K|z-w|$. On the other hand, a
bounded simply connected domain $\Omega$ is a John domain if there exists
$J>0$ so that every crosscut $\kappa$ of $\Omega$ satisfies
$\operatorname{diam}(H)\leq J\operatorname{diam}(\kappa)$, where $H$ is one of
the two components of $\Omega\setminus\kappa$. The inner domain of a piecewise
smooth Jordan curve is linearly connected if and only if it has no inward-
pointing cusps, and it is a John domain if and only if there are no outward
pointing cusps. (See Chapter 5 of [16].) Thus an analytic quadrilateral
$\Omega$, as defined in Definition 16, is a linearly connected John domain.
This geometric control gives us the Hölder continuity of some relevant
functions. Note that all the domains we consider are Jordan domains. Therefore
by Carathéodory’s Theorem, all the associated conformal maps extend as
homeomorphisms of the closures, and in particular they are well-defined on the
boundary.

######  Proposition 18.

Let $\Omega$ be an analytic quadrilateral with boundary arcs
$A,\tau_{1},B,\tau_{2}$. Let $\psi$ be a conformal map of $\Omega$ onto the
rectangle $R=(0,1)\times(0,m)$ taking the arcs $A$ and $B$ to the vertical
sides of the rectangle, and let $h_{c}:=\operatorname{Re}\psi$. Then there
exists $C>0$ so that

$|h_{c}(z)-h_{c}(w)|\leq C|z-w|^{1/4}\;\;\;\text{ for all
}z,w\in\overline{\Omega}.$

Further, there exists $p\in(0,1/2]$ and $C>0$ so that

$|\psi^{-1}(z)-\psi^{-1}(w)|\leq C|z-w|^{p}\;\;\;\text{ for all
}z,w\in\overline{R}.$

###### Proof.

Let $F_{\Omega}:\mathbb{D}\to\Omega$ and $F_{R}:\mathbb{D}\to R$ be conformal
maps with $\psi=F_{R}\circ F_{\Omega}^{-1}$. The conformal map $F_{R}$ is
known to be Hölder continuous with exponent $1/2$. Since $\Omega$ is a
linearly connected domain, Theorem 5.7 of [16] implies that $F_{\Omega}$
satisfies a lower Hölder condition with exponent $b<2$ on
$\partial\mathbb{D}$, that is,

$|F_{\Omega}(z)-F_{\Omega}(w)|\geq C|z-w|^{b}$

for $z,w\in\partial\mathbb{D}$. This is equivalent to a Hölder condition on
$F_{\Omega}^{-1}$ with exponent $1/b>1/2$ on $\partial\Omega$. Hence,

$|h_{c}(x)-h_{c}(y)|\leq C|x-y|^{1/4}$

on $\partial\Omega$. Then Theorem 1 in [12] proves the first result.

For the second statement, we use the same argument applied to
$\psi^{-1}=F_{\Omega}\circ F_{R}^{-1}$. Recall that every John domain is a
Hölder domain, which by definition means that $F_{\Omega}$ is Hölder
continuous on $\overline{\mathbb{D}}$. ∎

##  3 Non-crossing minimal subfamilies

In this section we assume that the finite network $G=(V,E,\sigma)$ is a plane
graph (i.e. $G$ is embedded in the plane with no edges crossing.) Further, we
assume that there are nonempty sets of boundary vertices $A$ and $B$ defined
as follows: $A=S_{1}\cap V$ and $B=S_{2}\cap V$, where $S_{1},S_{2}$ are two
disjoint arcs from a Jordan curve $\mathcal{C}$ that encircles $G$ in the
closed outer face of $G$. We will refer to $G=(V,E,\sigma,A,B)$ as a plane
network with boundary. Note that when we specialize to orthodiagonal maps
later, we take the primal graph as our network and the boundary of the outer
face as the Jordan curve.

Let $F$ be the conformal map from the interior of $\mathcal{C}$ to a rectangle
$(0,L)\times(0,1)$ that takes $S_{1}$ to the left side $\\{0\\}\times[0,1]$
and $S_{2}$ to the right side $\\{L\\}\times[0,1]$. Given a path $\gamma$ from
$A$ to $B$, the slit vertical strip $\\{z\,:\,\text{Re}(z)\in[0,L]\\}\setminus
F(\gamma)$ has two connected components. Let $U_{\gamma}$ be the connected
component containing $-i$ (i.e. the component “under” $F(\gamma)$). Then for a
subgraph $H$ of $G$ and a path $\gamma$ in $H$ from $A$ to $B$, we say that
$\gamma$ is the top path in $H$ from $A$ to $B$ if
$F(H)\subset\overline{U_{\gamma}}$. We say that two paths from $A$ to $B$ are
non-crossing when one path is the top path in the subgraph induced by these
two paths. A family of paths is non-crossing when every pair of paths in the
family is non-crossing. We note that two non-crossing paths are allowed to
share vertices and edges. Also we say that an edge $e$ is below a path
$\gamma$ if $\gamma$ is the top path of the graph induced by
$\\{e\\}\cup\gamma$.

######  Remark 19.

Two paths $\gamma_{1}$ and $\gamma_{2}$ are crossing if one can find
$e_{1}\in\gamma_{1}\setminus\gamma_{2}$ and
$e_{2}\in\gamma_{2}\setminus\gamma_{1}$, so that $e_{1}$ is below $\gamma_{2}$
and $e_{2}$ is below $\gamma_{1}$.

###  3.1 Non-crossing minimal subfamily algorithm

To establish the existence and uniqueness of the non-crossing minimal
subfamily in Proposition 1, we will use the following algorithm that outputs
the desired minimal subfamily given the extremal density $\rho^{*}$ for the
family of paths from $A$ to $B$.

Non-crossing Minimal Subfamily Algorithm.

Input: plane network with boundary $G=(V,E,\sigma,A,B)$ and the extremal
density $\rho^{*}$ for the family of paths from $A$ to $B$ in $G$.

Initialization: set $r(e)=\sigma(e)\rho^{*}(e)$ for each edge $e\in E$.

Iteration:

  1. 1.

Remove any edges $e$ for which $r(e)=0$ from $G$ (call such edges zero edges).

  2. 2.

If there is one, add the top path $\gamma$ in $G$ to a growing family
$\Gamma$; if not, terminate.

  3. 3.

For the top path $\gamma$ identified in step 2, set
$m(\gamma)=\min_{e\in\gamma}r(e)$.

  4. 4.

For each edge $e$ in the top path $\gamma$ identified in step 2, update $r(e)$
to be $r(e)-m(\gamma)$.

  5. 5.

Repeat.

Output: curve family $\Gamma$ and pmf $\mu^{*}(\gamma)=m(\gamma)/M$, where
$M:=\sum_{\gamma\in\Gamma}m(\gamma)=\operatorname{Mod}_{2,\sigma}(\Gamma)$.

In our proof of Proposition 1, we will show that this algorithm outputs a
minimal subfamily $\Gamma$ of non-crossing paths and that $\mu^{*}$ is the
optimal pmf guaranteed by Proposition 8. Before getting to the proof, we show
that we can direct the non-zero edges of $G$ to obtain a directed acyclic
graph (once we remove any edges with $\rho^{*}(e)=0$). Informally, the edges
are directed by the current flow defined by the solution of the discrete
Dirichlet problem.

######  Lemma 20.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary. Let $\rho^{*}$ be
the unique extremal density for the family of paths from $A$ to $B$ in $G$,
and let $E_{0}=\\{e\in E\,:\,\rho^{*}(e)=0\\}$. Then there exists an
orientation of all edges in $E\setminus E_{0}$ so that the following hold:

  1. (i)

At any vertex $v$ not in $A\cup B$, the sum of $\sigma(e)\rho^{*}(e)$ over the
edges leading into $v$ equals the sum of $\sigma(e)\rho^{*}(e)$ over the edges
leading out of $v$.

  2. (ii)

All edges in $E\setminus E_{0}$ with exactly one vertex in $A$ (resp. B) are
oriented away from $A$ (resp. towards $B$).

  3. (iii)

Let $\gamma$ be a path in $G\setminus E_{0}$ from $A$ to $B$. Then $\gamma$ is
a directed path from $A$ to $B$ if and only if $\gamma$ has $\rho^{*}$-length
1, i.e. $\ell_{\rho^{*}}(\gamma)=1$.

  4. (iv)

$G\setminus E_{0}$ is a directed acyclic graph.

###### Proof.

Let $h$ be the solution to the discrete Dirichlet problem on $V\setminus(A\cup
B)$ with $h|_{A}=0$ and $h|_{B}=1$, and recall, from (8), that
$\rho^{*}(vw)=|h(w)-h(v)|$. For each edge $e$ incident to vertices $v,w$ with
$\rho^{*}(e)\neq 0$, we orient $e$ from $v$ to $w$ when $h(w)-h(v)>0$.

Let $f$ be the corresponding current flow:
$f(vw)=\sigma(vw)\left[h(w)-h(v)\right]$ for $vw\in\vv{E},$ which we extend to
$V\times V$, by setting $f(wv)=-f(vw)$ when $vw\in\vv{E}$ and zero on pairs of
nodes not connected by an edge. For $v\in V\setminus(A\cup B)$, let
$E_{v}^{\text{in}}$ be the edges directed into $v$ and $E_{v}^{\text{out}}$ be
the edges directed out of $v$. Then, by harmonicity of $h$ at $v$,

$\displaystyle 0$ $\displaystyle=\sum_{w:\,vw\in
E}\sigma(vw)[h(w)-h(v)]=\sum_{w:\,vw\in E}f(vw)$ $\displaystyle=\sum_{e\in
E_{v}^{\text{out}}}\sigma(e)\rho^{*}(e)-\sum_{e\in
E_{v}^{\text{in}}}\sigma(e)\rho^{*}(e),$

which proves (i).

Statement (ii) follows from the discrete maximum principle, which implies that
$0\leq h\leq 1$. To show (iii), let $\gamma$ be a path from $A$ to $B$ in
$G\setminus E_{0}$ given by vertices $v_{0},v_{1},\cdots,v_{k}$. Then

$\ell_{\rho^{*}}(\gamma)=\sum_{j=1}^{k}\bigl{\lvert}h(v_{j})-h(v_{j-1})\bigr{\rvert}\geq\sum_{j=1}^{k}h(v_{j})-h(v_{j-1})=1.$

There is equality in the above equation if and only if $\gamma$ is a directed
path from $A$ to $B$.

To show that $G\setminus E_{0}$ has no cycles (respecting the edge
directions), we will assume for the sake of contradiction that there is a
cycle of directed edges $e_{1},e_{2},\cdots,e_{k}$ with each $e_{j}$ directed
from $v_{j-1}$ to $v_{j}$. Then

$0<\sum_{j=1}^{k}\rho^{*}(e_{j})=\sum_{j=1}^{k}h(v_{j})-h(v_{j-1})=h(v_{k})-h(v_{0})=0,$

since $v_{k}=v_{0}$. This is a contradiction. ∎

######  Lemma 21.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary, and let $E\setminus
E_{0}$ be oriented as in Lemma 20. Assume that $H$ is a connected subgraph of
$G\setminus E_{0}$ so that all the source vertices of $H$ are in $A$ and all
the sink vertices of $H$ are in $B$. Then the following hold:

  1. (i)

The top path of $H$ is a directed path from $A$ to $B$.

  2. (ii)

Let $\tilde{\Gamma}$ be a family of non-crossing paths that does not contain
the top path. Then there is an edge of the top path that is not contained in
any path in $\tilde{\Gamma}$.

###### Proof.

Let $\gamma$ be the top path of $H$. Suppose $\gamma$ is not a directed path
from $A$ to $B$, and let $e$ be an edge in $\gamma$ that is oriented “to the
left”. Since all the sources (resp. sinks) are in $A$ (resp. $B$), there must
be some directed path in $H$ from $A$ to $B$ that contains $e$. However, the
planarity of $H$ and the fact that $\gamma$ is the top path imply that $H$
must contain a directed cycle. This contradicts the fact that $H$ is a
directed acyclic graph from Lemma 20 (iv). Therefore, $\gamma$ is a directed
path from $A$ to $B$, which establishes (i).

We will prove (ii) by contradiction. Assume that every edge of $\gamma$ is
contained in some path in $\tilde{\Gamma}$. Let $\tilde{\gamma}$ be a path
from $\tilde{\Gamma}$ that contains the longest initial segment in common with
$\gamma$. Let $e_{0}$ be the edge in $\gamma$ immediately after this initial
segment (such an edge must exist since $\tilde{\gamma}\neq\gamma$). Further
$\tilde{\gamma}$ must contain an edge below $\gamma$ after the initial
segment. Now there must be some other path in $\tilde{\Gamma}$ that contains
$e_{0}$, and by the choice of $\tilde{\gamma}$, this path must have some edge
below $\tilde{\gamma}$. By Remark 19, this implies that $\tilde{\Gamma}$
contains two paths that cross, which is not possible.

