# Perfect Matchings and Essential Spanning Forests in Hyperbolic Double Circle
Packings

Zhongyang Li  Department of Mathematics, University of Connecticut, Storrs,
Connecticut 06269-3009, USA<EMAIL_ADDRESS>https://mathzhongyangli.wordpress.com

###### Abstract.

We investigate perfect matchings and essential spanning forests in planar
hyperbolic graphs via circle packings.

We prove the existence of nonconstant harmonic Dirichlet functions that vanish
in a closed set of the boundary, generalizing a result in [6]. We then prove
the existence of extremal infinite volume measures for uniform spanning
forests with partially wired boundary conditions and partially free boundary
conditions, generalizing a result in [5].

Using the double circle packing for a pair of dual graphs, we relate the
inverse of the weighted adjacency matrix to the difference of Green’s
functions plus an explicit harmonic Dirichlet function. This gives explicit
formulas for the probabilities of any cylindrical events.

We prove that the infinite-volume Gibbs measure obtained from approximations
by finite domains with exactly two convex white corners converging to two
distinct points along the boundary is extremal, yet not invariant with respect
to a finite-orbit subgroup of the automorphism group. We then show that under
this measure, a.s. there are no infinite contours in the symmetric difference
of two i.i.d. random perfect matchings.

As an application, we prove that the variance of the height difference of two
i.i.d. uniformly weighted perfect matchings under the boundary condition above
on a transitive nonamenable planar graph is always finite; in contrast to the
2D uniformly weighted dimer model on a transitive amenable planar graph as
proved in [39, 43], where the variance of height difference grows in the order
of $\log n$, with $n$ being the graph distance to the boundary. This also
implies that a.s. each point is surrounded by finitely many cycles in the
symmetric difference of two i.i.d. perfect matchings, again in contrast to the
2D Euclidean case.

##  1\. Introduction

A perfect matching, or a dimer cover of a graph is a subset of edges
satisfying the constraint that each vertex is incident to exactly one edge.
Perfect matchings are natural models in statistical mechanics, for example,
dimer configurations on the hexagonal lattice model the molecule structure of
graphite. See [40, 20] for more details.

Perfect matchings on 2-dimensional Euclidean lattice have been studied
extensively. Among a number of interesting topics including phase transition
([43]), limit shape ([15, 42, 12, 51, 44, 50, 54, 48, 57, 52, 11, 9]),
conformal invariance ([38, 39, 16, 47, 17, 4]), etc, one fundamental question
is to classify the infinite volume probability measures. It was proved in [62]
that for any given slope in the Newton polygon (which is the same as the set
of all the possible slopes), there is a unique translation-invariant, ergodic
Gibbs measure for the uniform dimer model on the 2-dimensional square grid.
More recently it was proved in [1] that for the uniformly weighted random
lozenge tilings of arbitrary domains approximating a closed, simply-connected
subset of ${\mathbb{R}}^{2}$ with piecewise smooth, simple boundary in the
scaling limit, around any point in the liquid region of its limit shape the
local statistics are given by the infinite-volume, translation-invariant,
extremal Gibbs measure of the appropriate slope.

Lately the dimer models on general lattices beyond the 2D Euclidean lattice
have attracted substantial interest; see for example [14] which proves a large
deviation principle for dimers on ${\mathbb{Z}}^{3}$; see also [46] for a
generalization of the honeycomb dimer model to higher dimensions. In this
paper we focus on the infinite-volume Gibbs measures for the dimer model on
hyperbolic planar graphs. Other statistical mechanical models in the
hyperbolic plane, such as the percolation model (see [21, 7, 49, 22, 23, 56,
55, 53]), have been studied extensively; the main aim of this paper is to
pursue this direction for the dimer model; and we aspire to open up a new
field in the research of the dimer model in the hyperbolic plane.

For statistical mechanical models on hyperbolic graphs, the structures of the
space of infinite-volume Gibbs measures are known to be significantly
different from the Euclidean case; for instance, although it is known from [2]
that any Gibbs measure for the 2D ferromagnetic Ising model is a convex
combination of the two extremal ones: the one with “$+$” boundary condition
and the one with “$-$” boundary condition (see also [19] for a similar result
for the 2D Potts model); however the Gibbs measure for the Ising model on
vertex-transitive tilings of the hyperbolic plane could be more complicated;
in particular it is known in [59, 67, 68] when the hyperbolic graph is self-
dual or when the vertex-degree is at least 35, under certain temperature,
unlike the Euclidean case, the free-boundary Gibbs measure is not $1/2$ of the
sum of the “$+$”-boundary Gibbs measure and “$-$” boundary Gibbs measure.

In the celebrated papers [38, 39], the scaling limit of the uniformly weighted
dimer model on the square grid with certain boundary conditions (Temperley
boundary conditions) is proved to be conformally invariant by relating the
inverse adjacency matrix to the differences of Green’s functions. The result
was later generalized to weighted dimer models on isoradial double graphs
([47]). When studying the hyperbolic graph, given that the hyperbolic plane
has nonzero curvature, the first question is to find a suitable embedding. It
turns out the double circle packing, which is known to exist for any
3-connected, transient, simple, proper planar graph with bounded vertex degree
and locally finite dual (see [31]) provides a nice framework, given that when
vertices of the pair of dual graphs are located at the circle centers, each
pair of dual edges are orthogonal. This critical property allows to obtain a
relation between the inverse adjacency matrices and the inverse Laplacian. For
the double circle packing graph, in the parabolic case (when the weighted
random walk on the graph is recurrent; see [25]) it has been proved in [63,
66, 24] that the discrete harmonic function converges to the continuous
harmonic function in the scaling limit. In this paper we shall focus on the
hyperbolic case.

Essentially we use a Temperley bijection between perfect matchings and
essential spanning forests, and study the distributions of perfect matchings
by studying the corresponding essential spanning forests. For the case of a
hypercubic graph ${\mathbb{Z}}^{d}$, this has been studied in [13], where the
celebrated transfer impedance theorem was proved. Cycle rooted spanning forest
on curved surface were studied in [32]. It was later proved in [33] that the
transfer current matrix converges to the differential of Green’s function. The
analysis of uniform spanning forest on non-amenable graphs was done in [5].

The main technique to study the dimer model in the hyperbolic plane is the
circle packing theorem [28, 29], the Kasteleyn-Temperley-Fisher theory [34,
35, 65] for the dimer model and the analysis of uniform spanning forest on
non-amenable graphs ([5]). The main results proved in this paper are
summarized below:

######  Theorem 1.1.

Let ${\mathbb{H}}^{2}$ be the Poincare unit disk model of the hyperbolic
plane. Let $G=(V,E)$ be a 3-connected, transient, simple, proper planar graph
with bounded vertex degree and locally finite dual.

  1. (1)

For any nonempty proper closed set
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$, if the subgraph of $G$ obtained
by removing an open neighborhood of $\mathcal{A}$ is transient, then there
exists a non-constant harmonic Dirichlet function whose values are constant
restricted to $\mathcal{A}$; see Theorem 3.14 for a precise statement.

  2. (2)

For any closed subset $\mathcal{A}\subset\partial{\mathbb{H}}^{2}$, there
exists an extremal probability measure on weighted essential spanning forests
of $G$, with wired boundary condition on $\mathcal{A}$ and free boundary
condition on $\partial{\mathbb{H}}^{2}\setminus\mathcal{A}$; see Theorem 3.17
for a precise statement.

  3. (3)

Consider perfect matchings on the bipartite graph $\overline{G}$ obtained from
the superpersion of $G$ and $G^{+}$. Under the Temperley boundary conditions,
we show that the limits of the entries of the inverse weighted adjacency
matrix on one class of black vertices are difference of the Dirichlet Green’s
functions; while limits of the entries of the inverse weighted adjacency
matrix on the other class of black vertices are difference of the Dirichlet
Green’s functions plus an explicit harmonic Dirichlet function; see Theorem
4.22 for a precise statement. We also show that this infinite volume Gibbs
measure is extremal under natural technical assumptions; see Propositions 6.18
and 6.20 for precise statements.

  4. (4)

Using a sequence of finite graphs with two convex white corners and no concave
white corners, such that the two boundaries portions divided by the two convex
corners converge to two closed subsets $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
of $\partial{\mathbb{H}}^{2}$, the limits of the entries of the inverse of the
weighted adjacency matrix converge to the difference of the Dirichlet Green’s
functions plus explicit harmonic Dirichlet functions; see Theorem 5.6 for a
precise statement. Moreover, this gives an extremal infinite-volume Gibbs
measure; see Proposition 6.5.

  5. (5)

For the infinite-volume Gibbs measure given in Theorem 5.6; or in Theorem
4.22, for uniformly weighted perfect matchings on $\overline{G}$, if $G$
satisfies certain isoperimetric inequalities, then the variance of the height
difference of two i.i.d. perfect matchings is always finite, given that the
the height difference on the boundary is 0. See Theorem 7.5 for precise
statements. We also prove the absence of doubly infinite self-avoiding paths
in the symmetric difference of two i.i.d. dimer configurations in the infinite
volume Gibbs measure given in Theorem 5.6, or the measure given in Theorem
4.22 under additional technical assumptions which is not necessarily
nonamenable. See Theorem 7.3 for a precise statement.

Theorem 1.1(1) generalizes results in [6]; Theorem 1.1(2) generalizes results
in [5]. Theorem 1.1(3) and (4), as far as we know, for the first time give
explicit formulas for the probabilities of cylinder events with respect to
infinite-volume Gibbs measures of perfect matchings in the hyperbolic plane.
It is worth noting that although the infinite-volume Gibbs measure in Theorem
1.1(4) is extremal, it is in general not invariant with respect to a finite-
orbit subgroup of the automorphism group of $G$, even if we assume that both
$G$ and $G^{+}$ are vertex transitive; see Remark 6.14.

Theorem 1.1(5) is in contrast with the 2D Euclidean case, where variance of
the difference at two points of height differences of two i.i.d. uniformly
weighted perfect matchings grows like $\log n$, where $n$ is the distance
between two points. While in the hypercubic lattice $d\geq 3$, whether the
associated flow has Gaussian fluctuation is an open problem ([14]). For the
infinite volume Gibbs measure obtained from Temperley boundary conditions, the
nonexistence of infinite double dimer contours was also obtained in [61] when
$G$ is a reversible random rooted graph which is one ended, bounded degree,
nonamenable triangulation almost surely; and exponential tail of height
functions were proved for nonamenable planar triangulations [61] by
considering the dimer height function as the winding of the branch of the
wired spanning forest.

The organization of the paper is as follows.

  * •

In Section 2, we first examine definitions and established properties in graph
theory; then introduce the Dirac operator on a bipartite graph formed through
the superposition of a pair of dual graphs characterized by perpendicular dual
edges; subsequently, several properties associated with this construction are
established.

  * •

In Section 3, we discuss harmonic Dirichlet functions and essential spanning
forest on transient planar graphs $G$ circle packed to the hyperbolic plane
${\mathbb{H}}^{2}$; and prove Theorem 1.1(1) (Lemma 3.14) and (2) (Theorem
3.17).

  * •

In Section 4, we introduce Temperley boundary conditions for finite subgraphs
derived from the superposition of a pair of dual graphs. We then establish
connections between Dirac operators, Laplacian operators, and random walks on
finite graphs with Temperley boundary conditions and prove Theorem 1.1(3).

  * •

In Section 5, we discuss the infinite volume Gibbs measure of perfect
matchings on $\overline{G}$ obtained by using sequences of finite graphs
exhausting the infinite graph with boundary conditions other than the
Temperley boundary conditions. We prove Theorem 1.1(4).

  * •

In Section 6, we prove that the limit measure for dimer configurations on
$\overline{G}$ obtained as in Theorem 1.1(4) is extreme; we also prove that
the limit measure for dimer configurations on $\overline{G}$ obtained as in
Theorem 1.1(3) is extreme under certain natural technical assumptions. We
first show that the extremity of the infinite-volume measure is equivalent to
the tail-triviality; then we show that the infinite-volume measure is tail-
trivial. The proof of the tail-triviality of the infinite volume measure is
inspired by the proof of Theorem 8.4 in [5]. One of the major differences is
instead of considering uniform spanning trees, we are considering perfect
matchings. The corresponding anti-symmetric function for the perfect matchings
is a “discrete integral” version of the anti-symmetric function for the
spanning trees. Although it is straightforward to see that the anti-symmetric
function for spanning trees has finite energy, the fact that the anti-
symmetric function for perfect matchings has finite energy is not so obvious;
but we still manage to prove it by using the transience property of the graph.

  * •

In Section 7, We consider the weak limit of probability measures on finite
graphs with the boundary conditions in Theorem 1.1(3)(4). We establish the
a.s. non-existence of infinite double dimer contours; and that each point is
almost surely enclosed by a finite number of cycles from the symmetric
difference of the two i.i.d. dimer configurations with either prescribed
distribution when the graph satisfies certain isoperimetric inequalities. The
proof depends on the fact that the decay rate of the Green’s functions on such
graphs with respect to the distance of two points due to isoperimetric
inequalities.

##  2\. Dirac Operators, Orthodiagonal Graphs and Double Circle Packings

In this section, we begin by examining definitions and established properties
in graph theory. Following that, we introduce the Dirac operator on a
bipartite graph formed through the superposition of a pair of dual graphs.
These dual graphs feature perpendicular dual edges. Subsequently, we establish
several properties associated with this construction.

###  2.1. Graph theory

We begin this section by reviewing definitions and established properties in
graph theory.

######  Definition 2.1.

  1. (1)

A graph is called vertex-transitive (resp. quasi-transitive) when there is a
unique orbit (at most finitely many orbits) of vertices under the action of
its automorphism group.

  2. (2)

A graph is called planar if it can be drawn in the plane such that vertices
are represented by points and edges are represented by continuous curves, and
no two edges can intersect at a point which is not a vertex.

We call a drawing of the graph into the plane ${\mathbb{R}}^{2}$ a proper
embedding in ${\mathbb{R}}^{2}$ if any compact subset $K$ in
${\mathbb{R}}^{2}$ intersects at most finitely many edges and vertices.

  3. (3)

The number of ends of a connected graph is the supreme over its finite
subgraphs of the number of infinite components that remain after removing the
subgraph.

  4. (4)

Let $G=(V,E)$ be a graph. Let $\Gamma$ be a subgroup of the automorphism group
of $G$ acting quasi-transitively on $G$. Let $\pi$ be a probability measure on
subgraphs of $G$.

    1. (a)

We say $\pi$ is $\Gamma$-invariant if for any event $\mathcal{A}$ and
$\gamma\in\Gamma$,

$\displaystyle\pi(\mathcal{A})=\pi(\gamma\mathcal{A}).$

    2. (b)

We say $\mu$ is $\Gamma$-ergodic if for any event $\mathcal{A}$ satisfying
$\mathcal{A}=\gamma\mathcal{A}$ for all $\gamma\in\Gamma$

$\displaystyle\pi(\mathcal{A})\in\\{0,1\\}.$

  5. (5)

A graph $G=(V,E)$ is called amenable if

$\displaystyle i_{E}(G):=\inf_{K\subset
V;0<|K|<\infty}\frac{|\partial_{E}K|}{|K|}=0$

where $\partial_{E}K$ consists of all the edges with exactly one vertex in $K$
and one vertex not in $K$. If $i_{E}(G)>0$, the graph is called non-amenable.

  6. (6)

Let $G=(V,E)$ be an infinite, connected, locally finite graph. Let
$\\{X_{n}\\}_{n\geq 0}$ be a simple random walk on $G$ and let
$p_{n}(y,x)=\mathbb{P}(X_{n}=y|X_{0}=x)$. Define the spectral radius $\rho$ of
$G$ by

$\displaystyle\rho(G):=\limsup_{n\rightarrow\infty}p_{n}(x,y)^{\frac{1}{n}}.$

Given that $G$ is connected $\rho(G)$ does not depend on the choices of $x,y$.

  7. (7)

(Theorem 6.7 in [58])Let $G=(V,E)$ be an infinite, connected, locally finite
graph, then $G$ is non-amenable if and only if $\rho(G)<1$.

###  2.2. Dirac operators

In this section, we introduce the Dirac operator on a bipartite graph
constructed from the superposition of a pair of dual graphs with perpendicular
dual edges and subsequently establish several of its properties.