∎

######  Lemma 22.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary, and let $E\setminus
E_{0}$ be oriented as in Lemma 20. After each completion of step 1 in the Non-
crossing Minimal Subfamily Algorithm, the connected components of the
resulting graph are isolated vertices or subgraphs with all their source
vertices in $A$ and all their sink vertices in $B$.

###### Proof.

The first time we complete step 1, we remove all edges with $\rho^{*}(e)=0$.
Let $H$ be one of the resulting connected components with at least one edge.
Since $H$ is a directed acyclic graph by Lemma 20, $H$ must contain at least
one source vertex and at least one sink vertex.111The fact that a connected
directed acyclic graph with at least one edge contains a source and sink is
straightforward to prove by contradiction. For instance, suppose there is no
sink. Then every vertex has at least one edge oriented away from the vertex.
Since the graph is acyclic, we can then create a directed path that contains
every vertex. However, we obtain a contradiction since the last vertex in the
path has an edge oriented outward, allowing us to create a cycle. The argument
showing there is a source vertex is similar. Properties (i) and (ii) of Lemma
20 imply that all the source vertices of $H$ must be in $A$ and all the sink
vertices of $H$ must be in $B$.

Now we proceed by induction. Suppose that the statement is true for the first
$k$ times we complete step 1, and consider a connected component $H$ obtained
after the $(k+1)$th iteration of step 1. We assume that $H$ has at least one
edge, and we will use contradiction to show that all the source vertices are
in $A$ and all the sink vertices are in $B$. For the sake of simplicity,
assume that $H$ contains a source vertex $v$ that is not in $A$ (as a similar
argument applies in the other case). In completing the first $k$ iterations of
the algorithm, we have identified the top paths
$\gamma_{1},\cdots,\gamma_{k}$. By the induction hypothesis and Lemma 21 (i),
these paths are all directed paths from $A$ to $B$. Let
$\Gamma_{v}\subset\\{\gamma_{1},\cdots,\gamma_{k}\\}$ be the collection of
these paths that pass through $v$. Let $E_{v}^{\text{in}}$ be the edges
directed into $v$ and $E_{v}^{\text{out}}$ be the edges directed out of $v$.
Then, since all of the edges directed towards $v$ have been removed by the end
of the $(k+1)$th iteration of step 1,

$\sum_{\gamma\in\Gamma_{v}}m(\gamma)=\sum_{e\in
E_{v}^{\text{in}}}\sigma(e)\rho^{*}(e).$

Namely, an edge $e$ gets removed once the sum of the weights of all the paths
in $\Gamma_{v}$ that are using $e$ has exhausted the initial quantity
$\sigma(e)\rho^{*}(e)$. Moreover, by Lemma 20 (i),

$\sum_{e\in E_{v}^{\text{in}}}\sigma(e)\rho^{*}(e)=\sum_{e\in
E_{v}^{\text{out}}}\sigma(e)\rho^{*}(e).$

However, this means that we have also removed all the edges directed out from
$v$, contradicting the fact that $H$ is a connected component larger than an
isolated vertex. ∎

###### Proof of Proposition 1.

The Non-crossing Minimal Subfamily Algorithm must terminate, because we remove
at least one edge from the graph during every iteration (except possibly the
first one). When the algorithm terminates, there are no edges remaining in the
graph by Lemma 22. Thus for each $e$ edge in $E\setminus E_{0}$, the
algorithm’s output path family $\Gamma$ must include a path containing $e$.
Furthermore, we must have

$\sigma(e)\rho^{*}(e)=\sum_{\gamma\in\Gamma}m(\gamma)\mathbbm{1}_{\\{e\in\gamma\\}}.$
(10)

We will apply Beurling’s Criterion (Theorem 5) to show that $\Gamma$ is a
Beurling subfamily. From Lemmas 22 and 21 (i), each path in $\Gamma$ is a
directed path from $A$ to $B$. Therefore Lemma 20 (iii) implies that the paths
in $\Gamma$ all have length 1. Now we will check the second condition for
$\Gamma$ to be a Beurling subfamily. Let $h\in\mathbb{R}^{E}$ with

$\sum_{e\in\gamma}h(e)\geq 0$

for all $\gamma\in\Gamma$. We apply this assumption and (10) to obtain the
needed condition:

$\displaystyle\sum_{e\in E}h(e)\rho^{*}(e)\sigma(e)$ $\displaystyle=\sum_{e\in
E}h(e)\sum_{\gamma\in\Gamma}m(\gamma)\mathbbm{1}_{\\{e\in\gamma\\}}$
$\displaystyle=\sum_{\gamma\in\Gamma}m(\gamma)\sum_{e\in\gamma}h(e)$
$\displaystyle\geq 0.$

Thus $\Gamma$ is a Beurling subfamily, which implies that
$\operatorname{Mod}_{2,\sigma}(\Gamma)$ is the same as
$\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$, where $\Gamma(A,B)$ is the
family of all paths from $A$ to $B$ in $G$. Moreover, if we set

$\mu^{*}(\gamma):=\frac{m(\gamma)}{\operatorname{Mod}_{2,\sigma}(\Gamma)}\;\text{
for all }\gamma\in\Gamma,$ (11)

then from (10) we have that $\mu^{*}$ satisfies (4), which implies that

$\operatorname{Mod}_{2,\sigma}(\Gamma)=\sum_{e\in
E}\sigma(e)\rho^{*}(e)^{2}=\sum_{e\in
E}\sigma(e)^{-1}\mathbb{P}_{\mu^{*}}(e\in\underline{\gamma})^{2}.$

Hence, in view of (1), we have shown that $\mu^{*}$ is an optimal pmf for
$\operatorname{Mod}_{2,\sigma}(\Gamma)$.

Finally, we want to show that $\Gamma$ is minimal for
$\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$. Note that $\Gamma$ must contain
a minimal subfamily $\tilde{\Gamma}$. By Proposition 8, there is a unique
optimal pmf $\tilde{\mu}$ supported on $\tilde{\Gamma}$, and by minimality
$\tilde{\mu}(\gamma)>0$ for all $\gamma\in\tilde{\Gamma}$. Then, Lemma 23
below shows that $\Gamma=\tilde{\Gamma}$. Hence, $\Gamma$ itself must be
minimal. Also, Lemma 23 shows that the pmf $\mu^{*}$ defined in (11) must be
the optimal pmf guaranteed by Proposition 8 for $\Gamma$. ∎

######  Lemma 23.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary and let
$\Gamma(A,B)$ be the family of all paths from $A$ to $B$ in $G$. Let
$\Gamma\subset\Gamma(A,B)$ be the path family output by the Non-crossing
Minimal Subfamily Algorithm, let $\tilde{\Gamma}\subset\Gamma(A,B)$ be any
subfamily with non-crossing paths, such that there is an optimal pmf
$\tilde{\mu}$ strictly supported on $\tilde{\Gamma}$, meaning that
$\tilde{\mu}(\gamma)>0$ for all $\gamma\in\tilde{\Gamma}$. Then
$\tilde{\Gamma}=\Gamma$ and

$\tilde{\mu}(\gamma)=\frac{m(\gamma)}{\operatorname{Mod}_{2,\sigma}(\Gamma)}\;\text{
for all }\gamma\in\Gamma.$

###### Proof.

Let $\gamma_{1},\gamma_{2},\cdots,\gamma_{n}$ be the paths of $\Gamma$ in the
order that they are identified in the algorithm. We will begin by proving that
$\gamma_{1}$ must be in $\tilde{\Gamma}$. Since the edges where $\rho^{*}=0$
are removed in step 1 of the algorithm, all the edges of $\gamma_{1}$ have
nonzero $\rho^{*}$. Therefore, by (4), all the edges of $\gamma_{1}$ must be
contained in paths of $\tilde{\Gamma}$. Thus, Lemmas 22 and 21 (ii) show that
$\gamma_{1}\in\tilde{\Gamma}$. Furthermore, Lemma 21 (ii) implies that there
must be some edge $e$ in $\gamma_{1}$ that is contained in no other path in
$\tilde{\Gamma}$. Using (4) for this edge $e$, we have

$\tilde{\mu}(\gamma_{1})\operatorname{Mod}_{2,\sigma}(\Gamma)=\sigma(e)\rho^{*}(e)=\min_{a\in\gamma_{1}}\sigma(a)\rho^{*}(a)=m(\gamma_{1}),$
(12)

and hence
$\tilde{\mu}(\gamma_{1})=m(\gamma_{1})/\operatorname{Mod}_{2,\sigma}(\Gamma)$.

We will prove by induction on $k$ that $\gamma_{1},\cdots,\gamma_{k}$ are in
$\tilde{\Gamma}$ with
$\tilde{\mu}(\gamma_{j})=\frac{m(\gamma_{j})}{\operatorname{Mod}_{2,\sigma}(\Gamma)}$
for $j=1,\cdots,k$. Assume this statement is true for $k-1$. Consider the
point in the algorithm where we have just identified $\gamma_{k}$ as the top
path, and consider an edge $e\in\gamma_{k}$. Since $e$ was not removed yet,

$r(e)=\sigma(e)\rho^{*}(e)-\sum_{j<k:\,e\in\gamma_{j}}m(\gamma_{j})>0.$

Therefore,

$\sum_{j<k:\,e\in\gamma_{j}}\tilde{\mu}(\gamma_{j})=\frac{1}{\operatorname{Mod}_{2,\sigma}(\Gamma)}\sum_{j<k:\,e\in\gamma_{j}}m(\gamma_{j})<\frac{\sigma(e)\rho^{*}(e)}{\operatorname{Mod}_{2,\sigma}(\Gamma)}=\sum_{\gamma:\,e\in\gamma}\tilde{\mu}(\gamma),$

where the last equality is from (2). This means that there must be a path in
$\tilde{\Gamma}\setminus\\{\gamma_{1},\cdots,\gamma_{k-1}\\}$ that contains
$e$, since we have not accumulated enough mass from the paths
$\gamma_{1},\cdots,\gamma_{k-1}$. Thus Lemmas 22 and 21 (ii) show that
$\gamma_{k}\in\tilde{\Gamma}$, and they further imply that there must be some
edge in $\gamma_{k}$ that is contained in no other path in
$\tilde{\Gamma}\setminus\\{\gamma_{1},\cdots,\gamma_{k-1}\\}$. Such edges
$e\in\gamma_{k}$ will satisfy

$r(e)=\min_{a\in\gamma_{k}}r(a),$

and therefore

$\tilde{\mu}(\gamma_{k})=\frac{m(\gamma_{k})}{\operatorname{Mod}_{2,\sigma}(\Gamma)}.$
(13)

This completes the induction proof. Therefore, we have shown that
$\Gamma\subset\tilde{\Gamma}$. Moreover, (13) implies that
$\tilde{\mu}(\gamma)=0$ for $\gamma\in\tilde{\Gamma}\setminus\Gamma$, and
therefore, $\Gamma=\tilde{\Gamma}$. ∎

###  3.2 Scaling limit of non-crossing minimal subfamilies

In this section, we will prove Theorem 2 (modulo Theorem 3). Assume the
hypotheses of Theorem 2 hold. Then by Proposition 8 there is a unique extremal
density $\rho_{n}$ on the edges of $G_{n}^{\bullet}$ and a unique optimal pmf
$\mu_{n}$ on $\Gamma_{n}$, which are related by

$\frac{\sigma_{n}(e)\rho_{n}(e)}{\operatorname{Mod}_{2,\sigma_{n}}(\Gamma_{n})}=\mathbb{P}_{\mu_{n}}\left[\gamma\text{
contains }e\right]$ (14)

for all $e\in G^{\bullet}_{n}$. We will prove that $\mu_{n}$ converges to the
transverse measure on the family of extremal curves in $\Omega$ as
$\epsilon\to 0$.

We begin with a lemma about a useful dual minimal subfamily. Since this lemma
only relates to the orthodiagonal map (and not to the underlying domain), for
ease of reading, we omit the subscript/superscript $n$.

######  Lemma 24.

Let $G$ be an orthodiagonal map with boundary arcs $S_{1},T_{1},S_{2},T_{2}$.
Let $\sigma$ be the canonical weights on the edges of $G^{\bullet}$ as in
Equation (9). Consider the family $\Gamma(A,B)$ of all paths in $G^{\bullet}$
from $A=V^{\bullet}\cap S_{1}$ to $B=V^{\bullet}\cap S_{2}$, and let
$\Gamma\subset\Gamma(A,B)$ be the unique non-crossing minimal subfamily
produced by Proposition 1. Likewise, consider the dual graph $G^{\circ}$ with
edge-weights as defined in Equation (9), and let $\Gamma^{\circ}$ be the
unique non-crossing minimal subfamily of all paths in $G^{\circ}$ from
$\alpha=V^{\circ}\cap T_{1}$ to $\beta=V^{\circ}\cap T_{2}$. Then

  1. (i)

$\operatorname{Mod}_{2,\sigma}(\Gamma)\cdot\operatorname{Mod}_{2,\sigma}(\Gamma^{\circ})=1.$

  2. (ii)

Let $\mu$ be the extremal density for $\Gamma$ and let $\eta$ be the extremal
density for $\Gamma^{\circ}$. Then

$\eta(e^{\circ})=\frac{\sigma(e)\rho(e)}{\operatorname{Mod}_{2}(\Gamma)}=\mathbb{P}_{\mu}\left[\gamma\text{
contains }e\right]=\sum_{\gamma\in\Gamma:\,e\in\gamma}\mu(\gamma),$ (15)

for all pairs of dual edges $(e,e^{\circ})\in E^{\bullet}\times E^{\circ}$.

  3. (iii)

Fix a curve $\gamma^{\circ}\in\Gamma^{\circ}$. Then, every path
$\gamma\in\Gamma$ contains exactly one edge $e$ whose dual edge $e^{\circ}$ is
in $\gamma^{\circ}.$

###### Proof.

From Fulkerson duality, see [3, Section 4.2], we have that

$\operatorname{Mod}_{2,\sigma}(\Gamma)\cdot\operatorname{Mod}_{2,\sigma^{-1}}(\hat{\Gamma})=1,$
(16)

where $\hat{\Gamma}$ is the family of all $AB$-cuts. Recall that $S\subset
V^{\bullet}$ is an $AB$-cut if $A\subset S$ and $B\cap S=\emptyset$. Moreover,
the usage of $S$ is defined by $\mathcal{N}(S,e)=\mathbbm{1}_{\\{e\in\delta
S\\}}$ where $\delta S=\\{e=xy\in E^{\bullet}:|S\cap\\{x,y\\}|=1\\}$. Note
that we can define a bijective map $r:E^{\bullet}\to E^{\circ}$ by setting
$r(e):=e^{\circ}$ which is the dual edge to $e$. Therefore, applying the
transformation $r$ to each edge in $\delta S$, we obtain an object $r(S)$ on
the dual graph $G^{\circ}$. We want to show that $r(S)$ contains a path
connecting $\alpha$ and $\beta$ and conversely every such path comes from a
cut $S$. First, every node in $V^{\bullet}\setminus\partial G$ is contained in
the interior of a closed dual face for $G^{\circ}$. In the case when the node
$v$ is in $\partial G$, let $C$ be the closure of the component of $v$ in the
intersection of the unbounded face for $G^{\circ}$ with $\hat{G}$. Take a
small ball $D_{v}$ centered at $v$ that is disjoint from any other vertex and
define $C\cup D_{v}$ to be the dual face containing $v$. By taking the union
of all the dual faces containing a node in $S$ and then taking the interior of
this set, we obtain an open set $U$ containing $S$ (and hence $A$) in its
interior, such that $B\subset{\rm int}(U^{c})$, where $U^{c}$ is the
complement of $U$. Now we can apply Lemma 2.4 in [9], where the quadrilateral
is our set $\hat{G}$, and obtain a simple curve $\gamma$ connecting $\alpha$
and $\beta$ with $\gamma\subset\partial U\cap\hat{G}$. By construction we see
that $\gamma$ is a union of dual edges, and hence is a simple path in
$G^{\circ}$. Conversely, every such path can be continued to get a Jordan
domain with $A$ in its interior and $B$ in its exterior.

Finally, note that, by Equation (9), for every pair of dual edges
$(e,e^{\circ})$ we have $\sigma(e^{\circ})=\sigma(e)^{-1}$. Therefore, we
conclude that
$\operatorname{Mod}_{2,\sigma^{-1}}(\hat{\Gamma})=\operatorname{Mod}_{2,\sigma}(\Gamma^{\circ})$
and (i) follows from (16).

Moreover, (15) follows from (5) and [3, Theorem 4 and Theorem 7], by letting
$\eta^{\bullet}$ be the extremal density for $\hat{\Gamma}$ and setting
$\eta(e^{\circ})=\eta^{\bullet}(r(e^{\circ}))$ (thinking of $r$ as an
idempotent operation).

Lastly, fix $\gamma^{\circ}\in\Gamma^{\circ}$. As in the previous paragraph,
$\gamma^{\circ}$ can be continued outside $\hat{G}$ to get a Jordan domain
with $A$ in its interior and $B$ in its exterior. Then each $\gamma\in\Gamma$
must cross this Jordan curve through an edge in $\gamma^{\circ}$, implying
that $\gamma$ must contain at least one such edge $e$ with
$e^{\circ}\in\gamma^{\circ}.$ Since every minimal subfamily is also a Beurling
subfamily, $\gamma^{\circ}$ has length 1 under $\eta$. Thus by (15),

$\displaystyle 1=\sum_{e\,:\,e^{\circ}\in\gamma^{\circ}}\eta(e^{\circ})$
$\displaystyle=\sum_{e\,:\,e^{\circ}\in\gamma^{\circ}}\sum_{\gamma\in\Gamma\,:\,e\in\gamma}\mu(\gamma)$
$\displaystyle=\sum_{\gamma\in\Gamma}\mu(\gamma)\left[\text{ \\# of edges
}e\in\gamma\text{ with }e^{\circ}\in\gamma^{\circ}\right]\geq 1.$

Therefore, the number of edges $e\in\gamma$ with $e^{\circ}\in\gamma^{\circ}$
must be identically equal to one. ∎

We are now ready to prove Theorem 2. This proof will rely on results that are
established in Section 4.

###### Proof of Theorem 2..

We begin by assuming that $\hat{G}_{n}=\overline{\Omega}$, and we will
complete the proof in this case. At the end of the proof, we will discuss how
to modify the proof for the more general case that
$\hat{G}_{n}\supset\overline{\Omega}$.

Let $\phi$ be the conformal map from $\Omega$ onto the rectangle
$R=(0,L)\times(0,1)$ so that (after extending $\phi$ to the boundary)
$\phi(A)$ is the left side of $R$ and $\phi(B)$ is the right side of $R$. Let
$h_{n}$ be the solution to the discrete Dirichlet problem on ${\rm
int}(V_{n}^{\bullet})$ with $h_{n}|_{A_{n}}=0$ and $h_{n}|_{B_{n}}=L$.
Similarly let $\tilde{h}_{n}$ be the solution to the discrete Dirichlet
problem on ${\rm int}(V_{n}^{\circ})$ with $\tilde{h}_{n}|_{\alpha_{n}}=0$ and
$\tilde{h}_{n}|_{\beta_{n}}=1$ for $\alpha_{n}=V^{\circ}\cap T_{1}^{n}$ and
$\beta_{n}=V^{\circ}\cap T_{2}^{n}$. Let $\Gamma_{n}^{\circ}$ be the unique
non-crossing minimal subfamily of all paths in $G_{n}^{\circ}$ from
$\alpha_{n}$ to $\beta_{n}$, and let $\rho_{n}$ (resp. $\eta_{n}$) be the
extremal density for $\Gamma_{n}$ (resp. $\Gamma_{n}^{\circ}$). Note that by
(8),

$\rho_{n}(v_{1}v_{2})=\frac{1}{L}|h_{n}(v_{1})-h_{n}(v_{2})|\;\text{ and
}\;\eta_{n}(w_{1}w_{2})=|\tilde{h}_{n}(w_{1})-\tilde{h}_{n}(w_{2})|$

for all $v_{1}v_{2}\in E^{\bullet}_{n}$ and $w_{1}w_{2}\in E^{\circ}_{n}$. By
Theorem 3, we know that $h_{n}$ approximates $\text{Re }\phi$ and
$\tilde{h}_{n}$ approximates $\text{Im }\phi$.

The non-crossing condition for $\Gamma_{n}$ allows us to order the paths in
$\Gamma_{n}$: $\gamma<\gamma^{\prime}$ means $\gamma\neq\gamma^{\prime}$ and
$\gamma^{\prime}$ is the top path in the subgraph consisting of these two
paths. Equivalently, $\gamma<\gamma^{\prime}$ means that $\gamma$ is
identified after $\gamma^{\prime}$ in the Non-crossing Minimal Subfamily
Algorithm.

Fix $\gamma^{\prime}\in\Gamma_{n}$ and set

$p=\sum_{\gamma\leq\gamma^{\prime}}\mu_{n}(\gamma).$

Let $e$ be an edge in $\gamma^{\prime}$, with dual edge $e^{\circ}$. Now
consider a curve $\gamma^{\circ}\in\Gamma_{n}^{\circ}$ that contains
$e^{\circ}$. Under the orientation of edges induced by $\tilde{h}_{n}$ in
Lemma 20, $\gamma^{\circ}$ is an oriented path from $\alpha_{n}$ to
$\beta_{n}$ by Lemma 20 (iii). Let $e^{\circ}=xy$, oriented with respect to
this orientation. Consider the portion of $\gamma^{\circ}$ before $x$ and let
this be given by vertices $u_{0},\cdots,u_{k}=x$. Recall that by Lemma
24(iii), every path $\gamma\in\Gamma_{n}$ contains exactly one edge $a$ with
$a^{\circ}\in\gamma^{\circ}.$ This along with (15) implies that

$\tilde{h}_{n}(x)=\sum_{j=1}^{k}\tilde{h}_{n}(u_{j})-\tilde{h}_{n}(u_{j-1})=\sum_{j=1}^{k}\eta_{n}(u_{j}u_{j-1})\leq\sum_{\gamma\leq\gamma^{\prime}}\mu_{n}(\gamma)=p.$

Similarly,

$\tilde{h}_{n}(y)\geq p.$

Furthermore, by Theorem 3 and Proposition 18,

$\displaystyle|\tilde{h}_{n}(y)-\tilde{h}_{n}(x)|$
$\displaystyle\leq|\tilde{h}_{n}(y)-\text{Im }\phi(y)|+|\text{Im
}\phi(y)-\text{Im }\phi(x)|+|\text{Im }\phi(x)-\tilde{h}_{n}(x)|$
$\displaystyle\leq\frac{C}{\log^{1/2}\left(\operatorname{diam}(\Omega)/\epsilon_{n}\right)}+C\epsilon_{n}^{1/4}$
(17) $\displaystyle\leq\delta_{n},$

where $\delta_{n}=\frac{C}{\log^{1/2}\left(1/\epsilon_{n}\right)}$ for some
constant $C$ depending on $\Omega$. So

$p\leq\tilde{h}_{n}(y)\leq p+\delta_{n}\;\;\text{ and
}\;\;p-\delta_{n}\leq\tilde{h}_{n}(x)\leq p.$

Now let $u$ be a point on $e$. Then $|u-y|\leq 2\epsilon_{n}$ and by arguing
as in (3.2) we obtain

$\displaystyle\text{Im }\phi(u)$ $\displaystyle\leq\tilde{h}_{n}(y)+|\text{Im
}\phi(y)-\tilde{h}_{n}(y)|+|\text{Im }\phi(u)-\text{Im }\phi(y)|$
$\displaystyle\leq p+3\delta_{n}.$

Similarly,

$\displaystyle\text{Im }\phi(u)$ $\displaystyle\geq\tilde{h}_{n}(x)-|\text{Im
}\phi(x)-\tilde{h}_{n}(x)|-|\text{Im }\phi(u)-\text{Im }\phi(x)|$
$\displaystyle\geq p-3\delta_{n}.$

Since this hold for all edges of $\gamma^{\prime}$, we have that for any point
$u$ on $\gamma^{\prime}$

$\text{Im }\phi(u)\in[p-3\delta_{n},\,p+3\delta_{n}].$ (18)

We are now ready to consider the limit points of $\Gamma_{n}$ as $n\to\infty$,
which are the curves $\gamma_{\star}$ such that there is an increasing
sequence $n_{k}$ and a sequence $\gamma_{k}\in\Gamma_{n_{k}}$ with
$\gamma_{k}$ converging to $\gamma_{\star}$ (using the Hausdorff distance as
the metric on curves). We claim that these limiting curves are the extremal
curves in $\Omega$:

$\Gamma:=\\{\phi^{-1}(\\{\text{Im}(z)=p\\})\,:\,p\in[0,1]\\}.$

This will be accomplished in two steps. We will first show that the limit
points of $\Gamma_{n}$ are contained in $\Gamma$, and then we will show that
every curve of $\Gamma$ is a limit point of $\Gamma_{n}$.