######  Definition 2.2.

  1. (1)

Let $G=(V,E)$ be a graph, and let $\Lambda\subset V$ be a subset of vertices.
The subgraph $G_{\Lambda}$ of $G$ induced by $\Lambda$ has vertex set
$\Lambda$; and two vertices in $\Lambda$ are joined by an edge in
$G_{\Lambda}$ if and only in if they are joined by an edge in $G$.

  2. (2)

Let $G=(V,E)$ be a graph. We say $G_{j}$ is a subgraph of $G$ consisting of
faces of $G$ if $G_{j}$ is the union of faces in $G$.

  3. (3)

Let $G=(V,E)$ be a one-ended, planar graph, properly embedded in to
${\mathbb{H}}^{2}$ with no infinite faces. Let $G^{+}$ be the dual graph of
$G$. Let $G_{j}$ be a finite subgraph of $G$. Let $\partial^{+}{G}_{j}$
consist of all the edges in $G^{+}$, whose dual edge joins exactly one vertex
in $\Lambda$ and one vertex in $V\setminus\Lambda$. We say $G_{j}$ is simply-
connected if it is connected and $\partial^{+}G_{\Lambda}$ is a simple closed
curve.

  4. (4)

Let $G=(V,E)$ be a graph. Let $G_{j}$ is a subgraph of $G$ consisting of faces
of $G$. Define $\partial G_{j}$ to be the set of all the vertices in $G_{j}$
adjacent to at least one vertices no in $G_{j}$; as well as edges joining two
vertices in $G_{j}$, each of which is adjacent to at least one vertices no in
$G_{j}$.

######  Definition 2.3.

Let $G=(V,E)$ be an infinite, connected graph in the hyperbolic plane in which
each face is bounded and finite. Let $G^{+}=(V^{+},E^{+})$ be the planar dual
graph of $G$; i.e. each vertex of $G^{+}$ corresponds to a face of $G$; two
vertices in $G^{+}$ are joined by an edge in $G^{+}$ if their corresponding
faces in $G$ share an edge.

Let $\overline{G}$ be a bipartite graph obtained from the superposition of $G$
and $G^{+}$. A black vertex of $\overline{G}$ is either a vertex of $G$ or a
vertex of $G^{+}$; a white vertex corresponds to an edge in $G$ (or
equivalently, a dual edge in $G^{+}$). A black vertex $b$ is joined to a white
vertex $w$ by an edge in $\overline{G}$ if and only if one of the following
two cases occur

  * •

$b$ is a vertex of $G$, and $w$ corresponds to edge of $G$ incident to $b$; or

  * •

$b$ is a vertex of $G^{+}$, and $w$ corresponds to an edge of $G^{+}$ incident
to $b$.

Throughout this paper, we make the following assumption

######  Assumption 2.4.

Let $G=(V,E)$ be an infinite, connected graph properly embedded into the
hyperbolic plane such that each face is bounded and finite, with edge weights
$\\{\nu(e)>0\\}_{e\in E}$. Let $G^{+}=(V^{+},E^{+})$ be the dual graph with
edge weights $\\{\nu(f)>0\\}_{f\in E^{+}}$. Assume there exists a proper
embedding of $(G,G^{+})$ into the hyperbolic plane, such that

  * •

each vertex of $G^{+}$ is placed in the interior of a face of $G$; and

  * •

each edge of $G$ and $G^{+}$ is represented by line segments in the Euclidean
plane joining adjacent vertices; and

  * •

each pair of dual edges $(e,e^{+})$ are perpendicular.

Let $\overline{G}$ be the bipartite graph obtained from the superposition of
$G$ and $G^{+}$ as in Definition 2.3.

Note that each face of $\overline{G}$ is a quadrangle and can be inscribed in
a circle. Therefore $\overline{G}$ is a circle pattern graph; see [41] for
more about dimer models on circle pattern graphs.

Define the $\overline{\partial}$ operator as a matrix with rows and columns
indexed by vertices of $\overline{G}$. Then entries of $\overline{\partial}$
are defined as follows:

  * •

If two vertices $u,v$ are not adjacent in $\overline{G}$, let
$\overline{\partial}(u,v)=0$.

  * •

If $w$ is a white vertex and $b$ is a vertex of $\overline{G}$ and $w$ and $b$
are adjacent vertices in $\overline{G}$, let $f$ be the edge of $G$ or $G^{+}$
containing $b$ as an endpoint and $w$ as an interior point; define
$\overline{\partial}(w,b)=\overline{\partial}(b,w)$ be the complex number with
length $\nu(f)$ and direction the same as the ray in the Euclidean plane
pointing from $w$ to $b$.

######  Lemma 2.5.

Suppose Assumption 2.4 holds. Let $w_{1},b_{1},w_{2},b_{2}$ be all the
vertices along a face of $\overline{G}$ in cyclic order. Then

$\displaystyle(-1)\prod_{i=1}^{2}\frac{\overline{\partial}(w_{i},b_{i})}{\overline{\partial}(b_{i},w_{i+1})}>0$

###### Proof.

Note that the polygon formed by $w_{1},b_{1},w_{2},b_{2}$ in cyclic order is
inscribed to a circle $\mathcal{C}$ in the Euclidean plane. Assume
$\mathcal{C}$, as a Euclidean circle has center $v_{0}$. Let $l_{1}$, $l_{2}$,
$l_{3}$, $l_{4}$ be the (oriented) perpendicular bisectors to the line
segments $w_{1}b_{1}$, $b_{1}w_{2}$, $w_{2}b_{2}$, $b_{2}w_{1}$, respectively
such that each one of $l_{1}$, $l_{2}$, $l_{3}$, $l_{4}$ has starting point
$v_{0}$. Then for each $i\in\\{1,2\\}$;

$\displaystyle\arg(\overline{\partial}(w_{i},b_{i}))=\arg(l_{2i-1})-\frac{\pi}{2}$
$\displaystyle\arg(\overline{\partial}(b_{i},w_{i+1}))=\arg(l_{2i})+\frac{\pi}{2}.$

Note also that

$\displaystyle\sum_{i=1}^{2}\left[\arg(l_{2i})-\arg(l_{2i-1})\right]=\pi.$

Then the lemma follows. ∎

###  2.3. Double circle packing graphs

In this section, we examine the established facts concerning how a double
circle packing graph inherently leads to an orthodiagonal graph.

######  Definition 2.6.

  1. (1)

A circle packing is a collection of discs $P=\\{C_{v}\\}$ in the plane
$\mathbb{C}$ such that any two distinct discs in $P$ have disjoint interiors.

  2. (2)

Given a circle packing $P$, the nerve $G(P)$ of $P$ is the graph with vertex
set $P$ and two vertices are joined by an edge if and only if their
corresponding circles are tangent. The tangency graph $G(P)$ have a planar
embedding by placing a vertex at the center of each circle, and each edge is
represented by a line segment between centers of tangent circles.

  3. (3)

A double circle packing of $G$ is a pair of circle packings $(P,P^{+})$, such
that

    1. (a)

$G$ is the nerve of $P$, and the dual graph $G^{+}$ is the nerve of $P^{+}$.

    2. (b)

Primal and dual circles are perpendicular. More precisely, for each vertex $v$
and face $f$ of G, the discs $P^{+}(f)$ and $P(v)$ have nonempty intersection
if and only if $f$ is incident to $v$, and in this case the boundary circles
of $P^{+}(f)$ and $P(v)$ intersect at right angles.

See Figure 2.1 for an example of double circle packing.

Figure 2.1. Double circile packing: the primal graph is represented by black
lines; the dual graph is represented by gray lines; the cirle packing for the
primal graph is represented by red circles; the circle packing for the dual
graph is represented by blue circles.

From Definition 2.6, it is straightforward to check the following

  * •

Let $(P,P^{+})$ be a double circle packing of $G$; let $e=(b_{3},b_{4})$ be an
edge of $G$ and $e^{+}=(b_{1},b_{2})$ be the dual edge of $G$ in $G^{+}$. For
$i\in\\{3,4\\}$ (resp. $j\in\\{1,2\\}$), let $C_{i}$ (resp. $C_{j}$) be the
circle in $P$ corresponding to $b_{i}$ (resp. $b_{j}$). Then the tangent point
of $C_{3}$ and $C_{4}$ in $P$ is the same as the tangent point of $C_{1}$ and
$C_{2}$ in $P^{+}$.

######  Proposition 2.7.

([27]; see also [31]) Every infinite, polyhedral (meaning that is it both
simple (i.e., not containing any loops or multiple edges) and 3-connected
(i.e. the subgraph induced by $V\ {u,v}$ is connected for each $u,v\in V$))
proper plane graph G with locally finite dual admits a double circle packing
in either the Euclidean plane (CP parabolic) or the hyperbolic plane (CP
hyperbolic), but not both, and that this packing is unique up to Möbius
transformations and reflections. In particular, in the hyperbolic case, the
packing is unique up to isometries of the hyperbolic plane; and $G$ is CP
hyperbolic if and only if it is transient for simple random walk.

##  3\. Essential Spanning Forest

In this section, we discuss harmonic Dirichlet functions and essential
spanning forest on transient planar graphs $G$ circle packed to the hyperbolic
plane ${\mathbb{H}}^{2}$. Generalizing results in [6], in Lemma 3.14 we show
that for any nonempty proper closed set
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$, if the subgraph of $G$ obtained
by removing an open neighborhood of $G$ is transient, then there exists a non-
constant harmonic Dirichet function whose values are constant restricted to
$\mathcal{A}$. Generalizing results in [5], we show that for any closed subset
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$, there exists an ergodic
probability measure on weighted essential spanning forests of $G$, with wired
boundary condition on $\mathcal{A}$ and free boundary condition on
$\partial{\mathbb{H}}^{2}\setminus\mathcal{A}$; see Theorem 3.17.

######  Definition 3.1.

  1. (1)

Let $G=(V,E)$ be a graph with a countable vertex set $V$ and edge set $E$. Let
$\vec{E}$ be the directed edge set; i.e., each edge in $E$ corresponds to two
edges in $\vec{E}$ with opposite orientations. For each directed edge
$e\in\vec{E}$, write $\underline{e}$ (resp. $\overline{e}$) for the tail
(resp. head) of $\vec{E}$; the edge is oriented from its tail to its head.
Write $-e$ for the edge with reversed orientation. Each edge occurs with both
orientations. We use $E$ to denote the set of unoriented edges.

  2. (2)

A network is a pair $(G,C)$, where $G$ is a connected graph with at least two
vertices, and $C:E\rightarrow(0,\infty)$ is a function from $E$ to positive
reals.

  3. (3)

Let $(G,C)$ be an infinite network, meaning that the graph $G=(V,E)$ has
infinitely many vertices; i.e., $|V|=\infty$. A loopless subgraph of $G$ that
contains every vertex of $G$ and is not necessarily connected is called a
spanning forest, and a spanning forest in which every vertex is connected to
infinitely many others is called an essential spanning forest.

  4. (4)

Let $(G,C)$ be an infinite network. Let $\Omega$ be the set consisting of all
the essential spanning forest. A Gibbs measure $\mu$ on $\Omega$ is a
probability measure on $\Omega$ satisfying the following condition: let
$\mathcal{F}$ be a randomly chosen essential spanning forest of $G$ according
to the distribution $\mu$; then for any finite subgraph
$\Lambda=(V_{\Lambda},E_{\Lambda})\subset G$ with undirected edge set
$E_{\Lambda}$, a subgraph $S\subset\Lambda$ and a subset of unoriented edges
$B\subset E\setminus E_{\Lambda}$

$\displaystyle\mathbb{P}(\mathcal{F}=S\cup B|B=\mathcal{F}\cap[E_{0}\setminus
E_{\Lambda,0}])\begin{cases}=0&\mathrm{if}\ S\cup B\notin\Omega\\\
\propto\prod_{e\in S}C(e)&\mathrm{If}\ S\cup B\in\Omega\end{cases}$

  5. (5)

An anti-symmetric function $\theta:\vec{E}\rightarrow{\mathbb{R}}$ is one
satisfying

$\displaystyle\theta(e)=-\theta(-e).$

Let

$\displaystyle R(e):=\frac{1}{C(e)}.$

For anti-symmetric functions
$\theta,\theta^{\prime}:\vec{E}\rightarrow{\mathbb{R}}$, define an inner
product

$\displaystyle\langle\theta,\theta^{\prime}\rangle_{R}:=\frac{1}{2}\sum_{e\in
E}R(e)\theta(e)\theta^{\prime}(e)=\sum_{e\in
E_{0}}R(e)\theta(e)\theta^{\prime}(e)$

Let $l_{-}^{2}(E)$ be the Hilbert space of all anti-symmetric functions
$\theta:\vec{E}\rightarrow{\mathbb{R}}$ with
$\mathcal{E}(\theta):=\langle\theta,\theta\rangle_{R}$ finite.

  6. (6)

Let $l^{2}(V)$ be the Hilbert space of functions on $V$ with inner product

$\displaystyle\langle f,g\rangle_{C}:=\sum_{v\in V}C_{v}f(v)g(v).$

where

$\displaystyle C_{v}:=\sum_{e=(v,w)\in E}C(e)$

  7. (7)

Define the gradient operator $\triangledown:l^{2}(V)\rightarrow l_{-}^{2}(E)$
by

$\displaystyle\triangledown F(e):=C(e)[F(\overline{e})-F(\underline{e})]$

  8. (8)

Define the divergence operator $div:l_{-}^{2}(E)\rightarrow l^{2}(V)$ by

$\displaystyle\mathrm{div}\theta(v):=\frac{1}{C_{v}}\sum_{\underline{e}=v}\theta(e).$

  9. (9)

A function $f:V\rightarrow{\mathbb{R}}$ is harmonic at a vertex $v$ if

$\displaystyle f(v)=\sum_{e=(v,w)\in E}\frac{C(e)f(w)}{C_{v}}$

or equivalently, if $\mathrm{div}(\triangledown f)(v)=0$

  10. (10)

A function $f:V\rightarrow{\mathbb{R}}$ is Dirichlet if it has finite
Dirichlet energy, i.e.

$\displaystyle\mathcal{E}(\triangledown f):=\sum_{e=(v,w)\in
E}C(e)(f(v)-f(w))^{2}<\infty$

The space of Dirichlet functions on $G$ is denoted by $\mathbf{D}(G)$.

  11. (11)

Let $\mathbf{L}(G)$ be the set of all real functions
$V\rightarrow{\mathbb{R}}$ and $\mathbf{L}_{0}(G)$ be the set of all
$u\in\mathbf{L}(G)$ with finite support. Let $\mathbf{D}_{0}(G)$ be the
closure of $\mathbf{L}_{0}(G)$ in $\mathbf{D}(G)$ with respect to the norm
$\|u\|:=\sqrt{\mathcal{E}(\triangledown u)+|u(b_{0})|^{2}}$, where $b_{0}$ is
a fixed vertex in $V$.

  12. (12)

We say $G$ is hyperbolic of type 2 if $\mathbf{D}(G)\neq\mathbf{D}_{0}(G)$.

  13. (13)

A harmonic Dirichlet function on $G$ is a Dirichlet function which is harmonic
at every vertex of $G$. The space of harmonic Dirichlet functions on $G$ is
denoted by $\mathbf{HD}(G)$.

  14. (14)

Given a directed edge $e\in E$, let $\chi^{e}:=\mathbf{1}_{e}-\mathbf{1}_{-e}$
be the unit flow along $e$. Let

$\displaystyle\bigstar:=\triangledown l^{2}(V);$

i.e. $\bigstar$ is the subspace in $l^{2}_{-}(E)$ spanned by the stars
$\sum_{\underline{e}=v}C(e)\chi^{e}=-\triangledown\mathbf{1}_{v}$.

  15. (15)

If $e_{1},e_{2},\ldots,e_{n}$ is an oriented cycle in $G$, then
$\sum_{i=1}^{n}\chi^{e_{i}}$ is called a cycle. Let $\diamondsuit\subset
l_{-}^{2}(E)$ be the subspace spanned by these cycles.

######  Definition 3.2.

  1. Let $G=(V,E)$ be a graph.

  2. (1)

Let $\gamma$ be a path in $G$ and $q:E\rightarrow(0,\infty)$ be a metric on
$G$. Define the $q$-length of a path $\gamma$ in $G$ is the sum $q(e)$ over
all edges in $\gamma$, i.e.