Suppose that $\gamma_{\star}$ is a limit point of $\Gamma_{n}$. Then there is
a sequence $\gamma_{k}\in\Gamma_{n_{k}}$ with $\gamma_{k}$ converging to
$\gamma_{\star}$. By Proposition 18, this implies that $\phi(\gamma_{k})$ must
converge to $\phi(\gamma_{\star})$. By (18), the oscillation of the imaginary
part of $\phi(\gamma_{k})$ must converge to zero, implying that
Im$\left[\phi(\gamma_{\star})\right]$ must be constant. This shows that
$\gamma_{\star}\in\Gamma$.

To show that every curve in $\Gamma$ is a possible limit, we must choose
$q\in[0,1]$ and find a $\gamma_{n}\in\Gamma_{n}$ that is close to
$\phi^{-1}(\\{\text{Im}(z)=q\\})$. Choose the unique
$\gamma_{n}=\gamma_{n}(q)\in\Gamma_{n}$ so that

$\sum_{\gamma<\gamma_{n}}\mu_{n}(\gamma)<q\;\;\text{ and
}\;\;q\leq\sum_{\gamma\leq\gamma_{n}}\mu_{n}(\gamma)=:p_{n}.$ (19)

Note that if $\gamma^{\prime}\in\Gamma_{n}$ and $e\in\gamma^{\prime}$, then by
(3.2) we have

$\mu_{n}(\gamma^{\prime})\leq\sum_{\gamma:e\in\gamma}\mu(\gamma)=\eta_{n}(e_{\circ})=|\tilde{h}_{n}(y)-\tilde{h}_{n}(x)|\leq\delta_{n}.$

Therefore $p_{n}-q\leq\delta_{n}$, and this together with (18) implies that

$\text{Im }\phi(\gamma_{n})\in[q-4\delta_{n},\,q+4\delta_{n}].$ (20)

Thus $\phi(\gamma_{n})$ converges to $\\{z\in R\,:\,\text{Im}(z)=q\\}$ and so
$\gamma_{n}$ converges to $\phi^{-1}(\\{\text{Im}(z)=q\\})$.

Let $X=\left(\bigcup_{n}\Gamma_{n}\right)\cup\Gamma$. To show $X$ is compact,
we show that every sequence in $X$ has a subsequence converging to an element
in $X$. A sequence in $X$ must satisfy one of the following two cases. In the
first case, there is a subsequence contained in $\bigcup_{n=1}^{N}\Gamma_{n}$
for some $N<\infty$. Since this set has a finite number of elements, this
means there must be a sub-subsequence that is constant, and hence it converges
to an element of $X$. In the second case, there is a subsequence
$\gamma_{n_{k}}$ with
$\gamma_{n_{k}}\in\left(\bigcup_{n=k}^{\infty}\Gamma_{n}\right)\cup\Gamma$. By
(18), there exists $p_{k}\in[0,1]$ so that
Im$\,\phi(\gamma_{n_{k}})\in[p_{k}-3\delta_{k},\,p_{k}+3\delta_{k}]$. Since
$[0,1]$ is compact, we can chose a subsequence of $p_{k}$ so that
$p_{{k_{j}}}\to p\in[0,1]$. This implies that the sub-subsequence
$\gamma_{n_{k_{j}}}$ converges to an element of $\Gamma\subset X$.

We are now ready to consider the convergence of the probability measures
$\mu_{n}$. Recall that $\mu_{n}$ are elements of the set $\mathcal{P}(X)$ of
probability measures on $(X,\mathcal{B}(X))$. We claim that the set of
probability measures $\\{\mu_{n}\\}$ is tight, i.e., for each $\delta>0$ there
exists a compact subset $K_{\delta}$ of $X$ such that $\mu_{n}(K_{\delta})\geq
1-\delta$ for all $n$. Since $X$ is compact, $\\{\mu_{n}\\}$ satisfies this
condition with $K_{\delta}=X$. Therefore, Prohorov’s Theorem (Theorem 83.10 in
[18]) implies that $\\{\mu_{n}\\}$ is conditionally compact (i.e. its closure
is compact) in the space $\mathcal{P}(X)$ equipped with the topology of weak
convergence. This implies the existence of subsequential limits of $\mu_{n}$.

It remains to show that the only possible subsequential limit of $\mu_{n}$ is
the transverse measure on $\Gamma$. To that end, let $\mu$ be a limit of
$\mu_{n_{k}}$, let $r\in(0,1)$, and set

$H_{r}=\\{\gamma\in X\,:\,\phi(\gamma)\subset[0,L]\times[0,r]\\}.$

To characterize $\mu$ as the transverse measure on $\Gamma$, we must simply
show that $\mu(H_{r})=r$ for all $r\in(0,1)$. Since $\mu$ is the limit of
$\mu_{n_{k}}$ under weak convergence, we have that
$\mu_{n_{k}}(H_{r})\to\mu(H_{r})$. Let $q_{1}=r-4\delta_{n}$ and
$q_{2}=r+4\delta_{n}$, and let $\gamma_{n}(q_{1}),\gamma_{n}(q_{2})$ be
defined by (19). Then by (20), $H_{r}$ contains the set

$\\{\gamma\in\Gamma_{n}\,:\,\gamma\leq\gamma_{n}(q_{1})\\},$

and $H_{r}\cap\Gamma_{n}$ is contained in the set

$\\{\gamma\in\Gamma_{n}\,:\,\gamma\leq\gamma_{n}(q_{2})\\}.$

Thus for $p_{n}(q_{1})$ defined by (19),

$\mu_{n}(H_{r})\geq p_{n}(q_{1})\geq q_{1}=r-4\delta_{n},$

implying that $\mu(H_{r})\geq r$. Similarly

$\mu_{n}(H_{r})\leq p_{n}(q_{2})\leq q_{2}+\delta_{n}=r+5\delta_{n},$

implying that $\mu(H_{r})\leq r$. Thus we have shown that $\mu(H_{r})=r$ for
all $r\in(0,1)$, which characterizes the limit measure as the transverse
measure on $\Gamma$.

We end by briefly discussing the modifications needed in the case that
$\hat{G}_{n}\neq\overline{\Omega}.$ This will rely on some of the results
which will be established in Section 4. By using a rotated version of Lemma
29, for $\epsilon_{n}$ small enough, we can analytically extend $\phi$ to a
domain that contains all vertices $z\in G$ that satisfy
$\mathrm{dist}(z,\tau_{1}\cup\tau_{2})\geq\epsilon_{n}^{1/4}$. Then in the
proof above, instead of using Theorem 3, we must utilize Proposition 42, which
is a version of Theorem 3 for the vertices satisfying
$\mathrm{dist}(z,\tau_{1}\cup\tau_{2})\geq\epsilon_{n}^{1/4}$, the extended
function $\phi$, and the harmonic function $\tilde{h}_{n}$. Further, for fixed
$\delta>0$, we define

$K_{\delta}=\left(\bigcup_{n}\\{\gamma^{\prime}\in\Gamma_{n}\,:\,\sum_{\gamma\leq\gamma^{\prime},\gamma\in\Gamma_{n}}\mu_{n}(\gamma)\in[\delta/2,1-\delta/2]\\}\right)\bigcup\Gamma.$

We use this definition of $K_{\delta}$ when we apply Prohorov’s theorem
(instead of the full set $X$), meaning that we must show that $K_{\delta}$ is
compact instead of $X$. ∎

###  3.3 Further consequences of the minimal non-crossing subfamily

As mentioned in the Introduction, Proposition 1 has an interpretation in terms
of current flow.

Let $G=(V,E,\sigma)$ be a finite weighted connected graph. Assume that there
are disjoint nonempty sets of boundary vertices $A$ and $B$. Solve the
(weighted) Dirichlet problem, as in Proposition 9, and find a function
$h:V\rightarrow\mathbb{R}$, so that $h|_{A}=0$, $h|_{B}=1$, and $h$ is
discrete harmonic on $V\setminus(A\cup B)$, as in (6). The function $h$
produces a current flow $f$ via Ohm’s law:

$f(xy)=\sigma(xy)\left(h(y)-h(x)\right)\;\;\;\text{ for all }xy\in\vv{E}.$
(21)

In particular, in the notations of Section 2.3, $f$ is a flow from $A$ to $B$,
meaning that it satisfies the node law in $V\setminus(A\cup B)$. Moreover, it
is well-known that ${\rm strength}(f)=M$ in this case, where
$M:=\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$ and $\Gamma(A,B)$ is the
family of all paths from $A$ to $B$ in $G$. To see this, use for instance [15,
Lemma 2.9] with their $\theta$ replaced by our flow $f$ and their gradient
flow $df$ replaced by our $dh$, then recall the connection (7). Finally,
Thomson’s principle [15, Page 40] states that among all flows from $A$ to $B$
with strength $M$, $f$ is the unique energy minimizer, where
$\mathcal{E}(f):=\sum_{e\in E}\sigma(e)^{-1}|f(e)|^{2}$.

Lemma 25 below describes the connection between current flow and the
probabilistic interpretation of modulus. Given a path $\gamma$ from A to B let
$x_{\gamma}$ be the unit flow along $\gamma$. Then, the current flow $f$ has a
path flow decomposition, if it can be written as a sum
$\sum_{\gamma\in\Gamma}a_{\gamma}x_{\gamma}$ of path flows, with coefficients
$a_{\gamma}>0$, for $\gamma$ in some family $\Gamma\subset\Gamma(A,B)$ and the
paths in $\Gamma$ all have the same direction of flow on every edge $e\in E$.

######  Lemma 25.

Let $f$ be the current flow defined in (21). Assume $\Gamma\subset\Gamma(A,B)$
strictly supports an optimal pmf $\mu^{*}$ for the probabilistic
interpretation of $\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$. Then, the
paths in $\Gamma$ all have the same direction of flow on every edge $e\in E$
and $g:=\sum_{\gamma\in\Gamma}M\mu^{*}(\gamma)x_{\gamma}$ is a path flow
decomposition for $f$, where $M:=\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$.
Conversely, if $\sum_{\gamma\in\Gamma}a_{\gamma}x_{\gamma}$ is a path flow
decomposition for $f$, then

$\sum_{\gamma\in\Gamma}a_{\gamma}={\rm strength}(f)=M,$

Moreover, $\tilde{\mu}(\gamma):=a_{\gamma}/M$ is an optimal pmf strictly
supported on $\Gamma$.

###### Proof.

First, we want to show that if $\mu^{*}$ is an optimal pmf strictly supported
on $\Gamma\subset\Gamma(A,B)$, then the paths in $\Gamma$ all have the same
direction of flow on each edge. Suppose not, then we can find an edge
$e=\\{x,y\\}$ and two paths $\gamma,\gamma^{\prime}\in\Gamma$, such that,
$\gamma=\gamma_{ax}\cup xy\cup\gamma_{yb}$ for some $a\in A$, $b\in B$ and
paths $\gamma_{ax}$ from $a$ to $x$ and $\gamma_{yb}$ from $y$ to $b$.
Likewise, $\gamma^{\prime}=\gamma^{\prime}_{a^{\prime}y}\cup
yx\cup\gamma^{\prime}_{xb^{\prime}}$ for some $a^{\prime}\in A$,
$b^{\prime}\in B$ and paths $\gamma^{\prime}_{a^{\prime}y}$ from $a^{\prime}$
to $y$ and $\gamma^{\prime}_{xb^{\prime}}$ from $x$ to $b^{\prime}$. Then,
find paths $\gamma_{1}\subset\gamma_{ax}\cup\gamma^{\prime}_{xb^{\prime}}$ and
$\gamma_{2}\subset\gamma^{\prime}_{a^{\prime}y}\cup\gamma_{yb}$ connecting $A$
and $B$, and define a new family of paths $\Gamma^{\prime}$ by adding
$\gamma_{1}$ and $\gamma_{2}$ to $\Gamma$. If
$0<t<\min(\mu^{*}(\gamma),\mu^{*}(\gamma^{\prime}))$, define a measure $\nu$
on $\Gamma^{\prime}$, by setting $\nu(\gamma):=\mu^{*}(\gamma)-t$,
$\nu(\gamma^{\prime}):=\mu^{*}(\gamma^{\prime})-t$,
$\nu(\gamma_{1})=\nu(\gamma_{2}):=t$, and keeping $\nu=\mu^{*}$ otherwise.
Then, $\nu$ is a pmf on $\Gamma^{\prime}$, and has the same edge probabilities
as $\mu^{*}$ at every edge except the edges in the set
$H:=(\gamma\cup\gamma^{\prime})\setminus(\gamma_{1}\cup\gamma_{2})$ where they
are smaller. In particular, $xy$ is guaranteed to be in $H$ and there

$\mathbb{P}_{\nu}(xy\in\underline{\gamma})=\mathbb{P}_{\mu^{*}}(xy\in\underline{\gamma})-2t<\mathbb{P}_{\mu^{*}}(xy\in\underline{\gamma}).$

This contradicts the optimality of $\mu^{*}$.