$\displaystyle\mathrm{length}_{q}(\gamma)=\sum_{e\in\gamma\cap E}q(e)$

  3. (2)

A metric $q:E\rightarrow(0,\infty)$ is an $L^{2}(E)$ metric if

$\displaystyle\|q\|^{2}:=\sum_{e\in E}C(e)[q(e)]^{2}<\infty$

  4. (3)

Let $q$ be a metric on $G$ and let $d_{q}$ be the associated distance
function. Let $C_{q}(G)$ be the completion of $(V,d_{q})$, and let the
$q$-boundary of $G$ be $\partial_{q}G=C_{q}(G)\setminus V$.

  5. (4)

Let $\Gamma$ be a collection of infinite paths in $G$. Define the extremal
length of $\Gamma$ as follows:

$\displaystyle\mathrm{EL}(\Gamma):=\sup_{q}\inf_{\gamma\in\Gamma}\frac{\mathrm{length}_{q}(\gamma)}{\|q\|^{2}}$

If $\mathrm{EL}(\Gamma)=\infty$, we say $\Gamma$ is null.

  6. (5)

We say that a property holds for almost every path of a family $\Gamma$ of
paths if it does for the member of $\Gamma$ except for those belonging to a
subfamily with infinite extremal length.

######  Definition 3.3.

Let $G=(V,E)$ be a graph. For a set of edges $K\subset E$, let
$\mathcal{F}(K)$ denote the $\sigma$-field of events depending only on $K$.
Define the tail $\sigma$-field to be the intersection of
$\mathcal{F}(E\setminus K)$ over all finite $K$. We say that a measure on
$2^{E}$ has trivial tail if every event in the tail $\sigma$-field has measure
either 0 or 1.

It is known that tail triviality of a measure $\mu$ is equivalent to

(3.1) $\displaystyle\forall A_{1}\in\mathcal{F}(E);\ \forall\epsilon>0;\
\exists K\ \mathrm{finite},\ \forall A_{2}\in\mathcal{F}(E\setminus K),\
|\mu(A_{1}\cap A_{2})-\mu(A_{1})\mu(A_{2})|<\epsilon.$

See, e.g. page 120 of [18].

######  Lemma 3.4.

Assume that G is of hyperbolic type of order 2. Then every $f\in\mathbf{D}(G)$
can be decomposed uniquely in the form: $f=g+h$, where $g\in\mathbf{D}_{0}(G)$
and $h\in\mathbf{HD}(G)$.

###### Proof.

See Theorem 2.1 in [71]. ∎

######  Lemma 3.5.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Denote by $P_{a,\infty}(G)$ the set of
all paths from $a\in V$ to $\partial{\mathbb{H}}^{2}$ and by $P_{\infty}(G)$
the union of $P_{a,\infty}(G)$ for all $a\in V$. Then

  1. (1)

For every $u\in\mathbf{D}(G)$, $u(x)$ has a limit as $x$ tends to
$\partial{\mathbb{H}}^{2}$ for almost every $P\in P_{\infty}(G)$. We denote
this limit by $u(P)$.

  2. (2)

Let $u\in\mathbf{D}(G)$. Then $u\in\mathbf{D}_{0}(G)$ if and only if $u(P)=0$
for almost every infinite path $P$.

###### Proof.

See Theorem 3.1 of [36] for part (1) of the lemma; see Theorem 3.2 of [71] for
part (2) of the lemma. ∎

######  Definition 3.6.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Let $m:E\rightarrow(0,\infty)$ be a
metric defined by

(3.2) $\displaystyle m(e=(u,v))=r_{u}+r_{v};$

where $r_{u}$ (resp. $r_{v}$) is the radius of the circle centered at $u$
(resp. $v$) in the circle packing of which $G$ is the nerve.

Let $\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary nonempty
proper closed subset of $\partial{\mathbb{H}}^{2}$. Let
$\mathcal{S}_{\mathcal{A}}$ be the subspace of $\triangledown\mathbf{HD}(G)$
defined as follows:

(3.3) $\displaystyle\mathcal{S}_{\mathcal{A}}:$
$\displaystyle=\triangledown\mathcal{R}_{\mathcal{A}}$ (3.4)
$\displaystyle\mathcal{R}_{\mathcal{A}}:$
$\displaystyle=\mathbf{HD}(G)\cap\left[\mathrm{closure\ of}\
\\{f\in\mathbf{D}(G):\forall z\in\mathcal{A},\ \exists\mathrm{open\ set}\
U_{z}\subset V,\ \right.$ $\displaystyle\left.\mathrm{s.t.}\ z\in U_{z},\
\mathrm{and}\ \mathrm{supp}(f)\cap U_{z}=\emptyset\\}\right].$

where the closure is with respect to the norm
$\|u\|:=\sqrt{\mathcal{E}(\triangledown u)+|u(b_{0})|^{2}}$ with $b_{0}$ a
fixed vertex in $V$, and open set $U_{z}$ is with respect to the topology
induced by the metric $m$.

######  Lemma 3.7.

Let $G$ be a locally finite connected graph, and let $\Gamma$ be the
collection of all infinite paths in $G$. Then G is recurrent if and only if
$\Gamma$ is null.

###### Proof.

See [69, 70]; see also Theorem 3.2 in [6]. ∎

######  Lemma 3.8.

The metric $m$ defined by (3.2) is an $L^{2}(E)$ metric.

###### Proof.

Note that

(3.5) $\displaystyle\|m\|^{2}=\sum_{e\in E}[m(e)]^{2}=\sum_{e=(u,v)\in
E}[r_{u}+r_{v}]^{2}\leq 2\sum_{e=(u,v)\in E}[r_{u}^{2}+r_{v}^{2}]<\infty,$

where the last inequality follows from the fact that $G$ has bounded vertex
degree, the circles have disjoint interior, and therefore the sum of area of
circles in the circle packing is bounded above by the area of the unit disk. ∎

######  Lemma 3.9.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Let
$\theta_{k}:\vec{E}\rightarrow{\mathbb{R}}$ be a sequence of $l^{2}_{-}(E)$
anti-symmetric function such that

  * •

For each $k$, there exists a neighborhood (with respect to the topology
induced by the metric $m$) $U_{k}$ of $\mathcal{A}$ satisfying

$\displaystyle\mathrm{supp}(\theta_{k})\cap U_{k}=\emptyset.$

  * •

there exists $\theta\in l^{2}_{-}(E)$,

(3.6) $\displaystyle\lim_{k\rightarrow\infty}\|\theta_{k}-\theta\|_{R}=0,$

where the norm $\|\cdot\|_{R}$ is induced by the inner product defined in
Definition 3.1(5).

Then for almost every singly infinite path
$\gamma=(\gamma(0),\gamma(1),\ldots)$ satisfying
$\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0$ for some $z\in\mathcal{A}$, we
have

(3.7)
$\displaystyle\lim_{k\rightarrow\infty}\sum_{i=1}^{\infty}|\theta_{k}(e_{i})-\theta(e_{i})|=0.$

where $e_{i}\in\vec{E}$ is the directed edge from $\gamma(i-1)$ to
$\gamma(i)$.

###### Proof.

Let

$\displaystyle\Gamma:=\\{\gamma=(\gamma(0),\gamma(1),\ldots)\in
P_{\infty}:\sum_{i=0}^{\infty}|\theta_{k}(e_{i})-\theta(e_{i})|=\infty\\}$

Since $\theta_{k}-\theta\in l^{2}_{-}(E)$, there exists an $L^{2}(E)$ metric
under which every path in $\Gamma$ has infinite length; and therefore $\Gamma$
is null. Therefore, for almost every singly infinite path
$\gamma=(\gamma(0),\gamma(1),\ldots)$ ,
$\sum_{i=0}^{\infty}|\theta_{k}(\gamma(i))-\theta(\gamma(i))|<\infty$.

From (3.6), we infer that

(3.8) $\displaystyle\lim_{k\rightarrow\infty}\theta_{k}(e)=\theta(e);\ \forall
e\in\vec{E}.$

For each $a\in V$, define

$\displaystyle P_{a,\infty,\mathcal{A}}:=\\{\gamma\in
P_{b_{0},\infty}:\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0,\mathrm{for\
some}\ z\in\mathcal{A}\\}.$

Since for any singly infinite path in $G$ starting from some vertex $a\in V$,
we may add finitely many different steps at the beginning to obtain an singly
infinite path in $P_{b_{0},\infty}$ without changing the limiting behavior, it
suffices to show that for almost every singly infinite path $\gamma\in
P_{b_{0},\infty,\mathcal{A}}$,
$\lim_{n\rightarrow\infty}\sum_{i=1}^{\infty}|\theta_{k}(e_{i})-\theta(e_{i})|=0$.

Let

(3.9) $\displaystyle\Gamma_{0}:=\\{\gamma\in
P_{b_{0},\infty,\mathcal{A}}(G):\sum_{i=0}^{\infty}|\theta_{k}(e_{i})-\theta(e_{i})|<\infty\\}.$

then it is straightforward to check that $P_{b,\infty}\setminus\Gamma_{0}$ is
null by the same arguments as above.

Let $\eta\in\Gamma_{0}$. Assume there exist $\epsilon_{1}>0$, such that for
any $N,K$, there exist $n>N$ and $k>K$, satisfying

(3.10) $\displaystyle\sum_{i=n}^{\infty}|\theta_{k}(e)|>\epsilon_{1}$

Then let $n_{1},k_{1}$ be such that (3.10) holds with $(n,k)$ replaced by
$(n_{1},k_{1})$ and

(3.11)
$\displaystyle\sqrt{\sum_{i=n_{1}}^{\infty}C(e_{i})|\theta(e)|^{2}}<\frac{1}{2};$

and

(3.12)
$\displaystyle\sqrt{\sum_{i=1}^{\infty}C(e_{i})|\theta_{k_{1}}(e_{i})-\theta(e_{i})|^{2}}<\frac{1}{2};$

This is possible since $\theta\in l_{-}^{2}(E)$, and
$\theta_{k}\rightarrow\theta$ with respect to the norm $\|\cdot\|_{R}$. Note
that (3.11) and (3.12) implies

$\displaystyle\sqrt{\sum_{i=n_{1}}^{\infty}C(e_{i})|\theta_{k_{1}}(e_{i})|^{2}}<1;$

Let $N_{2}>n_{1}$ be such that
$\mathrm{supp}(\theta_{k_{1}})\cap\\{\eta(n):n\geq N_{2}\\}=\emptyset$, this
is possible since $\lim_{n\rightarrow\infty}\eta(n)\in\mathcal{A}$. Let
$n_{2}>N_{2},k_{2}>k_{1}$ be such that (3.10) holds with $(n,k)$ replaced by
$(n_{2},k_{2})$, and

(3.13)
$\displaystyle\sqrt{\sum_{i=n_{2}}^{\infty}C(e_{i})|\theta(e_{i})|^{2}}<\frac{1}{2^{2}};$

and

(3.14)
$\displaystyle\sqrt{\sum_{i=1}^{\infty}C(e_{i})|\theta_{k_{2}}(e_{i})-\theta(e_{i})|^{2}}<\frac{1}{2^{2}};$

Then (3.11) and (3.12) implies

$\displaystyle\sqrt{\sum_{i=n_{2}}^{\infty}C(e_{i})|\theta_{k_{2}}(e_{i})|^{2}}<\frac{1}{2}.$

Continuing this process one can find a sequence $(n_{i},k_{i})_{i=1}^{\infty}$
satisfying all the following conditions

  * •

$n_{i}<n_{i+1}$, $k_{i}<k_{i+1}$ for each $i\in{\mathbb{N}}$; and

  * •

$\mathrm{supp}(\theta_{k_{i}})\cap\\{\eta(n):n\geq n_{i+1}\\}=\emptyset$; and

  * •

(3.10) holds with $(n,k)$ replaced by $(n_{i},k_{i})$.

  * •

$\sqrt{\sum_{j=n_{2}}^{\infty}C(e_{j})|\theta_{k_{i}}(e_{j})|^{2}}<\frac{1}{2^{i-1}}$,
for each $i\in{\mathbb{N}}$.

Then one can find an $L^{2}(E)$ metric $q$, such that
$\mathrm{length}_{q}(\eta)=\infty$, for any $\eta$ satisfying (3.10). Hence
the collection of all $\eta$ satisfying (3.10) is null. We infer that for
almost every $\gamma\in P_{b,\infty,\mathcal{A}}$, for any $\epsilon>0$, there
exists $N$, $K$ (depending only on $\epsilon$ but is independent of each
other), such that for all $n>N$ an $k>K$,

(3.15) $\displaystyle\sum_{i=n}^{\infty}|\theta_{k}(e_{i})|<\epsilon$

We shall prove (3.7) for almost every $\gamma\in\Gamma_{0}$. Assume there
exists $\phi\in\Gamma_{0}$ such that (3.15) holds and (3.7) does not hold;
then there exists $\epsilon_{0}>0$, such that for any $K$, there exists $k>K$,
with

(3.16)
$\displaystyle\sum_{i=1}^{\infty}|\theta_{k}(r_{i})-\theta(r_{i})|\geq\epsilon_{0}$

where $r_{i}$ is the directed edge from $\phi(i-1)$ to $\phi(i)$.

Since (3.15) holds for $\phi$, let $N_{1},K_{1}$ positive integers such that
for all $n\geq N_{1}$ and $k\geq K_{1}$,

(3.17)
$\displaystyle\sum_{i=n}^{\infty}|\theta_{k}(r_{i})|<\frac{\epsilon_{0}}{4}$

From the definition of $\Gamma_{0}$ in (3.9), we see that there exists
$N_{0}>0$, such that

(3.18)
$\displaystyle\sum_{i=N_{0}}^{\infty}|\theta(r_{i})|<\frac{\epsilon_{0}}{4}$

Since $N_{2}=\max\\{N,N_{0},N_{1}\\}$ is finite, by (3.8) there exists
$K_{0}>0$, such that for any $k\geq K_{0}$,

(3.19)
$\displaystyle\sum_{i=1}^{N_{2}}|\theta_{k}(r_{i})-\theta(r_{i})|<\frac{\epsilon_{0}}{4}$

Let $K_{2}=\max\\{K,K_{0},K_{1}\\}+1$. We have for any $n>N_{2}$

$\displaystyle\sum_{i=n}^{\infty}|\theta_{K_{2}}(r_{i})|$
$\displaystyle\geq\sum_{i=1}^{\infty}|\theta_{k}(r_{i})-\theta(r_{i})|-\sum_{i=1}^{N_{2}}|\theta_{K_{2}}(r_{i})-\theta(r_{i})|$
$\displaystyle-\sum_{i=N_{2}}^{\infty}|\theta(r_{i})|-\sum_{i=N_{2}}^{\infty}|\theta_{K_{2}}(r_{i})|$
$\displaystyle\geq\epsilon_{0}-\frac{\epsilon_{0}}{4}-\frac{\epsilon_{0}}{4}-\frac{\epsilon_{0}}{4}=\frac{\epsilon_{0}}{4}.$

where the last inequality follows from (3.16), (3.17), (3.18), (3.19). But
this is a contradiction to the fact that
$\lim_{n\rightarrow\infty}\phi(n)\in\mathcal{A}$, and
$\mathrm{supp}(\theta_{K_{2}})\cap U_{K_{2}}=\emptyset$, for some open set
$U_{K_{2}}\supset\mathcal{A}$, which implies
$\sum_{i=n}^{\infty}|\theta_{K_{2}}(r_{i})|=0$. This completes the proof. ∎

######  Corollary 3.10.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Let
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary nonempty proper
closed subset of $\partial{\mathbb{H}}^{2}$. Let $\mathcal{R}_{\mathcal{A}}$
be defined as in (3.4). Let $f\in\mathbf{HD}$. If
$f\in\mathcal{R}_{\mathcal{A}}$ then for almost every singly infinite path
$\gamma=(\gamma(0),\gamma(1),\ldots)$ satisfying
$\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0$ for some $z\in\mathcal{A}$,
$\lim_{n\rightarrow\infty}f(\gamma(n))=0$.