Consider $g:=\sum_{\gamma\in\Gamma}M\mu^{*}(\gamma)x_{\gamma}$. Then $g$
satisfies the node law on $V\setminus(A\cup B)$ and ${\rm strength}(g)=M$.
Also, since all the paths in $\Gamma$ flow in the same direction on every edge
$e=\\{x,y\\}\in E$, by (4), (7) and (21),

$|g(e)|=\sum_{\gamma\in\Gamma}M\mu^{*}(\gamma)\mathbbm{1}_{\\{e\in\gamma\\}}=M\mathbb{P}_{\mu^{*}}(e\in\underline{\gamma})=\sigma(e)\rho^{*}(e)=\sigma(e)|h(y)-h(x)|=|f(e)|,$

where $\rho^{*}$ is the extremal density for
$\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$. Therefore, $g$ is also a flow
energy minimizer, and by uniqueness in the Thomson’s principle we must have
$f=g$. So, we got a path flow decomposition for $f$.

Conversely, if $\sum_{\gamma\in\Gamma}a_{\gamma}x_{\gamma}$ is a path flow
decomposition for $f$, then $\sum_{\gamma\in\Gamma}a_{\gamma}={\rm
strength}(f)=M$. Therefore, setting $\tilde{\mu}(\gamma):=a_{\gamma}/M$ yields
a pmf on $\Gamma$. Moreover, every path in $\Gamma$ flows in the same
direction as $f$ on every edge. Thus,

$M\mathbb{P}_{\tilde{\mu}}(e\in\underline{\gamma})=\sum_{\gamma\in\Gamma}M\tilde{\mu}(\gamma)\mathbbm{1}_{\\{e\in\gamma\\}}=\sum_{\gamma\in\Gamma}a_{\gamma}\mathbbm{1}_{\\{e\in\gamma\\}}=|f(e)|=\sigma(e)\rho^{*}(e).$

Hence, by (1), $\tilde{\mu}$ is an optimal pmf for the probabilistic
interpretation of $\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$, that is
strictly supported on $\Gamma$. ∎

Proposition 1 and the non-crossing minimal subfamily algorithm imply that
there is a unique way of decomposing the current flow $f$ so that the paths
$\gamma$ in the path decomposition do not cross. Moreover, the corresponding
coefficients $a_{\gamma}$ are computed by the algorithm.

######  Proposition 26.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary and let $f$ be the
current flow from $A$ to $B$ as described above. Then, $f$ can be decomposed
as $\sum_{\gamma\in\Gamma}m_{\gamma}x_{\gamma}$, where the family $\Gamma$ and
the weights $\\{m_{\gamma}\\}$ are output by the non-crossing minimal
subfamily algorithm.

Moreover, if
$\sum_{\tilde{\gamma}\in\tilde{\Gamma}}a_{\tilde{\gamma}}x_{\tilde{\gamma}}$
is another path decomposition where $a_{\tilde{\gamma}}>0$ for every
$\tilde{\gamma}\in\tilde{\Gamma}$, and $\tilde{\Gamma}$ consists of non-
crossing paths from $A$ to $B$, then $\tilde{\Gamma}=\Gamma$ and the
$a_{\gamma}$-weights and $m_{\gamma}$-weights coincide.

###### Proof.

Let $\Gamma$ be the subfamily given by the non-crossing minimal subfamily
algorithm. For $\gamma\in\Gamma$, let $x_{\gamma}$ be the unit flow along
$\gamma$. Then, $\sum_{\gamma\in\Gamma}m_{\gamma}x_{\gamma}$ coincides with
the current flow $f$ by construction. Indeed, the algorithm routinely picks
the top path $\gamma$ and finds a weight $m_{\gamma}$ so that the
corresponding path flow satisfies $0\leq m_{\gamma}x_{\gamma}\leq f$ on every
edge. By Lemma 22, when the algorithm terminates the flow $f$ on every edge
has been exhausted.

Now suppose
$\sum_{\tilde{\gamma}\in\tilde{\Gamma}}a_{\tilde{\gamma}}x_{\tilde{\gamma}}$
is another path decomposition for $f$, where $\tilde{\Gamma}$ consists of non-
crossing paths from $A$ to $B$. By Lemma 25, defining
$\tilde{\mu}(\tilde{\gamma}):=a_{\tilde{\gamma}}/M$, with
$M:=\operatorname{Mod}_{2,\sigma}(\Gamma(A,B))$, yields an optimal pmf that is
strictly supported on $\tilde{\Gamma}$. Then, by Lemma 23, we obtain that
$\Gamma=\tilde{\Gamma}$, and $\tilde{\mu}(\gamma)=m_{\gamma}/M$, for every
$\gamma\in\Gamma$. Thus, the $a_{\gamma}$-weights and $m_{\gamma}$-weights
coincide. ∎

Another consequence of the non-crossing minimal subfamily algorithm is that it
gives an alternative way to draw a rectangle-packing given a planar electrical
network, as in the famous result from [8]. Namely, given a weighted plane
graph and two arcs of nodes $A$ and $B$ bordering the unbounded face, the
resulting current flow from $A$ to $B$, interpreting the edge-weights as
electric conductances, gives rise to a tiling of a rectangle by a collection
of smaller rectangles that have certain precise combinatorial properties
derived from the graph, see [13, Theorem 6.1] for a precise description of
these rectangle tilings. See also [17, Lemma 5.4] for similar tilings of more
general polygons in the plane.

######  Proposition 27.

Let $G=(V,E,\sigma,A,B)$ be a plane network with boundary. Let
$h:V\to\mathbb{R}$ be the discrete harmonic on $V\setminus(A\cup B)$ with
$h|_{A}=0$ and $h|_{B}=1$. Let $\Gamma(A,B)$ be the family of all paths from
$A$ to $B$ in $G$. Set
$M:=\mathcal{E}(h)=\operatorname{Mod}_{2,\sigma}\Gamma(A,B)$ and consider the
rectangle $\mathcal{R}=[0,1]\times[0,M]$. Then, $\mathcal{R}$ can be tiled
with a collection of smaller rectangles $\\{R_{e}\\}_{e\in E}$, so that each
maximal vertical segment composed of sides of rectangles is in bijection with
a node in the graph (up to resolving $4$-fold corners as explained in [13,
Section 6.1]).

Moreover, each rectangle $R_{e}$ is of the form
$[h(e),h(e)+\rho(e)]\times[\tilde{h}(e),\tilde{h}(e)+\sigma(e)\rho(e)]$, where
$\rho(e)$ is the extremal density for
$\operatorname{Mod}_{2,\sigma}\Gamma(A,B)$, $h(e):=\min\\{h(x),h(y):e=xy\\}$,
and where $\tilde{h}(e)$ is a height that can be computed using the non-
crossing minimal subfamily algorithm.

Figure 3: This is an example of Proposition 27 in the case of orthodiagonal
maps. For the map above the primal graph (in blue) is given by the nodes
$A,B,H,J,K,N,P,U$. Here we compute the modulus $M$ of the family of curves
connecting the set $\\{A,H\\}$ to the set $\\{U\\}$. In this case the quantity
$\tilde{h}(e)$ coincides with the harmonic function defined on the dual graph
with $\tilde{h}=0$ on the set $\\{C,L,T\\}$ and $\tilde{h}=M$ on the set
$\\{I,R\\}$.

######  Remark 28.

Note that the area of each rectangle $R_{e}$ is $\sigma(e)\rho^{2}(e)$, which
is the contribution of $e$ to the modulus:

$\sum_{e\in E}\sigma(e)\rho^{2}(e)=\operatorname{Mod}_{2,\sigma}\Gamma(A,B).$

Note also that the rectangle $R(e)$ degenerates when the corresponding
$\rho(e)$ is equal to zero.

Finally, Proposition 27 is stated for general planar graphs $G$. When, $G$ is
the primal graph of an othodiagonal map, then the quantity $\tilde{h}(e)$ can
be computed using the harmonic function defined on the dual graph with
$\tilde{h}=0$ on the bottom side and $\tilde{h}=M$ on the top side.

###### Proof.

Here we perform the non-crossing minimal subfamily algorithm from the bottom
up and write $\gamma_{1},\dots,\gamma_{n}$ for the output curves in the order
that they are added to $\Gamma$. So $\gamma_{1}$ is the bottom curve, etc…We
also write $m_{j}:=m(\gamma_{j})$ for the corresponding weights computed by
the algorithm.

To construct the rectangle tiling, we begin by mapping $\gamma_{1}$ to the
strip $(0,1)\times(0,m_{1})$. Writing $\gamma_{1}=x^{1}_{0}\ \cdots\
x^{1}_{k_{1}}$ we trace the horizontal segments
$[h(x^{1}_{j}),h(x^{1}_{j+1})]\times\\{0\\}$, for $j=0,\dots,k_{1}-1$, as well
as the vertical segments $\\{h(x^{1}_{j})\\}\times[0,m_{1}]$, for
$j=0,\dots,k_{1}$. For every edge $e=x^{1}_{j}x^{1}_{j+1}$,
$j=0,\dots,k_{1}-1$, set $\tilde{h}(e)=0$. Finally, for every edge
$e=x^{1}_{j}x^{1}_{j+1}$ that attains the minimum in step 3 of the algorithm,
trace the horizontal segment $[h(x^{1}_{j}),h(x^{1}_{j+1})]\times\\{m_{1}\\}$,
thus closing up the rectangle corresponding to these edges.

Rather than a formal proof by induction, we now describe the second step.
Consider the second curve $\gamma_{2}$ and add the strip
$(0,1)\times(m_{1},m_{1}+m_{2})$. Observe that $\gamma_{2}$ coincides with
$\gamma_{1}$ except for the edges that realized the minimum in the previous
step. More specifically, consider a maximal set of consecutive edges
$e_{1}\cdots e_{r}$ in $\gamma_{1}$, that attained the minimum in step 3 of
the algorithm, and form a subpath
$\gamma_{1}^{\prime}:=x_{j_{1}}^{1}x_{j_{1}+1}^{1}\cdots x_{j_{1}+r}^{1}$.
Note that the tops of the rectangles corresponding to the edges $e_{i}$ in
$\gamma_{1}^{\prime}$ were closed up in the previous step, and their union
forms a connected horizontal cut $K$. The second path $\gamma_{2}$ will
replace $\gamma_{1}^{\prime}$ with a new path of the form
$\gamma_{2}^{\prime}:=x^{2}_{k_{1}},\dots,x^{2}_{k_{l}},\dots,x^{2}_{k_{L}}$,
where the first and last node belong to $A\cup B\cup\\{\gamma_{1}\\}$
(depending on whether the face corresponding to
$\gamma_{1}^{\prime}\cup\gamma_{2}^{\prime}$ abuts $A\cup B$ or not). In
particular, we will have
$h(x^{2}_{k_{1}})<h(x^{2}_{k_{l}})<\cdots<h(x^{2}_{k_{L}})$. Also, for every
edge $e=x^{2}_{j}x^{2}_{j+1}$, $j=k_{1},\dots,k_{L}-1$, along the cut $K$, set
$\tilde{h}(e)=m_{1}$. Now further subdivide the cut $K$ and add vertical
segments of the form $\\{h(x^{2}_{k_{l}})\\}\times[m_{1},m_{1}+m_{2}]$.
Furthermore, extend the vertical segments from the previous step that were not
bounding an edge realizing the minimum. Finally, again close up the rectangles
corresponding to the edges in $\gamma_{2}$ that attain the minimum in step 3
of the algorithm. Thus, we are ready to repeat this process.

Note that each face in the plane network gives rise to a horizontal cut which
is a union of sides of smaller rectangles, while every node in the plane
network gives rise to a vertical segment which is also a union of sides of
smaller rectangles (these vertical segments may abut each other in some
special cases, namely when $h$ has the same value at a node in the subpaths
$\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$). When the algorithm ends we
will have packed the rectangle $[0,1]\times[0,M]$.

Both rectangle tilings in Figure 3 and Figure 6 were obtained this way. ∎

###  3.4 A numerical example

We have implemented the non-crossing minimal subfamily algorithm, and some of
its consequences, numerically. Consider the family of paths connecting the
arcs $ab$ and $cd$ of the cross domain on the left of Figure 4.

Figure 4: Left: a Delaunay triangulation of a cross domain; Center: the
triangulation shown together with its dual Voronoi graph; Right: the
corresponding orthodiagonal map.

The figure shows a Delaunay triangulation of the domain, produced by the
software package Triangle [19]. The circumcenters of the triangles form a dual
grid known as the Voronoi diagram. In the center of Figure 4, we show the same
triangulation along with its Voronoi dual. The edges of the triangulation are
blue, and the edges connecting two circumcenters are red. Note that the blue
and red edges are orthogonal, hence can be thought as the orthogonal diagonals
of a quadrilateral. In particular, an orthodiagonal map is obtained by
connecting each blue node with a neighboring red node. We have drawn the
corresponding orthodiagonal map on the right in Figure 4. With some minor
caveats (see below), the Delaunay triangulation can be thought as the primal
graph of the orthodiagonal map and the Voronoi diagram as the dual graph. See
also [21, Section 5.1].