###### Proof.

If $f\in\mathcal{R}_{\mathcal{A}}$, then there exist a sequence
$f_{k}\in\mathbf{D}(G)$ satisfying for any $k\in{\mathbb{N}}$ there exists a
neighborhood $U_{k}$ of $\mathcal{A}$, with $\mathrm{supp}(f_{k})\cap
U_{k}=\emptyset$; and $f_{k}$ converge to $f$ with respect to the norm
$\|u\|:=\sqrt{\mathcal{E}(\triangledown u)+|u(b_{0})|^{2}}$. This implies

$\displaystyle\lim_{k\rightarrow\infty}f_{k}(b_{0})=f(b_{0}).$

Then the corollary follows from Lemma 3.9 by letting

$\displaystyle\theta(e):=R(e)\triangledown f(e);\ \mathrm{and}\
\theta_{k}(e):=R(e)\triangledown f_{k}(e)\ \forall e\in\vec{E}\ \mathrm{and}\
k\geq 1.$

∎

######  Definition 3.11.

Given two subsets $A$ and $Z$ of vertices of a network, we call $\theta\in
l_{-}(E)$ a flow between $A$ and $Z$ if $\mathrm{div}\theta$ is 0 off of $A$
and $Z$; if it is nonnegative on $A$ and nonpositive on $Z$, then we say that
$\theta$ is a flow from $A$ to $Z$.

Let $\theta$ be a flow from $A$ to $Z$. We call

$\displaystyle\mathrm{Strength}(\theta):=\sum_{a\in A}\mathrm{div}\theta(a)$

the strength of the flow $\theta$. A flow of strength 1 is called a unit flow.

A voltage function is a function on the vertices of the network that is
harmonic at all $x\notin A\cup Z$. Given a voltage function $v$, we define the
associated current flow function $i$ on the edges by

$\displaystyle i(x,y)=c(x,y)[i(x)-i(y)].$

A unit current flow is current flow associated to a voltage function which
also has strength 1.

######  Lemma 3.12.

(Theorem 2.11 in [58])Let $G$ be a denumerable connected network. Random walk
on G is transient if and only if there is a unit flow on $G$ of finite energy
from some (every) vertex to $\infty$.

######  Corollary 3.13.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$.

Let $\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary nonempty
proper closed subset of $\partial{\mathbb{H}}^{2}$. Then the following two
statements are equivalent.

  1. (1)

There exists an open set $\mathcal{O}_{\mathcal{A}}$ satisfying
$\mathcal{A}\subset\mathcal{O}_{\mathcal{A}}$, such that for any open set
$U_{\mathcal{A}}$ satisfying $\mathcal{A}\subset
U_{\mathcal{A}}\subset\mathcal{O}_{\mathcal{A}}$, $G\setminus U_{\mathcal{A}}$
is transient.

  2. (2)

There exists an open set $U_{\mathcal{A}}$ satisfying
$\mathcal{A}\subset\mathcal{O}_{\mathcal{A}}$, such that, $G\setminus
U_{\mathcal{A}}$ is transient.

###### Proof.

It is straightforward to see that $(1)\Rightarrow(2)$. The fact that
$(2)\Rightarrow(1)$ follows from Lemma 3.12. ∎

######  Theorem 3.14.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Assume

(3.20) $\displaystyle\sup_{e\in E}C(e)=C_{0}<\infty.$

Let $\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary nonempty
proper closed subset of $\partial{\mathbb{H}}^{2}$ satisfying the following
conditions

  1. (a)

There exists an open set $U_{\mathcal{A}}\supseteq\mathcal{A}$, such that
$G\setminus U_{\mathcal{A}}$ is transient.

Then there exists a nonconstant harmonic Dirichlet function
$f:V\rightarrow{\mathbb{R}}$ such that $\triangledown
f\in\mathcal{S}_{\mathcal{A}}$.

###### Proof.

We identify each vertex of $G$ as a point in the unit disk which is the center
of the corresponding circle in the circle packing of $G$. By Corollary 3.13,
condition (a) implies that there exists an open set
$\mathcal{O}_{\mathcal{A}}$ satisfying
$\mathcal{A}\subset\mathcal{O}_{\mathcal{A}}$, such that for any open set
$U_{\mathcal{A}}$ satisfying $\mathcal{A}\subset
U_{\mathcal{A}}\subset\mathcal{O}_{\mathcal{A}}$, $G\setminus U_{\mathcal{A}}$
is transient.

Define $f:V\rightarrow{\mathbb{R}}$ by

$\displaystyle f_{0}(v)=d_{m}(v,\mathcal{A});\ \forall v\in V$

Then we claim that $f_{0}$ is a nonconstant Dirichlet function in the closure
(with respect to the norm $\|u\|:=\sqrt{\mathcal{E}(\triangledown u)}$) of the
set of functions

(3.21) $\displaystyle\\{g\in\mathbf{D}(G):\forall z\in\mathcal{A},\
\exists\mathrm{open\ set}\ U_{z}\subset V,\ \mathrm{s.t.}\ u\in U_{z},\
\mathrm{and}\ \mathrm{supp}(g)\cap U_{z}=\emptyset\\}.$

To see why that is true, note that the function $f_{0}$ has finite Dirichlet
energy because for each $e=(u,v)\in E$, we have

$\displaystyle|f_{0}(u)-f_{0}(v)|=|d_{m}(u,\mathcal{A})-d_{m}(v,\mathcal{A})|\leq
m(e);$

then

$\displaystyle\mathcal{E}(\triangledown f_{0})=\sum_{e=(u,v)\in
E_{0}}C(e)(f_{0}(u)-f_{0}(v))^{2}\leq C_{0}\sum_{e=(u,v)\in
E_{0}}[m(e)]^{2}<\infty;$

where the last inequality follows from (3.5). The fact that $f$ is nonconstant
follows directly from its definition.

Now we show that $f_{0}$ is in the closure of the set of functions defined by
(3.21). For each $\epsilon,\delta>0$, define

$\displaystyle U_{\mathcal{A},\epsilon}:$ $\displaystyle=\\{v\in
V:d_{m}(v,\mathcal{A})<\epsilon\\};\qquad
U_{\partial{\mathbb{H}}^{2},\delta}:=\\{v\in
V:d_{m}(v,\partial{\mathbb{H}}^{2})<\delta\\}.$

and

$\displaystyle
Q_{\mathcal{A},\epsilon}:=\\{z\in{\mathbb{H}}^{2},d_{{\mathbb{R}}^{2}}(z,\mathcal{A})<\epsilon\\}$

where $d_{{\mathbb{R}}^{2}}$ is the distance in the Euclidean plane. Then one
can see that

$\displaystyle U_{\mathcal{A},\epsilon}\subset Q_{\mathcal{A},\epsilon}$

For each $k\geq 1$, define $f_{k}:V\rightarrow{\mathbb{R}}$ by

$\displaystyle f_{k}(v):=d_{m}(v,U_{\mathcal{A},\frac{1}{k}})$

Then for each edge $e=(u,v)\in E\setminus
U_{\partial{\mathbb{H}}^{2},\delta}$,

$\displaystyle C(e)([f_{k}(u)-f_{k}(v)]-[f_{0}(u)-f_{0}(v)])^{2}\leq
C_{0}[|f_{k}(u)-f_{0}(u)|+|f_{k}(v)-f_{0}(v)|]^{2}\leq\frac{4C_{0}}{k^{2}}$

Choose $\delta_{k}>0$ such that

$\displaystyle\lim_{k\rightarrow\infty}\frac{4C_{0}}{k^{2}}|E\setminus
U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}|=0;\ \mathrm{and}\
\lim_{k\rightarrow\infty}\delta_{k}\rightarrow 0.$

where

$\displaystyle U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}=\\{v\in
U_{\partial{\mathbb{H}}^{2},\delta_{k}}:v^{\prime}\in
U_{\partial{\mathbb{H}}^{2},\delta_{k}};\forall v^{\prime}\sim v\ \mathrm{in}\
G\\}$

Then we have

$\displaystyle\|f_{k}-f\|^{2}$ $\displaystyle=\sum_{e=(u,v)\in
E}C(e)([f_{k}(u)-f_{k}(v)]-[f_{0}(u)-f_{0}(v)])^{2}+(f_{k}(b_{0})-f(b_{0}))^{2}$
$\displaystyle\leq\frac{C_{0}}{k^{2}}+\sum_{e=(u,v)\in E\setminus
U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}}C(e)([f_{k}(u)-f_{k}(v)]-[f_{0}(u)-f_{0}(v)])^{2}$
$\displaystyle+\sum_{e=(u,v)\in E\cap
U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}}C(e)([f_{k}(u)-f_{k}(v)]-[f_{0}(u)-f_{0}(v)])^{2}$
$\displaystyle\leq\frac{C_{0}}{k^{2}}+\frac{4C_{0}}{k^{2}}|E\setminus
U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}|+4C_{0}\mathrm{area}(Q_{\mathcal{A},\delta_{k}})$
$\displaystyle\leq\frac{C_{0}}{k^{2}}+\frac{4C_{0}}{k^{2}}|E\setminus
U^{\circ}_{\partial{\mathbb{H}}^{2},\delta_{k}}|+8\pi
C_{0}\delta_{k}\rightarrow 0$

as $k\rightarrow\infty$. It follows that $f_{0}$ is in the closure of the set
of functions defined by (3.21).

Since $\mathcal{A}$ is an non-empty, proper, closed subset of
$\partial\mathbb{H}^{2}$, from the definition of $f_{0}$, we see that $f_{0}$
is nonconstant on $\partial{\mathbb{H}}^{2}$.

Moreover, since $\mathcal{A}$ satisfies conditions (a), we claim that and
$f_{0}\notin\mathbf{D}_{0}(G)$. To see why that is true, let
$\mathcal{A}^{c}=\partial{\mathbb{H}}^{2}\setminus\mathcal{A}$. Note that by
Lemma 3.5(2), $f_{0}\in\mathbf{D}_{0}(G)$ if and only if for almost every
singly infinite path $\gamma=(\gamma(0),\gamma(1),\ldots,)$,
$\lim_{n\rightarrow\infty}f_{0}(\gamma(n))=0$. However, from the definition of
$f_{0}$ we see that for any $z\in\mathcal{A}^{c}$, and any path
$l:=(l(0),l(1),\ldots)$ satisfying $\lim_{n\rightarrow\infty}d_{m}(l(n),z)=0$,
we have $\lim_{n\rightarrow\infty}f_{0}(l(n))\neq 0$.

Let $\mathcal{O}_{\mathcal{A}}$ be a neighborhood of $\mathcal{A}$ given as in
condition (a). Then for any neighborhood $U_{\mathcal{A}}$ satisfying
$\mathcal{A}\subset U_{\mathcal{A}}\subset\mathcal{O}_{\mathcal{A}}$, and
almost every singly infinite path $\zeta=(\zeta(0),\zeta(1),\ldots,)$ in
$G\setminus U_{\mathcal{A}}$, $\zeta(n)$ must converge to a point
$x\in\mathcal{A}^{c}$ with respect to the metric $m$; since those do not
converge must have infinite lengths in $m$. By condition (a) and Lemma 3.7,
the collection of all the singly infinite paths in $G\setminus
U_{\mathcal{A}}$ is not null. Therefore the statement that for almost every
singly infinite path $\gamma=(\gamma(0),\gamma(1),\ldots,)$,
$\lim_{n\rightarrow\infty}f_{0}(\gamma(n))=0$ does not hold. We infer that
$f\notin\mathbf{D}_{0}(G)$.

Therefore $\mathbf{D}(G)\neq\mathbf{D}_{0}(G)$, and $G$ is hyperbolic type of
order 2. By Lemma 3.4, $f_{0}$ has the following decomposition

$\displaystyle f_{0}=g_{0}+f$

where $g_{0}\in\mathbf{D}_{0}(G)$, and $f\in\mathbf{HD}(G)$. Since
$f_{0}\notin\mathbf{D}_{0}(G)$, $f\neq 0$. From the identity $f=f_{0}-g_{0}$,
we see that for almost every singly infinite path $P$ converging to a point in
$\mathcal{A}$, $f(P)=0$, while it is not true that for almost every singly
finite path $P$ converging to a point in $\mathcal{A}^{c}$, $f(P)=0$.
Therefore $f$ is nonconstant. $f$ is in the closure of the set defined by
(3.21) because both $f_{0}$ and $g_{0}$ are in the closure of the set defined
by (3.21). Then the conclusion of the lemma follows. ∎

######  Assumption 3.15.

Let $G=(V,E)$ be a 3-connected, transient, simple planar graph with bounded
vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Suppose (3.20) holds. Let
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary closed subset of
$\partial{\mathbb{H}}^{2}$. Let

$\displaystyle V_{1}\subseteq V_{2}\subseteq\ldots\subseteq
V_{j}\subseteq\ldots$

be an increasing sequence of sets of vertices such that all the following
conditions hold

  1. (1)

$\cup_{j=1}^{\infty}V_{j}=V$; and

  2. (2)

let $G_{j}$ be the subgraph of $G$ induced by $V_{j}$; i.e. the vertex set of
$G_{j}$ is $V_{j}$, two vertices in $V_{j}$ are joined by an edge in $G_{j}$
if and only if they are joined by an edge in $G$. Then for each point
$z\in\mathcal{A}$, there exists a singly infinite path
$\gamma=(\gamma(0),\gamma(1),\ldots,\gamma(n),\ldots)$ in $G_{j}$ such that

$\displaystyle\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0.$

where $m$ is the metric defined as in (3.2). Let

$\displaystyle V_{j}\supseteq U_{j,1}\supseteq U_{j,2}\supseteq\ldots\supseteq
U_{j,s}\supseteq\ldots$

be a decreasing subset of vertices such that

  * •

$\cap_{s=1}^{\infty}\overline{U}_{j,s}=\mathcal{A}$; where
$\overline{U}_{j,s}$ is the closure of $U_{j,s}$ under the metric $m$; and

  * •

for each finite $s$, $V_{j}\setminus U_{j,s}$ is a finite set of vertices.

Let $G_{j,s}^{\bullet}$ be the graph obtained from $G_{j}$ by contracting all
the vertices in $U_{j,s}$ and edges joining two vertices in $U_{j,s}$ into one
vertex $z_{j,s}$.

######  Assumption 3.16.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual that can be circle packed in the
hyperbolic plane ${\mathbb{H}}^{2}$. Suppose (3.20) holds. Let
$\mathcal{A}\subset\partial{\mathbb{H}}^{2}$ be an arbitrary closed subset of
$\partial{\mathbb{H}}^{2}$. Let

$\displaystyle V\supseteq V_{1}\supseteq V_{2}\supseteq\ldots
V_{j}\supseteq\ldots$

be a decreasing subset of vertices such that

  * •

$\cap_{j=1}^{\infty}\overline{V}_{j}=\mathcal{A}$; where $\overline{V}_{j}$ is
the closure of $V_{j}$ under the metric $m$.

Let $G_{j}^{\bullet}$ be graph obtained from $G$ by contracting all the
vertices in $V_{j}$ and edges joining two vertices in $V_{j}$ into one vertex
denoted by $\mathbf{a}_{j}$. For each $j\in{\mathbb{N}}$, let

$\displaystyle U_{j,1}\subseteq U_{j,2}\subseteq\ldots\subseteq
U_{j,s}\subseteq\ldots$

be an increasing sequence of finite sets of vertices such that all the
following conditions hold

  1. (1)

$\mathbf{a}_{j}\in U_{j,s}$; and

  2. (2)

$\cup_{j=1}^{\infty}U_{j,s}=[V\setminus V_{j}]\cup\\{\mathbf{a}_{j}\\}$;

Let $G_{j,s}^{\bullet}$ be the subgraph of $G_{j}^{\bullet}$ induced by
$U_{j,s}$; i.e. the vertex set of $G_{j,s}$ is $U_{j,s}$, two vertices in
$U_{j,s}$ are joined by an edge in $G_{j,s}$ if and only if they are joined by
an edge in $G$.

######  Theorem 3.17.

Suppose either Assumption 3.15 or Assumption 3.16 holds. Let $\mu_{j,s}$ be
the Gibbs measure for spanning trees on $G_{j,s}$, then
$\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}$ converges weakly
to a measure $\mu_{\mathcal{A}}$ on essential spanning forest on $G$.
Moreover,

  1. (1)

$\mu_{\mathcal{A}}$ has trivial tail.

  2. (2)

If $\mathcal{A}=\emptyset$, $\mu_{\mathcal{A}}=FSF$, where FSF represents the
limit measure of free spanning trees on finite graphs;

  3. (3)

If $\mathcal{A}=\partial{\mathbb{H}}^{2}$, $\mu_{\mathcal{A}}=WSF$, where WSF
represents the limit measure of wired spanning trees on finite graphs, i.e.,
identifying all the vertices on the boundary as one vertex;

  4. (4)

Let $\mathfrak{F}$ be the random essential spanning forest on $G$, then for
any $e\in E$,

$\displaystyle
WSF(e\in\mathfrak{F})\leq\mu_{\mathcal{A}}(e\in\mathfrak{F})\leq
FSF(e\in\mathfrak{F}).$

  5. (5)

If $G$ is quasi-transitive, $\mu_{\mathcal{A}}$ is ergodic but not necessarily
automorhpism-invariant.