####  3.4.1 Approximating the modulus

In order to approximate the modulus of the family of curves, we first choose a
size parameter, $\epsilon>0$, which controls the size of the triangles in the
triangulation. The triangulation is constructed to satisfy the Delaunay
property as well as two additional constraints: no triangle may have area
larger than $\epsilon^{2}$ and no angle in any triangle may be smaller than
20∘. These additional constraints ensure that the edges of the triangulation
are of order $O(\epsilon)$, which is needed to ensure convergence of the
discrete modulus to the continuous modulus.

Next, we need to assign appropriate conductances to the edges in the
triangulation (Figure 4, left). As stated above, the triangulation is _almost_
the graph that is obtained from the blue vertices of the quadrilateral mesh
(Figure 4, right) by connecting vertices that belong to the same
quadrilateral. This does not quite reproduce the triangulation on the left;
notably absent are the edges on the boundary of the domain. As a consequence,
any vertex of the triangulation that is incident only upon boundary edges is
absent from the quadrilateral mesh. In the example shown, all convex “corners”
are excluded from the quadrilateral mesh. Fortunately, there is a relatively
simple way to address these issues at the boundary.

Most edges of the triangulation will correspond to diagonals in the
quadrilateral mesh. For these, we can use the formulas specified in Section
2.5. Namely, we assign the conductance, $\sigma$, on such an edge to be the
ratio of the length of the dual edge to the length of the edge in question. To
all boundary edges, we assign $\sigma=\epsilon/|E_{b}|$, where $E_{b}$ is the
set of boundary edges. Since the harmonic function $h$ is bounded between $0$
and $1$, this means, by Ohm’s law, that the contribution of all boundary edges
to the current flow is $O(\epsilon)$, ensuring that the contribution of the
extraneous boundary edges to the approximation is negligible as $\epsilon\to
0$. With this choice of conductances, we can then solve the discrete Dirichlet
problem to approximate the harmonic function $h$.

####  3.4.2 Non-crossing paths

Once the discrete harmonic function is computed, the non-crossing minimal
family can be determined using the algorithm described in Section 3.1. The
left side of Figure 5 shows the result of this numerical study for the choice
$\epsilon=0.005$. The algorithm produced a sequence of $34403$ non-crossing
paths from the boundary segment $ab$ to the boundary segment $cd$. We have
plotted 21 of these paths, plotting roughly every $1720$-th path in the
sequence. The first and last path trace the boundary segments between $a$ and
$d$ and between $b$ and $c$. In the middle of Figure 5, we show the
corresponding curves as computed by Driscoll’s Schwarz–Christoffel Toolbox for
Matlab. Using the toolbox code, we approximated a conformal mapping between
the cross and a rectangle with $ab$ and $cd$ sent to the vertical sides of the
rectangle. Inverting that map on horizontal lines produced the image. The
image on the right side of the figure shows the superposition of both sets of
curves.

Figure 5: Left: non-crossing paths for a triangulation of the cross domain;
Center: the corresponding curvess as computed by the Schwarz-Christoffel
Toolbox in Matlab; Right: the superposition of both sets of curves.

Using the non-crossing path decomposition, we are able to assign a two-
dimensional coordinate to each vertex in the graph. The $x$-coordinate of each
vertex is simply the value of the discrete harmonic function $h$ on that
vertex. To obtain the $y$-coordinate of a vertex $v$, we locate the first path
in the sequence passing through $v$ and sum the flows of all previous paths.
The rectangle tiling for this choice of $\epsilon$ is too dense to draw, but
the left side of Figure 6 shows a rectangle tiling for $\epsilon=0.075$. By
Theorem 3 and Theorem 4, this choice of coordinates provides a mapping on the
vertices of the triangulation that approximates the conformal map to the
rectangle. The right side of Figure 6 shows the non-crossing paths in these
coordinates (for the finer, $\epsilon=0.005$, triangulation). Except near $a$
and $d$, where there is large distortion, these curves are nearly horizontal,
as we would expect. Incidentally, the rectangle tiling (Figure 6, left) gives
an indication of where one may refine the original triangulation, since the
largest distortion corresponds to very large rectangles in the tiling.

Figure 6: Left: the rectangle tiling associated with a triangulation of the
cross domain shown in Figure 4; Right: the images of the non-crossing paths
from Figure 5 in coordinates given by the rectangle tiling.

##  4 Convergence of discrete harmonic functions

In this section, we prove Theorem 3 by adapting the arguments in [11] to our
setting and then we prove Theorem 4 using Fulkerson duality for modulus (Lemma
24).

We begin by recalling our assumptions:

  * •

The Jordan domain $\Omega$ is an analytic quadrilateral with boundary arcs
$A,\tau_{1},B,\tau_{2}$.

  * •

The orthodiagonal map $G=(V^{\bullet}\sqcup V^{\circ},E)$ with boundary arcs
$S_{1},T_{1},S_{2},T_{2}$ has maximal edge length at most $\epsilon$ and
approximates $\Omega$ within distance $\delta$. Recall that by Definition 17,
we ask that $\overline{\Omega}\subset\hat{G}$.

  * •

The continuous harmonic function $h_{c}$ is defined by $h_{c}=\text{Re }\psi$,
where $\psi$ is a conformal map of $\Omega$ onto the interior of the rectangle
$R=[0,1]\times[0,m]$, taking the arcs $A$ and $B$ to the vertical sides of the
rectangle.

  * •

The function $h_{d}:V^{\bullet}\to\mathbb{R}$ is discrete harmonic on ${\rm
int}(V^{\bullet})$ with $h_{d}(z)=0$ for $z\in S_{1}\cap V^{\bullet}$ and
$h_{d}(z)=1$ for $z\in S_{2}\cap V^{\bullet}$.

  * •

By scale invariance, we may assume diam$(\Omega)=1$.

Under these assumptions, our goal is to show for all $z\in
V^{\bullet}\cap\overline{\Omega}$

$|h_{d}(z)-h_{c}(z)|\leq\frac{C}{\log^{1/2}\left(1/\lambda\right)},$ (22)

where $\lambda:=\epsilon\vee\delta$ and $C$ depends only on the analytic
quadrilateral $\Omega$. Note that by assumption
$\lambda<\operatorname{diam}(\Omega)=1$. In our proof, we will show that this
equation holds whenever $0<\lambda<C^{\prime}$ for some $C^{\prime}>0$ which
depends only on $\Omega$. However, since $h_{d}$ and $h_{c}$ are both bounded
between 0 and 1 (by the discrete and continuous maximum principles), we may
adjust the constant $C$ in (22) so that it holds for all $\lambda\in(0,1)$.
Lastly, we note that we will use $C$ to refer to constants that are
independent of $\lambda$ but may depend on the analytic quadrilateral $\Omega$
(i.e. they depend on $\Omega$ and the boundary arcs $A,\tau_{1},B,\tau_{2}$).
The exact value of these constants may change from line to line.

###  4.1 Modified toolbox

We will be able to use some of the tools of [11] as they are stated in that
paper. However, we need to modify a handful of the results to apply to our
setting, and that is the goal of this section. Throughout this section, we
will use the terminology of flows as described in Section 2.3.

There is one last piece of terminology that we need to introduce before we
begin. Let $G=(V^{\bullet}\sqcup V^{\circ},E)$ be a finite orthodiagonal map
with boundary arcs $S_{1},T_{1},S_{2},T_{2}$. We say that $G_{1}$ is a sub-
orthodiagonal map of $G$ when $G_{1}$ is an othordiagonal map contained in $G$
and the topological boundary $\partial G_{1}$ intersects $T_{1}\cup T_{2}$.
The boundary $\partial V_{1}^{\bullet}$ are the points from $V_{1}^{\bullet}$
that are in $\partial G_{1}\setminus\left(T_{1}\cup T_{2}\right)$ and the
boundary $\partial V_{1}^{\circ}$ are the points in
$V_{1}^{\circ}\cap(T_{1}\cup T_{2})$. When we prove Theorem 3 in Section 4.2,
we will work with a sub-orthodiagonal map $G_{1}$ satisfying the hypotheses of
the following lemma. Since it may help to have a visual, see Figure 7 for a
sketch of this setting.

######  Lemma 29.

Let $G$ and $\psi$ satisfy the hypotheses of Theorem 3, and let $G_{1}$ be a
sub-orthodiagonal map of $G$ so that $\text{dist}(G_{1},A\cup
B)\geq\lambda^{1/4}$. Then for $\lambda$ small enough (depending on the
analytic quadrilateral $\Omega$), the map $\psi$ can be analytically extended
to a domain $\mathcal{O}$ containing $\hat{G}_{1}$ so that
$\mathrm{dist}(\hat{G}_{1},\partial\mathcal{O})\geq\frac{5}{16}\lambda^{1/4}$.

$A$$B$$\tau_{1}$$\tau_{2}$$S_{1}$$S_{2}$$T_{1}$$T_{2}$$\partial G_{1}$

$\partial\mathcal{O}$$\partial G_{1}$

Figure 7:  A sketch of the setting encountered in the proof of Theorem 3.
Left: An analytic quadrilateral $\Omega$ with boundary arcs $A,B$ (orange) and
$\tau_{1},\tau_{2}$ (black) is approximated by an orthodiagonal map $G$ with
boundary arcs $S_{1},T_{1},S_{2},T_{2}$. The boundary of the sub-orthodiagonal
map $G_{1}$ is dashed black and $\partial G\setminus\partial G_{1}$ is dashed
gray. Right: $G_{1}$ is contained within the extended domain $\mathcal{O}$ of
Lemma 29.

###### Proof.

By Schwarz reflection, we may extend $\psi$ across the arcs $\tau_{1}$ and
$\tau_{2}$. More precisely, recall from Definition 15, that since $\tau_{k}$
is an analytic arc, it is the image of the interval $[0,1]\times\\{0\\}$ under
a conformal map $F_{k}$ which is defined in a domain $\mathcal{D}_{k}$
containing $[0,1]\times\\{0\\}$. Then $\psi\circ F_{k}$ is defined in the
domain $\mathcal{D}_{k}\cap F_{k}^{-1}(\Omega)$ and extends continuously to
$[0,1]\times\\{0\\}$ mapping it to either the top or the bottom boundary arc
of $R$ (i.e. $[0,1]\times\\{0\\}$ or $[0,1]\times\\{m\\}$). Hence by Schwarz
reflection, we can extend this map over $[0,1]\times\\{0\\}$, and from this we
obtain an extension of $\psi$, which is equal to $(\psi\circ F_{k})\circ
F_{k}^{-1}$ near $\tau_{k}$. In particular, we have extended $\psi$ to be
conformal in an open set $\mathcal{O}$ containing $\Omega$ and the relative
interior of $\tau_{k}$. Let $z_{1},z_{2},z_{3},z_{4}$ be the four “corner”
points of $\Omega$, i.e. where the arcs $A,B$ and $\tau_{1},\tau_{2}$ meet.
Say $z_{1}=A\cap\tau_{1}$, then by Definition 16 the arcs $A$ and $\tau_{1}$
are not tangent at $z_{1}$ and by conformality they are also not tangent in
the image under $F_{1}^{-1}$. Thus, when we reflect, the boundary of
$F_{1}^{-1}(\mathcal{O})$ is also not tangent to $[0,1]\times\\{0\\}$, and by
conformality again we have shown that $\partial\mathcal{O}$ is not tangent
with $\tau_{1}$ at $z_{1}$. A similar argument works at the other corners.
Therefore, we can always find a small radius $r_{k}>0$, so that for $z\in
D_{r_{k}}(z_{k})\cap\overline{\Omega}$,
dist$(z,\partial\mathcal{O})\geq\frac{1}{2}\text{dist}(z,A\cup B)$. Set
$D=\cup_{k=1}^{4}D_{r_{k}}(z_{k})$. Then
$\left(\tau_{1}\cup\tau_{2}\right)\setminus D$ is contained compactly within
$\mathcal{O}$, and we may assume that

$\text{dist}\left(\left(\tau_{1}\cup\tau_{2}\right)\setminus
D,\,\partial\mathcal{O}\right)\geq\lambda^{1/4}$ (23)

by taking $\lambda$ small enough.

We wish to show that for all $p\in\hat{G}_{1}$

$\text{dist}(p,\partial\mathcal{O})\geq\frac{5}{16}\lambda^{1/4}.$ (24)

Let $p\in\hat{G}_{1}$. If the closest point to $p$ in $\partial\mathcal{O}$ is
in $A\cup B$, then (24) holds since dist$(p,A\cup B)\geq\lambda^{1/4}$. So we
will assume that the closest point to $p$ in $\partial\mathcal{O}$ is not
contained in $A\cup B$. First consider the case that $p\in\overline{\Omega}$.
The line segment connecting $p$ to a closest point in $\partial\mathcal{O}$
must cross out of $\Omega$ at some point $z_{0}\in\tau_{1}\cup\tau_{2}$. We
have two possible cases: (1)
$z_{0}\in\left(\tau_{1}\cup\tau_{2}\right)\setminus D$ or (2)
$z_{0}\in\left(\tau_{1}\cup\tau_{2}\right)\cap D$. In case (1), we use (23) to
obtain

$\text{dist}(p,\partial\mathcal{O})\geq\text{dist}(z_{0},\partial\mathcal{O})\geq\lambda^{1/4}.$

In case (2),

$\displaystyle\text{dist}(p,\partial\mathcal{O})$
$\displaystyle\geq\text{dist}(p,z_{0})+\text{dist}(z_{0},\partial\mathcal{O})$
$\displaystyle\geq\frac{1}{2}\left[\text{dist}(p,z_{0})+\text{dist}(z_{0},A\cup
B)\right]$ $\displaystyle\geq\frac{1}{2}\text{dist}(p,A\cup B)$
$\displaystyle\geq\frac{1}{2}\lambda^{1/4}.$

Next we assume that $p\notin\Omega$. Then there is some
$w_{0}\in\tau_{1}\cup\tau_{2}$ so that $\text{dist}(p,w_{0})\leq\delta$. If
$w_{0}\notin D$, then

$\text{dist}(p,\partial\mathcal{O})\geq\text{dist}(w_{0},\partial\mathcal{O})-\delta\geq\lambda^{1/4}-\delta\geq\frac{7}{8}\lambda^{1/4},$

where we used that $\lambda\leq\frac{1}{8}\lambda^{1/4}$. If $w_{0}\in D$,
then

$\text{dist}(w_{0},\partial\mathcal{O})\geq\frac{1}{2}\text{dist}(w_{0},A\cup
B)\geq\frac{1}{2}\left[\text{dist}(p,A\cup
B)-\text{dist}(w_{0},p)\right]\geq\frac{1}{2}\left(\lambda^{1/4}-\delta\right)\geq\frac{7}{16}\lambda^{1/4},$

and so

$\text{dist}(p,\partial\mathcal{O})\geq\text{dist}(w_{0},\partial\mathcal{O})-\text{dist}(w_{0},p)\geq\frac{7}{16}\lambda^{1/4}-\delta\geq\frac{5}{16}\lambda^{1/4}.$

This establishes (24).

∎

Next we show a modified version of Proposition 5.2 of [11], which tells us
that the energy of $h_{c}-h_{d}$ is small.

######  Proposition 30.

Assume we are in the setting of Theorem 3, and $G_{1}=(V^{\bullet}\sqcup
V^{\circ},E)$ is a sub-orthodiagonal map of $G$ as in Lemma 29, so that $\psi$
can be analytically extended to an open set $\mathcal{O}$ containing
$\hat{G}_{1}$.

Let $h_{c}=\text{Re}\,\psi$, and let $h_{d}:V^{\bullet}\to\mathbb{R}$ be
discrete harmonic on ${\rm int}(V^{\bullet})$ and satisfy $h_{d}=h_{c}$ on
$\partial V^{\bullet}$. Set

$L=||\nabla h_{c}||_{\infty,\hat{G}_{1}}\;\;\text{ and
}\;\;M=||Hh_{c}||_{\infty,\tilde{G}_{1}}.$

Then

$\mathcal{E}_{1}^{\bullet}(h_{c}-h_{d})\leq
C\left[M^{2}\epsilon^{2}+L^{2}(\epsilon\vee\delta)\right],$

where $C$ depends only on $\operatorname{Area}(\hat{G})$ and
$\operatorname{Length}(\tau_{1}\cup\tau_{2})$ and $\mathcal{E}_{1}^{\bullet}$
is the energy in the graph $G_{1}^{\bullet}$.

###### Proof.

We begin by following the initial steps in the proof of Proposition 5.2 in
[11]. In particular, we wish to define the bijective map $e\to e^{\dagger}$
from $\vv{E}^{\bullet}$ to $\vv{E}^{\circ}$. For $e\in\vv{E}^{\bullet}$, let
$Q=[v_{1},w_{1},v_{2},w_{2}]$ be the inner face of $G_{1}$ containing $e$. We
set $e^{\dagger}=e^{\circ}_{Q}$, and we choose its orientation based on the
orientation of $e$: if $e^{-}=v_{1}$ (i.e. $e$ is oriented from $v_{1}$ to
$v_{2}$), then choose the orientation with $(e^{\dagger})^{-}=w_{1}$, and in
the other case choose the opposite orientation.

Define

$\phi(e):=\sigma(e)[h_{c}(e^{+})-h_{c}(e^{-})].$

We wish to define a flow $\theta$ on $G^{\bullet}_{1}$ using the harmonic
conjugate $\tilde{h}_{c}$ of $h_{c}$ (i.e. $\tilde{h}_{c}=\text{Im}\,\psi$).
However, we must take a little more care near the boundary arcs $\tau_{k}$. To
this end, let $U_{k}$ be the component of $\hat{G}_{1}\setminus\Omega$ that
intersects $\tau_{k}$, for $k=1,2$, see Figure 7. Next, we define a projection
$p:V^{\circ}\rightarrow\overline{\Omega}$ as follows: for $w\in
V^{\circ}\cap\overline{\Omega}$, set $p(w)=w$; for $w\in U_{k}\cap V^{\circ}$,
set $p(w)$ to be a point in $\tau_{k}$ that is closest to $w$. For
$e\in\vv{E}^{\bullet}$, define

$\theta(e)=\tilde{h}_{c}(p((e^{\dagger})^{+}))-\tilde{h}_{c}(p((e^{\dagger})^{-})).$

Note that if $e^{\dagger}$ is not incident to a vertex in $U_{1}\cup U_{2}$,
then our definition gives

$\theta(e)=\tilde{h}_{c}((e^{\dagger})^{+})-\tilde{h}_{c}((e^{\dagger})^{-})$

matching the definition in [11].

Now we want to show that $\theta$ satisfies the node law on ${\rm
int}(V^{\bullet})$. Let $v\in{\rm int}(V^{\bullet})$. Suppose first $v$ is
contained within an interior face $f_{v}$ of the dual graph $G^{\circ}_{1}$,
i.e. there is a cycle of dual edges surrounding $v$. Let
$w_{1},w_{2},\cdots,w_{k}$ be the vertices incident to $f_{v}$ in
counterclockwise order. Then (taking indices mod $k$)

$\sum_{e\in\vv{E}^{\bullet}\,:\,e^{-}=v}\theta(e)=\sum_{j=1}^{k}\left[\tilde{h}_{c}(p(w_{j+1}))-\tilde{h}_{c}(p(w_{j}))\right]=0.$

If $v$ is not in an interior face $f$, then the edges $e^{\dagger}$ such that
$e\in\vv{E}^{\bullet}$ with $e^{-}=v$ form a path with both endpoints in
$\partial V^{\circ}$ (and in fact both endpoints will be in the same boundary
arc $T_{1}$ or $T_{2}$, see Figure 7). Let $w_{1},w_{2},\cdots,w_{k}$ be the
vertices of this path, indexed so that connecting $w_{k}$ to $w_{1}$ in the
outer face of $G_{1}^{\circ}$ would give a counterclockwise cycle. Then

$\sum_{e\in\vv{E}^{\bullet}\,:\,e^{-}=v}\theta(e)=\sum_{j=1}^{k-1}\left[\tilde{h}_{c}(p(w_{j+1}))-\tilde{h}_{c}(p(w_{j}))\right]=\tilde{h}_{c}(p(w_{k}))-\tilde{h}_{c}(p(w_{1}))=0,$

where we used that $\tilde{h}_{c}=\text{Im}\,\psi$ is constant on $\tau_{k}$.

Next, we apply Proposition 4.9 in [11] with $f=h_{c}$ and $h=h_{d}$ to obtain
that

$\mathcal{E}_{1}^{\bullet}(h_{c}-h_{d})\leq\mathcal{E}_{1}^{\bullet}(h_{c}-h_{d})+\mathcal{E}_{1}^{\bullet}(\theta-\sigma\,dh_{d})=\mathcal{E}_{1}^{\bullet}(\phi-\theta),$

and we note that it suffices to bound
$\mathcal{E}_{1}^{\bullet}(\phi-\theta).$ The contribution to
$\mathcal{E}_{1}^{\bullet}(\phi-\theta)$ from a face
$Q=[v_{1},w_{1},v_{2},w_{2}]$ is

$\frac{|v_{1}v_{2}|}{|w_{1}w_{2}|}\left(\frac{|w_{1}w_{2}|}{|v_{1}v_{2}|}[h_{c}(v_{2})-h_{c}(v_{1})]-[\tilde{h}_{c}(p(w_{2}))-\tilde{h}_{c}(p(w_{1}))]\right)^{2}\;\;\;\;\;\;\;\;\;\;\;\;$
$\;\;\;\;\;\;\;\;\;\;\;\;=|v_{1}v_{2}|\cdot|w_{1}w_{2}|\cdot\left(\frac{h_{c}(v_{2})-h_{c}(v_{1})}{|v_{1}v_{2}|}-\frac{\tilde{h}_{c}(p(w_{2}))-\tilde{h}_{c}(p(w_{1}))}{|w_{1}w_{2}|}\right)^{2}.$
(25)

Therefore, assuming that neither $w_{1}$ nor $w_{2}$ are in $U_{1}\cup U_{2}$,
the quantity in the parentheses (using the Cauchy-Riemann equations) can be
rewritten as

$\frac{h_{c}(v_{2})-h_{c}(v_{1})-\langle\nabla h_{c}(q),{\bf
v}\rangle|v_{1}v_{2}|}{|v_{1}v_{2}|}-\frac{\tilde{h}_{c}(w_{2})-\tilde{h}_{c}(w_{1})-\langle\nabla\tilde{h}_{c}(q),{\bf
w}\rangle|w_{1}w_{2}|}{|w_{1}w_{2}|},$ (26)

where ${\bf v}$ and ${\bf w}$ are the unit vectors in the directions of
$\vv{v_{1}v_{2}}$ and $\vv{w_{1}w_{2}}$, respectively, and $q\in Q$ is the
intersection point of $e_{Q}^{\bullet}$ and $e_{Q}^{\circ}$. Note that
$||H\tilde{h}_{c}||_{\infty,\tilde{G}_{1}}=||Hh_{c}||_{\infty,\tilde{G}_{1}}=M$
by the Cauchy-Riemann equations. Hence both terms in (26) can be bounded using
Lemma 5.3 in [11] to obtain

$\displaystyle\left|\frac{h_{c}(v_{2})-h_{c}(v_{1})-\langle\nabla
h_{c}(q),{\bf v}\rangle|v_{1}v_{2}|}{|v_{1}v_{2}|}\right|\leq 2M\epsilon,$
$\displaystyle\left|\frac{\tilde{h}_{c}(w_{2})-\tilde{h}_{c}(w_{1})-\langle\nabla\tilde{h}_{c}(q),{\bf
w}\rangle|w_{1}w_{2}|}{|w_{1}w_{2}|}\right|\leq 2M\epsilon.$

Therefore if $w_{1},w_{2}$ are not in $U_{1}\cup U_{2}$, the contribution to
$\mathcal{E}^{\bullet}(\phi-\theta)$ is at most

$|v_{1}v_{2}|\cdot|w_{1}w_{2}|\cdot(4M\epsilon)^{2}=32\,\operatorname{Area}(Q)M^{2}\epsilon^{2}.$

Here, however, we need to deal with some extra cases. Assume that at least one
of $w_{1}$ and $w_{2}$ are in $U_{1}\cup U_{2}$. We will bound the terms in
the parentheses of (25) directly. Note first that

$\left|\frac{h_{c}(v_{2})-h_{c}(v_{1})}{|v_{1}v_{2}|}\right|\leq L.$

If $w_{1}$ and $w_{2}$ are both in $U_{k}$, then
$\tilde{h}_{c}(p(w_{2}))-\tilde{h}_{c}(p(w_{1}))=0$. If $w_{1}$ is in $U_{k}$,
but $w_{2}$ is not, then the line segment connecting $w_{1}$ to $w_{2}$ must
cross $\tau_{k}$ and $|p(w_{1})-w_{1}|\leq|w_{1}-w_{2}|$, by the definition of
$p$. (And a similar statement is true if $w_{2}$ is in $U_{k+1}$, taking the
indices mod 2). Then

$|p(w_{1})-p(w_{2})|=|p(w_{1})-w_{1}|+|w_{1}-w_{2}|+|w_{2}-p(w_{2})|\leq
3|w_{1}-w_{2}|,$

which implies that

$\left|\frac{\tilde{h}_{c}(p(w_{2}))-\tilde{h}_{c}(p(w_{1}))}{|w_{1}w_{2}|}\right|\leq
L\frac{|p(w_{2})-p(w_{1})|}{|w_{2}-w_{1}|}\leq 3L.$

Thus if $Q$ is incident to a vertex in $U_{1}\cup U_{2}$, then the
contribution to $\mathcal{E}_{1}^{\bullet}(\phi-\theta)$ is at most

$|v_{1}v_{2}|\cdot|w_{1}w_{2}|\cdot(4L)^{2}\leq
32\,\operatorname{Area}(Q)L^{2}.$

The faces $Q$ that are incident to a vertex in $U_{1}\cup U_{2}$ must be in a
narrow band containing $\tau_{k}$ for either $k=1$ or $k=2$, and the area of
these bands is bounded by
$2\,\operatorname{Length}(\tau_{1}\cup\tau_{2})(\epsilon\vee\delta)$. Thus
summing over all inner faces $Q$ of $G_{1}$ gives

$\mathcal{E}_{1}^{\bullet}(\phi-\theta)\leq
32\,\operatorname{Area}(\hat{G})M^{2}\epsilon^{2}+64\,\operatorname{Length}(\tau_{1}\cup\tau_{2})L^{2}(\epsilon\vee\delta).$

∎

The next proposition generalizes Proposition 6.1 from [11] to our setting. In
this and the rest of this section, we will use the notation $D_{r}(z)$ to be
the closed disk of radius $r$ centered at $z$.