  6. (6)

If $\mathcal{A}$ satisfies condition (a) in Lemma 3.14, then
$\mu_{\mathcal{A}}\neq WSF$.

###### Proof.

We only prove the conclusion under Assumption 3.15; the conclusion under
Assumption 3.16 can be proved similarly.

Note that $\triangledown\mathbf{HD}(G)$ is a subspace of $l_{-}^{2}(E)$. It is
straightforward to check that

$\displaystyle\triangledown\mathbf{HD}(G)\perp\bigstar;\qquad\triangledown\mathbf{HD}(G)\perp\diamondsuit;\qquad\diamondsuit\perp\bigstar;$

where $\perp$ is with respect to the inner product
$\langle\cdot,\cdot\rangle_{R}$ in $l_{-}^{2}(E)$. Indeed, $l^{2}_{-}(E)$ has
a decomposition as follows:

$\displaystyle
l_{-}^{2}(E)=\bigstar\oplus\diamondsuit\oplus\triangledown\mathbf{HD}(G)$

For each $e\in E$, define

(3.22) $\displaystyle
I_{e,\mathcal{A}}=P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e};$

i.e. $I_{e}$ is the orthogonal projection of $\chi_{e}$ to the subspace
$\bigstar\oplus\mathcal{S}_{\mathcal{A}}$ of $l_{-}^{2}(E)$. Let
$E_{\frac{1}{2}}\subset\vec{E}$ be a subset of $\vec{E}$ such that each edge
in $E$ appears exactly once in $E_{\frac{1}{2}}$ with a fixed orientation.
Then define a matrix $Y$ with rows and columns indexed by $E_{\frac{1}{2}}$ by

$\displaystyle Y_{\mathcal{A}}(e,f)=I_{e,\mathcal{A}}(f).$

Then the weak convergence of
$\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}$ follows from
Lemmas 3.20 and 3.22.

Let $\langle\chi_{F}\rangle$ be the linear space spanned by $\\{\chi_{f}:f\in
F\\}$. The fact that $\mu_{\mathcal{A}}$ has trivial tail follows from Lemma
3.23 by letting $s\rightarrow\infty$ and then letting $j\rightarrow\infty$,
and the fact that

$\displaystyle\|P_{\langle\chi_{F}\rangle}I_{e,j,s}\|^{2}\rightarrow 0$

given that $d_{G}(F,K)\rightarrow 0$ and $e\in K$. Then (1) follows.

Since tail triviality implies mixing, when $G$ is quasi-transitive and
infinite, $\mathrm{Aut}(G)$ has an infinite orbit, we infer that in this case
$\mu_{\mathcal{A}}$ is ergodic. If $\mathcal{A}$ is not automorphism
invariant, then $\mu_{\mathcal{A}}$ is not automorphism invariant. Then (5)
follows.

(2)-(3) are straightforward to verify from the deterministic expression of
$\mu_{\mathcal{A}}$. To see why (4) is true, note that by Lemma 3.20 we have

$\displaystyle\mu_{\mathcal{A}}(e\in\mathfrak{F})=I_{e,\mathcal{A}}(e)=C(e)\langle
P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e},\chi_{e}\rangle_{R}=C(e)\|P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}\|_{R}^{2}.$

Moreover, it is known from [5] that

$\displaystyle
FSF(e\in\mathfrak{F})=C(e)\|P_{\bigstar}\chi_{e}\|_{R}^{2};\qquad
WSF(e\in\mathfrak{F})=C(e)\|P^{\perp}_{\diamondsuit}\chi_{e}\|_{R}^{2};$

Then (4) follows from the fact that

$\displaystyle\bigstar\subset[\bigstar\oplus\mathcal{S}_{\mathcal{A}}]\subset[\bigstar\oplus\triangledown\mathbf{HD}(G)]=[\diamondsuit^{\perp}].$

When $\mathcal{A}$ satisfies condition (a) in Lemma 3.14, then by Lemma 3.14
we see that $\mathcal{S}_{\mathcal{A}}$ is non-trivial linear space. Therefore
$\bigstar$ is a proper subset of $\bigstar\oplus\mathcal{S}_{\mathcal{A}}$.
Then (6) follows. ∎

Let $G$ be an infinite graph. We define two possibly different currents,

$\displaystyle I_{e,F}:=P^{\perp}_{\diamondsuit}\chi_{e},$

the free current between the endpoints of $e$ (also called the “limit
current”), and

$\displaystyle I_{e,W}:=P_{\bigstar}\chi_{e},$

the wired current between the endpoints of $e$ (also called the “minimal
current”). Recall the following two propositions

######  Proposition 3.18.

(Free Currents) Let G be an infinite graph exhausted by subgraphs
$\\{G_{n}=(V_{n},E_{n})\\}_{n=1}^{\infty}$. Let e be an edge in $G_{1}$ and
let

$\displaystyle\bigstar_{G_{n}}:=\triangledown l^{2}(V_{n})$

$I_{n}:=P_{\bigstar{G_{n}}}\chi_{e}$ . Then $\|I_{n}-I_{e,F}\|_{R}\rightarrow
0$ as $n\rightarrow\infty$ and $\mathcal{E}(I_{e,F})=I_{e,F}(e)$.

###### Proof.

See Corollary 3.17 of [64]; see also Proposition 7.1 of [5]; in the latter
further assumption that each subgraph $G_{n}$ is finite was made; while in
Corollary 3.17 of [64] $G_{n}$ is not required to be finite. ∎

######  Proposition 3.19.

Let $G$ be an infinite graph exhausted by subnetworks
$\\{G_{n}\\}_{n=1}^{\infty}$. Let $G_{n,W}=(V_{n,W},E_{n,W})$ be formed by
identifying the complement of $G_{n}$ in $G$ to a single vertex. Let $e$ be an
edge in $G_{1}$ and

$\displaystyle\bigstar_{G_{n,W}}:=\triangledown l^{2}(V_{n,W})$

Define $I_{n}:=P_{\bigstar_{G_{n,W}}}\chi_{e}$. Then
$\|I_{n}-I_{e,W}\|_{R}\rightarrow 0$ as $n\rightarrow\infty$ and
$\mathcal{E}(I_{e,W})=I_{e,W}(e)$, which is the minimal energy among all
$\theta\in l^{2}_{-}(E)$ satisfying $\mathrm{div}\theta=\mathrm{div}\chi_{e}$.

###### Proof.

See Theorem 3.25 of [64]; see also Proposition 7.2 of [5] with the further
assumption that each $G_{n}$ is finite. ∎

######  Lemma 3.20.

Suppose (3.20) holds. Let $\mathcal{T}$ be the weighted spanning tree on
$G_{j,s}$, and let $e\in E_{\frac{1}{2}}$. Then

$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}(e\in\mathcal{T}):=I_{e,\mathcal{A}}(e);$

where $I_{e,\mathcal{A}}$ is defined by (3.22).

###### Proof.

Consider the weighted spanning tree model on the finite graph
$G_{j,s}=(V_{j,s},E_{j,s})$. By the transfer current theorem on finite graphs
(see [13]; see also Theorem 4.1 of [5]), we obtain

$\displaystyle\mu_{j,s}(e\in\mathcal{T})=I_{e,j,s}(e)$

where we let $\diamondsuit_{j,s}$ be the subspace of $l^{2}_{-}(E_{j,s})$
generated by $\sum_{i=1}^{n}\chi^{e_{i}}$ for all the oriented cycles
$e_{1},e_{2},\ldots,e_{n}$ in $G_{j,s}$, and assume

$\displaystyle I_{e,j,s}:=P_{\bigstar_{j,s}}\chi_{e}.$

and $P_{\bigstar_{j,s}}\chi_{e}$ is the orthogonal projection of $\chi_{e}\in
l^{2}_{-}(E_{j,s})$ to the subspace $\bigstar_{j,s}:=\triangledown
l^{2}(V_{j,s})$. Note that for each fixed $e,j$ by Proposition 3.19 we obtain

$\displaystyle\lim_{s\rightarrow\infty}\|I_{e,j,s}-I_{e,j,W_{j}}\|_{R}=0;$

where

$\displaystyle I_{e,j,W_{j}}=P_{\bigstar_{j}}\chi_{e};$

and $\bigstar_{j}:=\triangledown l^{2}(V_{j})$.

Moreover, by Proposition 3.18 we obtain

$\displaystyle\lim_{j\rightarrow\infty}\|I_{e,j,W_{j}}-I_{e,\mathcal{A}}\|_{R}=0;$

then the lemma follows.

∎

Let $F$ be a set of edges. The contracted graph $G/F$ is defined by by
identifying every pair of vertices that are joined by edges in $F$. We
identify the set of edges in $G$ and in $G/F$. Let $T_{G}$ (resp. $T_{G/F}$)
denote a subtree of $G$ (resp. $G/F$). Recall that $\langle\chi_{F}\rangle$ be
the linear space spanned by $\\{\chi_{f}:f\in F\\}$. Let

$\displaystyle
Z:=P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\langle\chi_{F}\rangle.$

Let

$\displaystyle\bigstar\oplus\mathcal{S}_{\mathcal{A}}=\widehat{\bigstar}\oplus
Z;$

Let $e$ be an edge that does not form a cycle together with edges in $F$. Let
$\widehat{I}_{e}:=P_{\widehat{\bigstar}}\chi_{e}$. Then

(3.23)
$\displaystyle\widehat{I}_{e}:=P_{\widehat{\bigstar}}\chi_{e}=P_{Z}^{\perp}P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}=P_{\langle\chi_{F}\rangle}^{\perp}P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}=P_{\langle\chi_{F}\rangle}^{\perp}I_{e,\mathcal{A}}$

where the identity
$P_{Z}^{\perp}P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}=P_{\langle\chi_{F}\rangle}^{\perp}P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}$
follows from the fact that
$P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e}\in\bigstar\oplus\mathcal{S}_{\mathcal{A}}$.

Recall the following lemma:

######  Lemma 3.21.

(Proposition 4.2 in [5])Let $G$ be a finite connected graph. Assuming that
there is no cycle of $G$ in $F$, the distribution of $T_{G}$ conditioned on
$F\subset T_{G}$ is equal to the distribution of $T_{G/F}\cup F$ when we think
of $T_{G}$ and $T_{G/F}$ as sets of edges.

######  Lemma 3.22.

Suppose (3.20) holds. Let $\mathcal{T}$ be the weighted random spanning tree
on $G_{j,s}$, and let $e_{1},\ldots,e_{k}\in E_{\frac{1}{2}}$. Then

$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}(e_{1},\ldots,e_{k}\in\mathcal{T}):=\det[Y_{\mathcal{A}}(e_{i},e_{j})]_{1\leq
i,j\leq k}$

###### Proof.

The proof is an adaptation of the proof of (4.4) in [5].

If some cycle can be formed from the edges $e_{1},\ldots,e_{k}$, then the
linear combination of corresponding columns of
$[Y_{\mathcal{A}}(e_{i},e_{j})]$ is zero. Therefore, we assume that there are
no such cycles for the remainder of the proof.

Note that

$\displaystyle Y_{\mathcal{A}}(e,f)=I_{e,\mathcal{A}}(f)$
$\displaystyle=C(e)\langle I_{e,\mathcal{A}},\chi_{f}\rangle=C(e)\langle
P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e},\chi_{f}\rangle_{R}$
$\displaystyle=C(e)\langle
P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{e},P_{\bigstar\oplus\mathcal{S}_{\mathcal{A}}}\chi_{f}\rangle_{R}=C(e)\langle
I_{e,\mathcal{A}},I_{f,\mathcal{A}}\rangle_{R}$

Therefore

$\displaystyle\det[Y_{\mathcal{A}}(e_{i},e_{j})]_{1\leq i,j\leq
k}=\left[\prod_{1\leq i\leq k}C(e_{i})\right]\det Y_{k};$

where $Y_{k}$ is the Gram matrix with entries $\langle
I_{e_{i},\mathcal{A}},I_{e_{j},\mathcal{A}}\rangle_{R}$. The determinant of a
Gram matrix is the squared volume of the parallelepiped spanned by its
determining vectors; therefore

$\displaystyle\det[Y_{\mathcal{A}}(e_{i},e_{j})]_{1\leq i,j\leq
k}=\prod_{i=1}^{k}\left[C(e_{i})\|P_{Z_{i}}^{\perp}I_{e_{i},\mathcal{A}}\|^{2}_{R}\right]$

where $Z_{i}$ is the linear space spanned by
$\\{I_{e_{1},\mathcal{A}},\ldots,I_{e_{i-1},\mathcal{A}}\\}$.

By Lemmas 3.20, 3.21, we have

$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}(e_{i}\in\mathcal{T}|e_{1},\ldots,e_{i-1}\in\mathcal{T})=\widehat{I}_{e_{i}}(e_{i})$

with $F=\\{e_{1},\ldots,e_{i-1}\\}$. Moreover

$\displaystyle\widehat{I}_{e_{i}}(e_{i})=C(e_{i})\langle\widehat{I}_{e_{i}},\chi_{e_{i}}\rangle_{R}=C(e_{i})\langle
P_{\widehat{\bigstar}}\chi_{e},P_{\widehat{\bigstar}}\chi_{e}\rangle_{R}=C(e_{i})\|P_{Z_{i}}^{\perp}I_{e,\mathcal{A}}\|^{2}.$

where the last identity follows from (3.23). Therefore we have

$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}(e_{1},\ldots,e_{k}\in\mathcal{T})$
$\displaystyle=\prod_{i=1}^{k}\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\mu_{j,s}(e_{i}\in\mathcal{T}|e_{1},\ldots,e_{i-1}\in\mathcal{T})$
$\displaystyle=\prod_{i=1}^{k}\|P_{Z_{i}}^{\perp}\left[C(e_{i})I_{e_{i},\mathcal{A}}\|^{2}\right].$

Then the lemma follows. ∎

######  Lemma 3.23.

Let $T$ be the weighted random spanning tree on the finite graph $G_{j,s}$.
Let $F$ and $K$ be disjoint nonempty sets of edges. If
$A_{1}\in\mathcal{F}(K)$ and $A_{2}\in\mathcal{F}(F)$, then

$\displaystyle|\mu_{j,s}[A_{1}\cap
A_{2}]-\mu_{j,s}[A_{1}]\mu_{j,s}[A_{2}]|\leq\left(2^{2|K|}|K|\sum_{e\in
K}C(e)\|P_{\langle\chi_{F}\rangle}I_{e,j,s}\|_{R}^{2}\right)^{\frac{1}{2}}$

###### Proof.

See Theorem 8.4 of [5]. ∎

##  4\. Dimers and Temperley Boundary Conditions

In this section, we introduce Temperley boundary conditions for finite
subgraphs derived from the superposition of a pair of dual graphs. We then
establish connections between Dirac operators, Laplacian operators, and random
walks on finite graphs with Temperley boundary conditions. We prove explicit
expressions of the limits of entries of the inverse Dirac operator under
Temperley boundary conditions, which give a unique infinite-volume Gibbs
measure for perfect matchings on the bipartite graph obtained from the
superposition of a transient planar graph and its dual graph; see Lemma 4.22.

###  4.1. Perfect matchings on finite subgraphs: probability measure and local
statistics

######  Definition 4.1.

Let $G=(V,E)$ be a graph. A perfect matching, or dimer configuration $M$ on
$G$ is a subset of edges such that each vertex in $V$ is incident to exactly
one edge in $M$.

We define a probability measure for perfect matchings on finite subgraphs of
$\overline{G}$ as follows:

######  Definition 4.2.

Suppose Assumption 2.4 holds. Let $\overline{G}_{0}$ be a finite subgraph of
$\overline{G}$. Let $\mathcal{M}$ be the collection of all the perfect
matchings on $\overline{G}_{0}$. Define the partition function for perfect
matchings on $\overline{G}_{0}$ by

$\displaystyle Z(\overline{G}_{0}):=\sum_{M\in\mathcal{M}}\prod_{e=(w,b)\in
M}|\overline{\partial}(w,b)|$

For each $M\in\mathcal{M}$, let the probability of $M$ be

$\displaystyle\mathbb{P}(M):=\frac{1}{Z(\overline{G}_{0})}\prod_{e=(w,b)\in
M}|\overline{\partial}(w,b)|$

From Lemma 2.5, we obtain the following corollary:

######  Corollary 4.3.

Suppose Assumption 2.4 holds. Let $\overline{G}_{0}$ be the finite subgraph of
$\overline{G}$ admitting a perfect matching. Then

  1. (1)

$\displaystyle
Z(\overline{G}_{0})=\left|\det\left[\overline{\partial}|_{W_{\overline{G}_{0}},B_{\overline{G}_{0}}}\right]\right|$

where $\overline{\partial}|_{W_{\overline{G}_{0}},B_{\overline{G}_{0}}}$ is
the submatrix of $\overline{\partial}$ with rows restricted to white vertices
of $\overline{G}_{0}$ and columns restricted to black vertices of
$\overline{G}_{0}$.