######  Proposition 31.

Assume we are in the setting of Theorem 3. Let $G_{1}=(V^{\bullet}\sqcup
V^{\circ},E)$ be a sub-orthodiagonal map of $G$ as in Lemma 29. Then, $\psi$
can be analytically extended to an open set $\mathcal{O}$ containing
$\hat{G}_{1}$, and, for $\epsilon$ small enough,
$\mathrm{dist}(\hat{G}_{1},\partial\mathcal{O})\geq\frac{5}{16}\lambda^{1/4}\geq
4\epsilon$.

Fix $z_{0}\in{\rm int}(V^{\bullet})\cap\Omega$, and let $r\geq 4\epsilon$.
Assume that $D_{r}(z_{0})\cap\partial V^{\bullet}=\emptyset$. Then there is a
unit flow $\theta$ in $G_{1}^{\bullet}$ from $V^{\bullet}\cap D_{r}(z_{0})$ to
$\partial V^{\bullet}$ so that

$\mathcal{E}_{1}^{\bullet}(\theta)\leq
C\log\left(\frac{\operatorname{diam}(\Omega)}{\epsilon}\right),$

for $\epsilon\leq\epsilon_{0}$, where $C$ and $\epsilon_{0}$ depend only on
the analytic quadrilateral $\Omega$.

To prove this proposition, we would like to define the flow over an edge to be
the change in argument over its dual edge (taking care with the edge
orientations). Then take a fixed vertex $v$ in the topological interior, and
consider the collection of all the edges incident to $v$; their dual edges
form a cycle around $v$. Thus, the sum of the flow over these edges will be
the change in argument over the cycle of dual edges, which will be zero, as
long as the cycle does not contain the origin. However, if $v$ is on $T_{1}$
or $T_{2}$, then instead of a cycle, we will obtain a path, and the beginning
and ending argument may not be equal. To compensate for this, we will first
transform the region conformally.

###### Proof.

Recall that $\psi$ is a conformal map of $\Omega$ onto the rectangle
$R=[0,1]\times[0,m]$ taking the arcs $\tau_{1}$ and $\tau_{2}$ to the
horizontal sides of $R$. Let $\psi(z_{0})=x_{0}+iy_{0}$. We may assume that
$y_{0}\geq m/2$, by rotating the rectangle $180^{\circ}$ around its midpoint
if needed. Let

$g(z)=\exp\left(\frac{\pi}{y_{0}}z\right)+\exp\left(\frac{\pi
x_{0}}{y_{0}}\right).$

Note that $g$ maps $R$ conformally strictly into the annulus $A$ with inner
radius 1 and outer radius $e^{\pi/y_{0}}$ centered at $e^{\pi x_{0}/y_{0}}$,
and $g$ maps $R^{\prime}=\\{z\in R\,:\,\text{Im}(z)<y_{0}\\}$ onto
$A\cap\mathbb{H},$ where
$\mathbb{H}:=\\{z\in\mathbb{C}:\operatorname{Im}z>0\\}$. Further, $g$ takes
the horizontal sides of $R^{\prime}$ to intervals in $\mathbb{R}$, and

$g(\psi(z_{0}))=g(x_{0}+iy_{0})=e^{\pi\frac{x_{0}}{y_{0}}}e^{i\pi}+e^{\pi\frac{x_{0}}{y_{0}}}=0.$

Let $F=g\circ\psi$. We will define a flow $\phi$ on $\vv{E}^{\bullet}$ from
$z_{0}$ to $\partial V^{\bullet}$ as follows. Given a face
$Q=[v_{1},w_{1},v_{2},w_{2}]$, let $e$ be the orientation of $e_{Q}^{\bullet}$
with $e^{-}=v_{1}$. If $F(e_{Q}^{\circ})\subset\overline{\mathbb{H}}$, set

$\phi(e)=\arg(F(w_{2}))-\arg(F(w_{1}))\;\;\;\text{ and
}\;\;\;\phi(-e)=\arg(F(w_{1}))-\arg(F(w_{2})),$

where we use a branch of the argument that is well-defined on $\mathbb{H}$. If
$F(e_{Q}^{\circ})$ does not intersect $\mathbb{H}$, then we define
$\phi(e)=0=\phi(-e)$. If $F(e_{Q}^{\circ})$ intersects $\mathbb{H}$ and
$\mathbb{C}\setminus\overline{\mathbb{H}}$, we define $\phi(e)$ to be the
change of argument along $F(e_{Q}^{\circ})\cap\overline{\mathbb{H}}$,
respecting the orientation above.

Let $v\in{\rm int}(V^{\bullet})$. Consider the set $N$ of edges
$e\in\vv{E}^{\bullet}$ with $e^{-}=v$. Recall that $\vv{E}^{\bullet}$ is the
set of directed edges, which contains both possible orientations for all the
edges of $E^{\bullet}$. The edges dual to the edges in $N$ either form a cycle
around $v$ (in the case $v$ is in the topological interior of $\hat{G}$) or a
path starting and ending on a boundary arc $T_{k}$. In the latter case, we can
form a cycle by connecting the endpoints of the path with an arc that is in
the complement of $\hat{G}_{1}$ but still in the domain of $\psi$. Now in both
cases, the image of the cycle under $F$ forms a cycle in $\mathbb{C}$, which
we will call $\gamma_{v}$. When $v=z_{0}$, then $\gamma_{v}$ winds once around
the origin, and in all other cases, the winding number of $\gamma_{v}$ around
the origin is zero. Since $\sum_{e\in\vv{E}^{\bullet}\,:\,e^{-}=v}\phi(e)$
gives the sum of the change in argument over the path(s) of
$\gamma_{v}\cap\overline{\mathbb{H}}$, this is equal to $\pi$ when $v=z_{0}$
and it is equal to 0 otherwise. Thus $\phi$ is a flow of strength $\pi$ from
$z_{0}$ to $\partial V^{\bullet}$. Define

$\theta(e)=\begin{cases}0&\quad\text{if }e^{+}\text{ and }e^{-}\in
D_{r}(z_{0})\\\ \frac{1}{\pi}\phi(e)&\quad\text{else}\\\ \end{cases}.$

Then $\theta$ is a unit flow from from $V^{\bullet}\cap D_{r}(z_{0})$ to
$\partial V^{\bullet}$.

It remains to bound the energy of $\theta$. Let $Q=[v_{1},w_{1},v_{2},w_{2}]$
be a face of $G_{1}$ so that $v_{1},v_{2}$ are not both in $D_{r}(z_{0})$. We
first consider the case that $F(e_{Q}^{\circ})$ is contained in
$\overline{\mathbb{H}}$. Then the contribution of $e_{Q}^{\bullet}$ to
$\mathcal{E}_{1}^{\bullet}(\theta)$ is

$\frac{|v_{1}v_{2}|}{|w_{1}w_{2}|}|\theta(e^{\bullet}_{Q})|^{2}=\frac{1}{\pi^{2}}|v_{1}v_{2}|\cdot|w_{1}w_{2}|\left(\frac{\arg(F(w_{2}))-\arg(F(w_{1}))}{|w_{1}w_{2}|}\right)^{2}.$

Note that $|v_{1}v_{2}|\cdot|w_{1}w_{2}|=2\,\text{area}(Q)$. Since $Q$ has
diameter at most $2\epsilon$ and since
$\mathrm{dist}(\hat{G}_{1},\partial\mathcal{O})\geq 4\epsilon$, we see that
$Q$ is contained in a disk $D_{t}(q)$ with the property that
$D_{2t}(q)\subset\mathcal{O}$. Hence, by the Koebe distortion theorem [10,
Theorem I.4.5 and Equation I.(4.18)], the derivative of $\log\circ F$ in $Q$
is comparable (with absolute constants) to any one of its values in $Q$. Thus

$\displaystyle\frac{|v_{1}v_{2}|}{|w_{1}w_{2}|}|\theta(e^{\bullet}_{Q})|^{2}$
$\displaystyle=\frac{2}{\pi^{2}}\,\text{area}(Q)\left(\frac{\arg(F(w_{2}))-\arg(F(w_{1}))}{|w_{2}-w_{1}|}\right)^{2}$
$\displaystyle\leq\frac{2}{\pi^{2}}\,\text{area}(Q)\max_{z\in
e^{\circ}_{Q}}\left|\frac{d}{dz}\log\circ F(z)\right|^{2}$ $\displaystyle\leq
10\int_{Q}\frac{1}{|F(z)|^{2}}|F^{\prime}(z)|^{2}\,dA$
$\displaystyle=10\int_{F(Q)}\frac{1}{|z|^{2}}\,dA.$

If $F(e_{Q}^{\circ})$ intersects $\mathbb{H}$ but is not fully contained in
it, the above estimate still holds, since there are points
$w_{1}^{\prime},w_{2}^{\prime}$ on the edge $e_{Q}^{\circ}$ so that the
contribution of $e_{Q}^{\bullet}$ to $\mathcal{E}_{1}^{\bullet}(\theta)$ is

$\frac{2}{\pi^{2}}\text{area}(Q)\left(\frac{\arg(F(w_{2}^{\prime}))-\arg(F(w_{1}^{\prime}))}{|w_{1}w_{2}|}\right)^{2}\leq\frac{2}{\pi^{2}}\text{area}(Q)\left(\frac{\arg(F(w_{2}^{\prime}))-\arg(F(w_{1}^{\prime}))}{|w_{2}^{\prime}-w_{1}^{\prime}|}\right)^{2}.$

Also, for $|z-z_{0}|=2\epsilon$, we have

$|F(z)|\geq\frac{\epsilon}{2}|F^{\prime}(z_{0})|\geq
C_{0}\epsilon|\psi^{\prime}(z_{0})|\geq
C_{0}\frac{\epsilon}{4}\frac{\text{dist}(\psi(z_{0}),\partial\psi(\mathcal{O}))}{\operatorname{diam}(\mathcal{O})}\geq
C_{0}\epsilon\frac{\text{dist}(\psi(z_{0}),\partial\psi(\mathcal{O}))}{\operatorname{diam}(\Omega)}.$

The first inequality follows from the Koebe One-Quarter Theorem [10, Theorem
I.4.1, Equation I.(4.9)]. The second inequality follows from the chain rule,
where $C_{0}=(\pi/m)e^{2\pi/m}$ and therefore depends only on the analytic
quadrilateral $\Omega$. The third inequality is another application of the
Koebe One-Quarter Theorem [10, Corollary I.4.4]. And the last inequality
follows because without loss of generality we can assume that
$\operatorname{diam}(\mathcal{O})\leq 4\operatorname{diam}(\Omega)$, since
every point in $\hat{G}_{1}$ is within distance $\delta$ of $\Omega$ by
assumption.

Let $w_{0}\in\partial\mathcal{O}$ so that
$\text{dist}(\psi(z_{0}),\partial\psi(\mathcal{O}))=|\psi(z_{0})-\psi(w_{0})|$.
Then by Proposition 18, there is $p\in(0,1/2]$ so that

$\text{dist}(\psi(z_{0}),\partial\psi(\mathcal{O}))=|\psi(z_{0})-\psi(w_{0})|\geq
C_{1}|z_{0}-w_{0}|^{1/p}\geq
C_{1}\left(\text{dist}(z_{0},\partial\mathcal{O})\right)^{1/p}\geq