  2. (2)

Let $t\geq 1$ be a positive integer, and $e_{1},\ldots,e_{t}$ be $t$ distinct
edges of $\mathcal{G}_{j}$ with no common vertices. Let $e_{1},\ldots,e_{t}\in
M$ be the event that all the edges $e_{1},\ldots,e_{t}$ appear in a random
perfect matching. Then

$\displaystyle\mathbb{P}(e_{1},\ldots,e_{t}\in
M)=\prod_{i=1}^{t}\nu(e_{i})\left|\det[\overline{\partial}|_{W_{\overline{G}_{0}},B_{\overline{G}_{0}}}]^{-1}_{\\{w_{1},\ldots,w_{t}\\},\\{b_{1},\ldots,b_{t}\\}}\right|$

where $e_{i}=(w_{i},b_{i})$ has white vertex $w_{i}$ and black vertex $b_{i}$
for all $1\leq i\leq t$.

###### Proof.

Part (1) of the corollary is inspired by the Kasteley-Temperley-Fisher
algorithm to count the number of perfect matchings in a finite planar graph;
see [34, 65, 35]. Assume

$\displaystyle|W_{\overline{G}_{0}}|=|B_{\overline{G}_{0}}|=k$

and write

$\displaystyle
A:=\overline{\partial}|_{W_{\overline{G}_{0}},B_{\overline{G}_{0}}}.$

Let $S_{k}$ be the symmetric group of $k$ elements. Then we have

(4.1) $\displaystyle\det\left[A\right]:=\sum_{\sigma\in
S_{k}}(-1)^{I(\sigma)}\prod_{i=1}^{k}A_{i,\sigma(i)};$

where $I(\sigma)$ is the inversion number of the permutation $\sigma$. Each
nonzero term on the right hand side of (4.1) corresponds to a dimer
configuration; Lemma 2.5 implies that each nonzero term in the right hand side
of (4.1) has the same argument and the modulus equal to the product of weights
of edges present in the dimer configuration.

Part (2) of the corollary is inspired by the work of Kenyon to compute local
statistics of lattice dimers; see [37]. ∎

Let $\overline{D}$ be obtained from $\overline{\partial}$ by multiplying the
weight of each edge $e$ by $\frac{1}{\sqrt{\nu(e)\nu(e^{+})}}$, where $e^{+}$
is the dual edge of $e$. More precisely we have

$\displaystyle\overline{D}=S\overline{\partial}S^{*}$

where for each black vertex $b$ we have

$\displaystyle S(b,b)=1;$

and for each white vertex $w$, let $e$, $e^{+}$ be the pair of edges in
$\overline{G}$ and $\overline{G}^{+}$ in which $w$ lies, we have

$\displaystyle S(w,w)=\frac{1}{\sqrt{\nu(e)\nu(e^{+})}};$

and $S^{*}$ is the transpose conjugate of $S$.

###  4.2. Temperley Boundary Conditions

Suppose Assumption 2.4 holds. Assume that $G_{j}$ is a finite simply-connected
subgraph of $G$ drawn on ${\mathbb{H}}^{2}$ consisting of faces of $G$ and is
induced by a finite subset of vertices.

Let $G_{j}^{+}$ be the interior dual graph of $G_{j}$, i.e., each vertex of
$G_{j}^{+}$ corresponds to a finite face of $G_{j}$; and two vertices of
$G_{j}^{+}$ are joined by an edge $e^{+}$ of $G_{j}^{+}$ if and only if the
corresponding faces in $G$ share an edge $e$. Let $\overline{G}_{j}$ be the
bipartite graph obtained from the superposition of $G_{j}$ and $G_{j}^{+}$ as
in Definition 2.3.

Figure 4.1. Temperley Boundary Condition: the primal graph is represented by
black lines; the dual graph is represented by gray lines.

######  Lemma 4.4.

Suppose Assumption 2.4 holds. Let $\mathcal{G}_{j}$ be the graph obtained from
$\overline{G}_{j}$ by removing a a fixed vertex $b_{0}$ on the boundary. Then
$\mathcal{G}_{j}$ admits a perfect matching.

###### Proof.

A directed spanning tree of a graph is a connected acyclic collection of edges
of the graph, where each edge has a chosen direction such that each vertex but
one has exactly one outgoing edge. The single vertex with no outgoing edges is
called the root of the tree.

Recall that $b_{0}$ be the unique black vertex of $\overline{G}_{j}$ that is
not in $\mathcal{G}_{j}$. Consider a directed spanning tree $T_{b_{0}}$ of
$G_{j}$ rooted at $b_{0}$. Let $R_{b_{0}}^{+}$ be the collection of dual edges
of $\mathcal{G}_{j}$, such that a dual edge is present in $R_{b_{0}}^{+}$ if
and only if its corresponding primal edge is absent in $T_{b_{0}}$.

We claim that each connected component of $R_{b_{0}}^{+}$ contains exactly one
dual vertex not in the interior dual graph $\overline{G}_{j}^{+}$. Indeed, if
a component of $R_{b_{0}}^{+}$ contains at least two dual vertices not in
$\overline{G}_{j}^{+}$, then $T_{b_{0}}$ will not be connected. If a component
of $R_{b_{0}}^{+}$ contains no dual vertices not in $\overline{G}_{j}^{+}$,
then $T_{b_{0}}$ will contain at least one cycle. Also because $T_{b_{0}}$ is
connected, $R_{b_{0}}^{+}$ is acyclic; hence $R_{b_{0}}^{+}$ is a forest;
i.e., each component of $R_{b_{0}}^{+}$ is a tree. Moreover, $R_{b_{0}}^{+}$
contains all the dual vertices in $\overline{G}_{j}^{+}$ because $T_{b_{0}}$
is acyclic.

We may let the unique dual vertex in one component of $R_{b_{0}}^{+}$ not in
$\overline{G}_{j}^{+}$ be the root of the tree, and orient the edges of
$R_{b_{0}}^{+}$ such that every dual vertex in
$R_{b_{0}}^{+}\cap\overline{G}_{j}^{+}$ has exactly one outgoing edge; and
each dual vertex in $R_{b_{0}}^{+}\setminus\overline{G}_{j}^{+}$ has no
outgoing edges.

We can obtain a dimer configuration $M$ on $\mathcal{G}_{j}$ from $T_{b_{0}}$
as follows: an edge $e=(u,v)\in M$ if and only if one of the following two
cases occurs

  1. (1)

there exists an edge $f=(u,w)\in T_{b_{0}}$, such that $e\subset f$, and $f$
is the unique outgoing edge at $u$ in $T_{b_{0}}$; or

  2. (2)

there exists an edge $g=(z,v)\in R_{b_{0}}^{+}$ such that $e\subset g$, and
$g$ is the unique outgoing edge at $v$ in $R_{b_{0}}^{+}$.

∎

######  Remark 4.5.

The finite graph $\mathcal{G}_{j}$ constructed as above is said to exhibit a
Temperley boundary condition; see Figure 4.1 for an example.

###  4.3. Discrete Laplacian Operator and Discrete Harmonic Functions

######  Definition 4.6.

Suppose Assumption 2.4 holds. Define

(4.2) $\displaystyle\Delta:=\overline{D}^{*}\overline{D}$

Let

$\displaystyle\overline{D}_{j}:=\overline{D}|_{W(\mathcal{G}_{j}),B(\mathcal{G}_{j})}$

i.e., $\overline{D}_{j}$ is the submatrix of $\overline{D}$ with rows
restricted to white vertices of $\mathcal{G}_{j}$ and columns restricted to
black vertices of $\mathcal{G}_{j}$. Note also that the white vertices of
$\mathcal{G}_{j}$ are identified with edges of $G_{j}$ or edges of
$G_{j}^{+}$, and black vertices of $\mathcal{G}_{j}$ are identified with
$[V(G_{j})\cup V(G_{j}^{+})]\setminus\\{b_{0}\\}$, where $b_{0}$ be the vertex
removed from $\overline{G}_{j}$ to obtain $\mathcal{G}_{j}$. Define the
discrete Laplacian operator $\Delta_{G_{j}^{+}}$ on $G_{j}^{+}$ by

(4.3)
$\displaystyle\Delta_{G_{j}^{+}}:=\left.\overline{D}_{j}^{*}\overline{D}_{j}\right|_{V(G_{j}^{+}),V(G_{j}^{+})}=\left.\Delta\right|_{V(G_{j}^{+}),V(G_{j}^{+})}.$

Define the discrete Laplacian operator $\Delta_{G_{j}}$ on $G_{j}$ by

(4.4)
$\displaystyle\Delta_{G_{h}}:=\left.\overline{D}_{j}^{*}\overline{D}_{j}\right|_{V(G_{j})\setminus\\{b_{0}\\},V(G_{j})\setminus\\{b_{0}\\}};$

Here $V(G_{j})$ (resp. $V(G_{j}^{+})$) is the set consisting of all the
vertices of $G_{j}$ (resp. $G_{j}^{+}$).

The next two lemma gives the asymptotics of the entries of $\Delta_{G_{j}}$
and $\Delta_{G_{j}^{+}}$ as $\epsilon\rightarrow 0$. We first make the
following assumption:

######  Lemma 4.7.

Suppose Assumption 2.4 holds.

  1. (1)

If $b_{1},b_{2}$ are adjacent vertices in $G$, we have

$\displaystyle\Delta(b_{1},b_{2})=\Delta_{G}(b_{1},b_{2})=-\frac{\nu(e)}{\nu(e^{+})};$

where $e=\langle b_{1},b_{2}\rangle$ is the edge in $G$ with endpoints
$b_{1},b_{2}$.

  2. (2)

If $b_{1},b_{2}$ are adjacent vertices in $G^{+}$, we have

$\displaystyle\Delta(b_{1},b_{2})=\Delta_{G^{+}}(b_{1},b_{2})=-\frac{\nu(e)}{\nu(e^{+})};$

where $e=\langle b_{1},b_{2}\rangle$ is the edge in $G^{+}$ with endpoints
$b_{1},b_{2}$.

  3. (3)

If $d_{\overline{G}}(b_{1},b_{2})>2$, then

$\displaystyle\Delta(b_{1},b_{2})=0.$

where $d_{\overline{G}}(\cdot,\cdot)$ is the graph distance between two
vertices.

  4. (4)

If $d_{\overline{G}}(b_{1},b_{2})=2$ and $b_{1}\in V$, $b_{2}\in V^{+}$, then

$\displaystyle\Delta(b_{1},b_{2})=0.$

  5. (5)

For each vertex $b$ of $G$

$\displaystyle\Delta(b,b)=\Delta_{G}(b,b)=\sum_{e\in E(G):e\sim
b}\frac{\nu(e)}{\nu(e^{+})}$

where the sum is over all the incident edges of $b$ in $G$.

  6. (6)

For each vertex $b$ of $G^{+}$

$\displaystyle\Delta(b,b)=\Delta_{G^{+}}(b,b)=\sum_{e\in E(G^{+}):e\sim
b}\frac{\nu(e)}{\nu(e^{+})}$

where the sum is over all the incident edges of $b$ in $G^{+}$.

###### Proof.

(1), (2) and (3) (5) (6) are straightforward to be verified; it remains to
prove (4).

Assume $d_{\overline{G}}(b_{1},b_{2})=2$ and $b_{1},b_{2}$ are not adjacent in
$G$. In this case exactly one of $b_{1}$ and $b_{2}$ is a vertex of $G$ and
the other is a vertex of $G^{+}$. Without loss of generality, assume $b_{1}$
is a vertex of $G$ and $b_{2}$ is a vertex of $G^{+}$. Then we can find two
white vertices $w_{1}$ and $w_{2}$, such that each one of $w_{1}$ and $w_{2}$
is adjacent to both $b_{1}$ and $b_{2}$ in $\overline{G}$. Assume
$(b_{1},w_{1})$ is part of an edge $(b_{1},b_{1}^{\prime})$ in $G$,
$(b_{2},w_{1})$ is part of an edge $(b_{2},b_{2}^{\prime})$ in $G^{+}$,
$(b_{1},w_{2})$ is part of an edge $(b_{1},b_{1}^{\prime\prime})$ in $G$,
$(b_{2},w_{2})$ is part of an edge $(b_{2},b_{2}^{\prime\prime})$ in $G^{+}$.
Let $\alpha_{1}$ (resp. $\alpha_{2}$) be the unit vector pointing from $b_{1}$
to $b_{1}^{\prime}$ (resp. $b_{2}$ to $b_{2}^{\prime}$); let
$\alpha_{1}^{\prime}$ (resp. $\alpha_{2}^{\prime}$) be the unit vector
pointing from $b_{1}$ to $b_{1}^{\prime\prime}$ (resp. $b_{2}$ to
$b_{2}^{\prime\prime}$).

Then we have

$\displaystyle\Delta(b_{1},b_{2})$
$\displaystyle=[\overline{D}^{*}(b_{1},w_{1})][\overline{D}(w_{1},b_{2})]+[\overline{D}^{*}(b_{1},w_{2})][\overline{D}(w_{2},b_{2})]$
$\displaystyle=\sqrt{\frac{\nu(b_{1},b_{1}^{\prime})}{\nu(b_{2},b_{2}^{\prime})}}\cdot\sqrt{\frac{\nu(b_{2},b_{2}^{\prime})}{\nu(b_{1},b_{1}^{\prime})}}\overline{\alpha}_{1}\alpha_{2}+\sqrt{\frac{\nu(b_{1},b_{1}^{\prime\prime})}{\nu(b_{2},b_{2}^{\prime\prime})}}\cdot\sqrt{\frac{\nu(b_{2},b_{2}^{\prime\prime})}{\nu(b_{1},b_{1}^{\prime\prime})}}\overline{\alpha}_{1}^{\prime}\alpha_{2}^{\prime}$
$\displaystyle=\overline{\alpha}_{1}\alpha_{2}+\overline{\alpha}_{1}^{\prime}\alpha_{2}^{\prime}$

Note also that either

$\displaystyle\alpha_{2}=\mathbf{i}\alpha_{1};\qquad\alpha_{2}^{\prime}=-\mathbf{i}\alpha_{1}^{\prime}$

or

$\displaystyle\alpha_{2}=-\mathbf{i}\alpha_{1};\qquad\alpha_{2}^{\prime}=\mathbf{i}\alpha_{1}^{\prime}.$

where $\mathbf{i}^{2}=-1$ is the imaginary unit. Then (4) follows. ∎

######  Lemma 4.8.

Suppose Assumption 2.4 holds. Let $\Delta_{G_{j}}$, $\Delta_{G_{j}^{+}}$ be
defined as in (4.3) and (4.4). Then $\Delta_{G_{j}}$ is the Neumann Laplacian
operator on $G_{j}$ and $\Delta_{G_{j}^{+}}$ is the Dirichlet Laplacian
operator on $G_{j}^{+}$.

###### Proof.

By Lemma 4.7, it suffices to verify the boundary conditions. Let $f$ be a
function defined on the vertices of $G_{j}$. Let $b$ be a boundary vertex of
$G_{j}$. Explicit computations show that

$\displaystyle[\Delta_{G_{j}}f](b)=\sum_{b^{\prime}\in V(G_{j}):b^{\prime}\sim
b}\frac{\nu(bb^{\prime})}{\nu([bb^{\prime}]^{+})}[f(b)-f(b^{\prime})]$

Hence in order to make $[\Delta_{G_{j}}f]=0$, it suffices that $f$ is harmonic
at every vertex of $G_{j}$ and $f(u)=f(b)$ for all vertices $u\sim b$ and
$u\in V(G)\setminus V(G_{j})$. It follows that $\Delta_{G_{j}}$ is the Neumann
Laplacian operator on $G_{j}$.

Let $g$ be a function defined on the vertices of $G_{j}^{+}$. Let $s$ be a
boundary vertex of $G_{j}^{+}$, which corresponds to a boundary face of
$G_{j}$. Explicit computations show that

$\displaystyle[\Delta_{G_{j}^{+}}f](b)=\sum_{s^{\prime}\in
V(G_{j}^{+}):s^{\prime}\sim
s}\frac{\nu(ss^{\prime})}{\nu([ss^{\prime}]^{+})}[f(b)-f(b^{\prime})]+\sum_{s^{\prime\prime}\in
V(G^{+})\setminus V(G_{j}^{+}):s^{\prime\prime}\sim
s}\frac{\nu(ss^{\prime\prime})}{\nu([ss^{\prime\prime}]^{+})}f(b)$

Hence in order to make $[\Delta_{G_{j}^{+}}f]=0$, it suffices that $f$ is
harmonic at every vertex of $G_{j}^{+}$ and $f(u)=0$ for all vertices $u\sim
s$ and $u\in V(G^{+})\setminus V(G_{j}^{+})$. It follows that
$\Delta_{G_{j}^{+}}$ is the Dirichlet Laplacian operator on $G_{j}^{+}$. ∎

######  Definition 4.9.

Let $G_{\Lambda}\subseteq G$ be a simply-connected subgraph of $G$ induced by
a set of vertices $\Lambda\subseteq V$. Let $\Lambda_{o}$ consist of all the
vertices in $\Lambda$ in which all of whose adjacent vertices in $G$ are in
$\Lambda$ as well. A discrete harmonic function on $G_{\Lambda}$ is a function
$f:\Lambda\rightarrow\mathbb{C}$ satisfying

(4.5) $\displaystyle\Delta f(v)=0,\qquad\forall v\in\Lambda_{o}.$

We can similarly define the discrete harmonic function on $G^{+}$.

######  Lemma 4.10.

(Maximal modulus principle for discrete harmonic functions)Let
$G_{\Lambda}\subseteq G$ be a finite simply-connected subgraph of $G$ induced
by a set of vertices $\Lambda\subseteq V$. Let
$f:\Lambda\rightarrow\mathbb{C}$ be a discrete harmonic function on
$G_{\Lambda}$. Then

$\displaystyle\max_{v\in\Lambda}|f(v)|=\max_{u\in[\Lambda\setminus\Lambda_{o}]}|f(u)|$

###### Proof.

Note that for each $v\in\Lambda_{o}$, by (4.5) we have

$\displaystyle f(v)=\sum_{u\in V:u\sim v\ \mathrm{in}\
G}-\frac{\Delta(v,u)}{\Delta(v,v)}f(u)$

with

$\displaystyle-\frac{\Delta(v,u)}{\Delta(v,v)}>0,\ \mathrm{and}\ \sum_{u\in
V:u\sim v\ \mathrm{in}\ G}-\frac{\Delta(v,u)}{\Delta(v,v)}=1.$

Then we obtain for each $v\in\Lambda_{o}$,

$\displaystyle|f(v)|\leq\max_{u\in V:u\sim v\ \mathrm{in}\ G}|f(u)|.$

Then the lemma follows. ∎

######  Lemma 4.11.

There exists at most one $\overline{D}^{-1}:E\times[V\cup
V^{+}]\rightarrow\mathbb{C}$ satisfying both of the following conditions

  1. (1)

$\overline{D}[\overline{D}]^{-1}=I$; and

  2. (2)

there exists a function $f:{\mathbb{N}}\rightarrow{\mathbb{R}}^{+}$, such that

$\displaystyle\lim_{m\rightarrow\infty}f(m)=0;$

and

$\displaystyle\left|\overline{D}^{-1}(b,w)\right|\leq
f(d_{\overline{G}}(b,w))$

###### Proof.

Assume there exist two functions $K_{1}:E\times[V\cup
V^{+}]\rightarrow\mathbb{C}$ and $K_{2}:E\times[V\cup
V^{+}]\rightarrow\mathbb{C}$ satisfying both (1) and (2). Then

$\displaystyle\Delta(K_{1}-K_{2})=\overline{D}^{*}[\overline{D}(K_{1}-K_{2})]=0;$

which implies that $K_{1}-K_{2}$ is discrete harmonic. Then $K_{1}-K_{2}=0$
follows from (2) and Lemma 4.10. ∎

###  4.4. Discrete Green’s Function on an Infinite Transient Graph

######  Definition 4.12.

We say a random walk on a graph is recurrent if it visits its starting
position infinitely often with probability one and transient if it visits its
starting point finitely often with strictly positive probability.

Let $x$ be the starting point of the random walk $\\{X_{n}\\}_{n=0}^{\infty}$.
Let $N_{x}$ be the total time of visits of the random walk at $x$ and
$\theta_{x}$ be the first return time to $x$, i.e.

$\displaystyle\theta_{x}:=\inf\\{n\geq 1:X_{n}=x,\mathrm{if}\ X_{0}=x\\}$

By writing

$\displaystyle\mathbb{P}(N_{x}=1)=\mathbb{P}(\theta_{x}=\infty);$
$\displaystyle\mathbb{P}(N_{x}=n)=\mathbb{P}(\theta_{x}<\infty)^{n-1}\mathbb{P}(\theta_{x}=\infty).$

We see that a random walk starting at $x$ is transient if and only if
$\mathbb{E}N_{x}<\infty$, in this case the random walk a.s. visits $x$
finitely many times; see also [10].

######  Definition 4.13.

For $u,v\in V(G^{+})$, define the discrete Green’s function by

$\displaystyle
G(u,v)=\sum_{n=0}^{\infty}\mathbb{P}^{u}(X_{n}=v)=\mathbb{E}N_{u}$

where $X_{n}$ is the random walk on $G^{+}$ starting at $u$ with transition
probability

(4.6)
$\displaystyle\mathbb{P}^{u}(X_{n+1}=v|X_{n}=y)=\begin{cases}0&\mathrm{If}\ y\
\mathrm{and}\ v\ \mathrm{are\ not\ adjacent\ in}\ G^{+}\\\
\left|\frac{\Delta(y,v)}{\Delta(y,y)}\right|&\mathrm{otherwise}\end{cases}$

Similarly we can define a random walk and discrete Green’s function on $G$.

One can verify the following lemma by straightforward computations.

######  Lemma 4.14.

Let the discrete Green’s function on $G^{+}$ be defined as in Definition 4.13;
write

(4.7) $\displaystyle F^{v}(u):=\frac{G(u,v)}{|\Delta(v,v)|}.$

then for each fixed u

$\displaystyle[\Delta F^{v}](u)=\delta_{v}(u)$

where

$\displaystyle\delta_{v}(u)=\begin{cases}1&\mathrm{If}\ u=v\\\
0&\mathrm{Otherwise}.\end{cases}$

###  4.5. Dirichlet Green’s Function on a Finite Graph

######  Definition 4.15.

(See also Definition 1.6 in [60] and Definition 1.1 in [8])Let the random walk
on $G^{+}=(V^{+},E^{+})$ be defined as in (4.6). For $u\in V(G_{j}^{+})$, let
$\tau^{u}_{j}$ be the first time when a random walk on $G^{+}$ visits a vertex
in $V^{+}\setminus V(G_{j}^{+})$

For $u,v\in V(G_{j}^{+})$, define the discrete Dirichlet Green’s function on
$G_{j}^{+}$ to be

$\displaystyle
G_{j,d}(u,v)=\sum_{n=0}^{\infty}\mathbb{P}^{u}(X_{n}=v;\tau^{u}_{j}>n)$

######  Assumption 4.16.

Suppose the random walk on $G^{+}$ defined by (4.6) satisfies

$\displaystyle\sum_{n=0}^{\infty}\mathbb{P}^{u}(X_{n}=v)<\infty,\ \forall
u,v\in V(G^{+})$

######  Lemma 4.17.

Suppose Assumption 4.16 holds. Let $\\{G_{n}^{+}\\}_{n=1}^{\infty}$ be a
sequence of simply-connected finite subgraphs of $G^{+}$ induced by subset of
vertices such that

$\displaystyle G_{1}^{+}\subseteq G_{2}^{+}\subseteq\ldots\subseteq
G_{n}^{+}\subseteq\ldots$

and

(4.8) $\displaystyle\cup_{n=1}^{\infty}G_{n}^{+}=G^{+}$

Then for any $u,v\in V(G_{\epsilon}^{+})$

$\displaystyle\lim_{n\rightarrow\infty}G_{n,d}(u,v)=G(u,v)$

where $G(u,v)$ is defined as in (4.13).

###### Proof.

Note that

$\displaystyle
G(u,v)-G_{n,d}(u,v)=\sum_{j=0}^{\infty}\mathbb{P}^{u}(X_{j}=v;\tau^{u}_{n}\leq
j)$

Let $\mathcal{N}_{v,[t,\infty)]}$ be the number of visits to $v$ of a random
walk starting from $u$ in $[t,\infty)$ steps. Then

(4.9) $\displaystyle
G(u,v)-G_{n,d}(u,v)=\mathbb{E}\mathcal{N}_{v,[\tau_{n}^{u},\infty)}$

By (4.8) we have $\tau_{n}^{u}\leq\tau_{n+1}^{u}$, and therefore
$\mathcal{N}_{v,[\tau_{n^{u},\infty)}}\geq\mathcal{N}_{v,[\tau_{n+1^{u},\infty)}}$.
Moreover, (4.8) also implies

(4.10) $\displaystyle\lim_{n\rightarrow\infty}\tau_{n}^{u}=\infty.$

Then

(4.11)
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\mathcal{N}_{v,[\tau_{n}^{u},\infty)}=0.$

follows from Assumption 4.16 and (4.10). Then the lemma follows from (4.9) and
(4.11). ∎

######  Lemma 4.18.

Let the Dirichlet Green’s function on $G_{j}^{+}$ be defined as in Definition
4.15; write

(4.12) $\displaystyle F_{j,d}^{v}(u):=\frac{G_{j,d}(u,v)}{\Delta(v,v)}.$

then for each fixed $u\in V(G_{j}^{+})$

$\displaystyle[\Delta_{G_{j}^{+}}F_{j,d}^{v}](u)=\delta_{v}(u).$

###### Proof.

Note that

$\displaystyle[\Delta_{G_{j}^{+}}F_{j,d}^{v}](u)$
$\displaystyle=\frac{1}{\Delta(v,v)}\left[\Delta(u,u)F^{v}(u)+\sum_{y\in
V(G_{h}^{+}):y\sim u)}\Delta(u,y)F^{v}(y)\right]$
$\displaystyle=\frac{1}{\Delta(v,v)}\sum_{n=0}^{\infty}\left[\Delta(u,u)\mathbb{P}^{u}(X_{n}=v,\tau_{j}^{u}>n)+\sum_{y\in
V(G_{h}^{+}):y\sim u}\Delta(u,y)\mathbb{P}^{y}(X_{n}=v,\tau_{j}^{u}>n)\right]$
$\displaystyle=\sum_{n=0}^{\infty}\frac{\Delta(u,u)}{\Delta(v,v)}[\mathbb{P}^{u}(X_{n}=v,\tau_{h}^{u}>n)-\mathbb{P}^{u}(X_{n+1}=v,\tau_{j}^{u}>n+1)]$
$\displaystyle=\begin{cases}0&\mathrm{If}\ u\neq v\\\ 1&\mathrm{If}\
u=v.\end{cases}$

∎

###  4.6. Neumann Green’s Function on a Finite Graph

######  Definition 4.19.

Let $b_{0}\in V(G_{j})$ be a fixed vertex on the boundary of $G_{j}$. For
$u,v,x,y\in V(G_{j})$ and $u\neq b_{0}$, let $W_{n,j}$ is the random walk on
$T_{j}$ starting from $u$ defined by

$\displaystyle\mathbb{P}_{h}^{u}(W_{n+1,j}=v|W_{n,j}=y)=\begin{cases}0&\mathrm{If}\
y\ \mathrm{and}\ v\ \mathrm{are\ not\ adjacent\ in}\ T_{h}\\\
\left|\frac{\Delta(y,v)}{\sum_{x\in V(G_{j}):x\sim
y}\Delta(y,x)}\right|&\mathrm{otherwise}\end{cases}$

Let $\eta$ be the first hitting time of $W_{n,j}$ to $b_{0}$, i.e,

$\displaystyle\eta=\inf\\{m\geq 0:W_{m,j}=b_{0}\\}.$

Define the discrete Neumann Green’s function on $G_{j}$ to be

$\displaystyle
G_{j,N}(u,v)=\sum_{n=0}^{\infty}\mathbb{P}^{u}_{j}(W_{n,j}=v;n<\eta).$

It is straightforward to see from the definition that

$\displaystyle G_{j,N}(u,b_{0})=0.$

######  Lemma 4.20.

Let the Neumann Green’s function on $G_{h}$ be defined as in Definition 4.19;
write

(4.13) $\displaystyle F_{j,N}^{v}(u):=\frac{G_{j,N}(u,v)}{\sum_{y\in
V(G_{j}):y\sim v}|\Delta(v,y)|}.$

then for each fixed $u\in V(G_{h})\setminus\\{b_{0}\\}$

$\displaystyle[\Delta_{G_{j}}F_{j,N}^{v}](u)=\delta_{v}(u)$

###### Proof.

Note that

$\displaystyle[\Delta_{G_{j}}F_{j,N}^{v}](u)$
$\displaystyle=\frac{1}{\sum_{y\in V(G_{j}):y\sim
V}|\Delta(v,y)|}\left[\sum_{y\in V(G_{j}):y\sim
u}|\Delta(u,y)|F_{j,N}^{v}(u)+\sum_{y\in V(G_{h})\setminus\\{b_{0}\\}:y\sim
u}\Delta(u,y)F_{j,N}^{v}(y)\right]$ $\displaystyle=\frac{1}{\sum_{y\in
V(G_{h}):y\sim v}|\Delta(v,y)|}\sum_{n=0}^{\infty}\left[\sum_{y\in
V(G_{h}):y\sim u}|\Delta(u,y)|\mathbb{P}^{u}(W_{n,j}=v,\eta>n)\right.$
$\displaystyle+\left.\sum_{y\in V(G_{h})\setminus\\{b_{0}\\}:y\sim
u}\Delta(u,y)\mathbb{P}^{y}(W_{n,j}=v,\eta>n)\right]$
$\displaystyle=\sum_{n=0}^{\infty}\frac{\sum_{y\in
V(G_{h})}|\Delta(u,y)|}{\sum_{y\in
V(G_{h})}|\Delta(v,y)|}[\mathbb{P}^{u}(W_{n,j}=v,\eta>n)-\mathbb{P}^{u}(W_{n+1,j}=v,\eta>n+1)]$
$\displaystyle=\begin{cases}0&\mathrm{If}\ u\neq v\\\ 1&\mathrm{If}\
u=v.\end{cases}$

∎

###  4.7. Inverse matrices

######  Lemma 4.21.

Suppose Assumption 2.4 holds. Let $G_{j}$ be a finite, connected subgraph of
$G$ consisting of faces of $G$.

  1. (1)

$\displaystyle\overline{D}_{j}^{-1}(b,w)=\frac{1}{\xi}(F^{b_{1}}_{j,d}(b)-F^{b_{2}}_{j,d}(b))\sqrt{\frac{\nu(b_{1}b_{2})}{\nu([b_{1}b_{2}]^{+})}};\qquad\forall
b\in V(G_{j}^{+}).$

where

     * •

$b_{1},b_{2}\in V^{+}$ such that $w$ is the white vertex of $G$ corresponding
to the edge $(b_{1},b_{2})$;

     * •

$\xi$ is the unit vector pointing from $b_{2}$ to $b_{1}$ in the Euclidean
plane;

     * •

$F^{b}_{j,d}(\cdot)$ is defined as in (4.12).

  2. (2)

$\displaystyle\overline{D}_{j}^{-1}(b,w)=\frac{1}{\zeta}(F_{j,N}^{b_{3}}(b)-F_{j,N}^{b_{4}}(b))\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}};\qquad\forall
b\in V(G_{j})\setminus\\{b_{0}\\}.$

where

     * •

$b_{3},b_{4}\in V$ such that $w$ is the white vertex of $\overline{G}$
corresponding to the edge $(b_{3},b_{4})$;

     * •

$\zeta$ is the unit vector pointing from $b_{4}$ to $b_{3}$ in the Euclidean
plane;

     * •

$F^{b}_{j,N}(\cdot)$ is defined as in (4.13).

###### Proof.

Explicit computations show that

$\displaystyle\overline{D}_{j}^{*}\overline{D}_{j}\overline{D}_{j}^{-1}(b,w)=\overline{D}_{j}^{*}(b,w)=\begin{cases}0&\mathrm{If}\
b\ \mathrm{and}\ w\ \mathrm{are\ not\ adjacent}\\\
\frac{1}{\mathbf{i}\xi}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu(b_{1}b_{2})}}&\mathrm{If}\
b\sim w,\ \mathrm{and}\ b\in V,\ \ \mathrm{and}\ b=b_{3}\\\
-\frac{1}{\mathbf{i}\xi}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu(b_{1}b_{2})}}&\mathrm{If}\
b\sim w,\ \mathrm{and}\ b\in V,\ \ \mathrm{and}\ b=b_{4}\\\
\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu(b_{3}b_{4})}}&\mathrm{If}\ b\sim
w,\ \mathrm{and}\ b\in V^{+},\ \mathrm{and}\ b=b_{1},\\\
-\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu(b_{3}b_{4})}}&\mathrm{If}\
b\sim w,\ \mathrm{and}\ b\in V^{+},\ \mathrm{and}\ b=b_{2}\end{cases}$

where the relative locations of $b_{1},b_{2},w$ are described in the lemma;
and $(b_{3},b_{4})$ is the dual edge of $b_{1}b_{2}$; the unit vector pointing
from $b_{4}$ to $b_{3}$ in the Euclidean plane is $\mathbf{i}\xi$, given that
the unit vector pointing from $b_{2}$ to $b_{1}$ in the Euclidean plane is
$\xi$. Then by (4.3), we obtain

$\displaystyle\Delta_{G_{j}^{+}}\overline{D}_{j}^{-1}(b,w)=\overline{D}_{j}^{*}(b,w);\forall
b\in V^{+}$

Let $b\in V^{+}$. Define a matrix

$\displaystyle
B(b,w):=\frac{1}{\xi}(F^{b_{1}}(b)-F^{b_{2}}(b))\sqrt{\frac{\nu(b_{1}b_{2})}{\nu(b_{3}b_{4})}};\qquad\forall
b\in V^{+}.$

By Lemma 4.14, we obtain

$\displaystyle[\Delta_{G_{j}^{+}}B](b,w)$ $\displaystyle=\sum_{b^{\prime}\in
V^{+}}\Delta_{G_{j}^{+}}(b,b^{\prime})B(b^{\prime},w)$
$\displaystyle=\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu(b_{3}b_{4})}}\left[\sum_{b^{\prime}\in
V^{+}}\Delta_{G_{j}^{+}}(b,b^{\prime})F^{b_{1}}(b^{\prime})-\sum_{b^{\prime}\in
V^{+}}\Delta_{G_{j}^{+}}(b,b^{\prime})F^{b_{2}}(b^{\prime})\right]$
$\displaystyle=\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu(b_{3}b_{4})}}\left[\delta_{b}(b_{1})-\delta_{b}(b_{2})\right];$

where the last identity follows from Lemma 4.18. Then Part (1) of the Lemma
follows. Parts (2) of the lemma follows from Lemma 4.20 similarly. ∎

######  Theorem 4.22.

Suppose that $G$ is an infinite, 3-connected, simple, proper plane graph with
locally finite dual and admitting a double circle packing (see Proposition
2.7) in the hyperbolic plane. Suppose Assumption 2.4 holds. Assume
$\\{G_{j}\\}_{j=1}^{\infty}$ is a sequence of finite, connected subgraphs of
$G$ exhausting $G$, (i.e.$\cup_{j=1}^{\infty}G_{j}=\infty$). Let $b_{0,j}$ be
the removed black vertex to obtain $\mathcal{G}_{j}$ from $\overline{G}_{j}$.
Assume

$\displaystyle\lim_{j\rightarrow\infty}b_{0,j}=b_{0}\in\partial{\mathbb{H}}^{2};$

where $b_{0}$ is an arbitrary point along $\partial{\mathbb{H}}^{2}$ except
for

  * •

the limit points of a null family of singly infinite paths; or

  * •

the limit points of simple random walks with probability 0.

Let $\overline{D}_{j}$ be defined as in Definition 4.6. Then

  1. (1)

$\displaystyle\lim_{j\rightarrow\infty}\overline{D}_{j}^{-1}(b,w)=\frac{1}{\xi}\left(\frac{G_{G^{+}}(b,b_{1})}{|\Delta(b_{1},b_{1})|}-\frac{G_{G^{+}}(b,b_{2})}{|\Delta(b_{2},b_{2})|}\right)\sqrt{\frac{\nu(b_{1}b_{2})}{\nu([b_{1}b_{2}]^{+})}};\qquad\forall
b\in V(G_{j}^{+}).$

where $b_{1},b_{2},\xi$ are given as in Lemma 4.21(1), and $G_{G^{+}}$ is the
Green’s function on $G^{+}$ as defined in Definition 4.13.

  2. (2)

(4.14)
$\displaystyle\lim_{j\rightarrow\infty}\overline{D}_{j}^{-1}(b,w)=\frac{1}{\zeta}\left(\frac{G_{G}(b,b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(b,b_{4})}{|\Delta(b_{4},b_{4})|}+H_{b_{3}b_{4}}(b)\right)\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}};\qquad\forall
b\in V(G_{j}).$

where $b_{3},b_{4},\zeta$ are given as in Lemma 4.21(2), $G_{G}$ is the
Green’s function on $G$ as defined in Definition 4.13 and $H_{b_{3}b_{4}}$ is
a Dirichlet harmonic function on $G$ satisfying

    1. (a)

$H_{b_{3}b_{4}}(b_{0})=0$;

    2. (b)

$\displaystyle\triangledown
H_{b_{3}b_{4}}(e)=P_{\triangledown\mathbf{HD}}\chi_{b_{3}b_{4}}(e);$

i.e. the gradient of $H_{b_{3}b_{4}}$ is the orthogonal projection of
$\chi_{b_{3}b_{4}}$ on the subspace $\triangledown\mathbf{HD}$.

    3. (c)

Let $Z_{n}$ be the random walk on $G$ as defined in Definition 4.13 with
$G^{+}$ replaced by $G$, then almost surely

$\displaystyle\lim_{n\rightarrow\infty}\lim_{j\rightarrow\infty}\overline{D}_{j}^{-1}(Z_{n},w)=\lim_{n\rightarrow\infty}\frac{1}{\zeta}\left(\frac{G_{G}(Z_{n},b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(Z_{n},b_{4})}{|\Delta(b_{4},b_{4})|}+H_{b_{3}b_{4}}(Z_{n})\right)\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}};$

###### Proof.

Part(1) of the lemma follows from Lemma 4.21(1) and Lemma 4.17.

Now we prove Part(2) of the lemma. Let

$\displaystyle
F_{j,d,G}^{b_{i}}(b):=\frac{G_{j,d,G}(b,b_{i})}{|\Delta(b_{i},b_{i})|};\qquad
i\in\\{3,4\\};$

where $G_{j,d,G}$ is the Dirichlet Green’s function on $G_{j}$ as defined in
Definition 4.15 with $G_{j}^{+}$ replaced by $G_{j}$. Let

$\displaystyle F_{j,N,G}^{b_{i}}(b):=F_{j,N}^{b_{i}}(b);\qquad i\in\\{3,4\\}.$

Define

$\displaystyle
H_{b_{3}b_{4},j}(b):=F_{j,N,G}^{b_{3}}(b)-F_{j,N,G}^{b_{4}}(b)-F_{j,d,G}^{b_{3}}(b)-F_{j,d,G}^{b_{4}}(b);$

then it satisfies

$\displaystyle\Delta H_{b_{3},b_{4},j}(z)=0;\qquad\forall z\in V(G_{j}).$

Moreover, let $e\in\vec{E}_{j}$ be a directed edge of $G_{j}$, then

$\displaystyle\triangledown H_{b_{3}b_{4},j}(e)$
$\displaystyle=\frac{\nu(e)}{\nu(e^{+})}\left\\{\left[F_{j,N,G}^{b_{3}}(\overline{e})-F_{j,N,G}^{b_{4}}(\overline{e})-F_{j,N,G}^{b_{3}}(\underline{e})+F_{j,N,G}^{b_{4}}(\underline{e})\right]\right.$
$\displaystyle-\left.\left[F_{j,d,G}^{b_{3}}(\overline{e})-F_{j,d,G}^{b_{4}}(\overline{e})-F_{j,d,G}^{b_{3}}(\underline{e})+F_{j,d,G}^{b_{4}}(\underline{e})\right]\right\\}.$

By Lemma 4.17, we have

$\displaystyle\lim_{j\rightarrow\infty}\left[F_{j,d,G}^{b_{3}}(\overline{e})-F_{j,d,G}^{b_{4}}(\overline{e})-F_{j,d,G}^{b_{3}}(\underline{e})+F_{j,d,G}^{b_{4}}(\underline{e})\right]$
$\displaystyle=\frac{G_{G}(\overline{e},b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(\overline{e},b_{4})}{|\Delta(b_{4},b_{4})|}-\frac{G_{G}(\underline{e},b_{3})}{|\Delta(b_{3},b_{3})|}+\frac{G_{G}(\underline{e},b_{4})}{|\Delta(b_{4},b_{4})|};$

where $G_{G}(\cdot,\cdot)$ is the Green’s function on the infinite transient
graph $G$ as defined in Definition 4.13.

Let $f$ be the directed edge with $\underline{f}=b_{4}$ and
$\overline{f}=b_{3}$. Note that

$\displaystyle
I_{e,\partial{\mathbb{H}}^{2}}(f)=\frac{\nu(f)}{\nu(f^{+})}\langle
P_{\bigstar}\chi_{e},\chi_{f}\rangle_{R};\qquad$ $\displaystyle
I_{e,\emptyset}(f)=\frac{\nu(f)}{\nu(f^{+})}\langle
P_{\diamondsuit^{\perp}}\chi_{e},\chi_{f}\rangle_{R}.$

For $s\in\vec{E}$, define

$\displaystyle J_{e}(s):$
$\displaystyle=\frac{\nu(s)}{\nu(s^{+})}\left[\frac{G_{G}(\overline{e},\overline{s})}{|\Delta(\overline{s},\overline{s})|}-\frac{G_{G}(\overline{e},\underline{s})}{\left|\Delta(\underline{s},\underline{s})\right|}-\frac{G_{G}(\underline{e},\overline{s})}{\left|\Delta(\overline{s},\overline{s})\right|}+\frac{G_{G}(\underline{e},\underline{s})}{\left|\Delta(\underline{s},\underline{s})\right|}\right]$
$\displaystyle=K_{\underline{e}}(s)-K_{\overline{e}}(s)$

where for each $v\in V$ and $s\in\vec{E}$,

$\displaystyle
K_{v}(s):=\frac{\nu(s)}{\nu(s^{+})}\left[-\frac{G_{G}(v,\overline{s})}{\left|\Delta(\overline{s},\overline{s})\right|}+\frac{G_{G}(v,\underline{s})}{\left|\Delta(\underline{s},\underline{s})\right|}\right]$

is the expected number of times that a weighted random walk (as defined in
4.6) starting from $v$ uses $s$ minus the expected number of times that it
uses $-s$. Then

$\displaystyle\mathrm{div}K_{v}(w)=\frac{\delta_{v}(w)}{|\Delta(v,v)|}$

Then

$\displaystyle\mathrm{div}J_{e}(v)=\frac{1}{|\Delta(v,v)|}[\delta_{v}(\underline{e})-\delta_{v}(\overline{e})]=\mathrm{div}I_{e,\partial{\mathbb{H}}^{2}}(v).$

Therefore $\mathrm{div}(J_{e}-I_{e,\partial{\mathbb{H}}^{2}})=0$, which
implies $(J_{e}-I_{e,\partial{\mathbb{H}}^{2}})\perp\bigstar$. Since
$I_{e,\partial{\mathbb{H}}^{2}}\in\bigstar$, we deduce that
$P_{\bigstar}J_{e}=I_{e,\partial{\mathbb{H}}^{2}}$.

For any $v\in V$ and $s\in\vec{E}$, let $\theta_{v}(f)$ be the probability
that the first step of the random walk starting at $v$ will use the edge $f$
minus the probability that it will use the edge $-f$. Then
$\Delta(v,v)\theta_{v}=-\triangledown\mathbf{1}_{v}$. Hence
$\theta_{v}\in\bigstar$. Note that

$\displaystyle
J_{e}=\sum_{v}[G_{G}(\underline{e},v)\theta_{v}-G_{G}(\overline{e},v)\theta_{v}];$

then $J_{e}\in\bigstar$; therefore $J_{e}=I_{e,\partial{\mathbb{H}}^{2}}$.

For $s\in\vec{E}$, define

$\displaystyle J_{e,j,N}(s):$
$\displaystyle=\frac{\nu(s)}{\nu(s^{+})}\left[\frac{G_{j,N,G}(\overline{e},\overline{s})}{\left|\sum_{x\in
V(G_{j}):x\sim
b_{0,j}}\Delta(\overline{s},x)\right|}-\frac{G_{j,N,G}(\overline{e},\underline{s})}{\left|\sum_{x\in
V(G_{j}),x\sim\underline{s}}\Delta(\underline{s},x)\right|}-\frac{G_{j,N,G}(\underline{e},\overline{s})}{\left|\sum_{x\in
V(G_{j}),x\sim\overline{s}}\Delta(\overline{s},x)\right|}+\frac{G_{j,N,G}(\underline{e},\underline{s})}{\left|\sum_{x\in
V(G_{j}),x\sim\underline{s}}\Delta(\underline{s},x)\right|}\right]$
$\displaystyle=K_{\underline{e},j,N}(s)-K_{\overline{e},j,N}(s).$

where for each $v\in V$ and $s\in\vec{E}$,

$\displaystyle
K_{v,j,N}(s):=\frac{\nu(s)}{\nu(s^{+})}\left[-\frac{G_{j,N,G}(v,\overline{s})}{\left|\sum_{x\in
V(G_{j}),x\sim\overline{s}}\Delta(\overline{s},x)\right|}+\frac{G_{j,N,G}(v,\underline{s})}{\left|\sum_{x\in
V(G_{j}),x\sim\underline{s}}\Delta(\underline{s},x)\right|}\right]$

is the expected number of times that a weighted random walk on $G_{j}$
absorbed at $b_{0,j}$ (as defined in Definition 4.19) starting from $v$ uses
$s$ minus the expected number of times that it uses $-s$. Then

$\displaystyle\mathrm{div}K_{v,j,N}(w)=\begin{cases}\frac{\delta_{v}(w)}{|\sum_{x\in
V(G_{j}):x\sim v}\Delta(v,x)|};&\mathrm{If}\ v\neq b_{0,j}\ \mathrm{and}\
w\neq b_{0,j};\\\ 0;&\mathrm{If}\ v=b_{0,j};\\\ -\frac{1}{|\sum_{x\in
V(G_{j}):x\sim b_{0,j}}\Delta(b_{0,j},x)|};&\mathrm{If}\ v\neq b_{0,j}\
\mathrm{and}\ w=b_{0,j}.\end{cases}$

Assume neither endpoints of $e$ coincide with $b_{0,j}$. Then

$\displaystyle\mathrm{div}J_{e,j,N}(v)$
$\displaystyle=\frac{1}{\left|\sum_{x\in V(G_{j}):x\sim
v}\Delta(v,x)\right|}[\delta_{v}(\underline{e})-\delta_{v}(\overline{e})]=\mathrm{div}[P_{\bigstar_{j}}\chi_{e}](v).$

where $\bigstar_{j}:=\triangledown l^{2}(V_{j})$. Therefore
$\mathrm{div}(J_{e,j,N}-P_{\bigstar_{j}}\chi_{e})=0$, which implies
$(J_{e,j,N}-P_{\bigstar_{j}}\chi_{e})\perp\bigstar_{j}$. Since
$P_{\bigstar_{j}}\chi_{e}\in\bigstar_{j}$, we deduce that
$P_{\bigstar_{j}}J_{e,j,N}=P_{\bigstar_{j}}\chi_{e}$.

For any $v\in V_{j}$ and $s\in\vec{E}$, let $\theta_{j,N,v}(f)$ be the
probability that the first step of the weighted random walk on $G_{j}$
absorbed at $b_{0,j}$ starting at $v$ will use the edge $f$ minus the
probability that it will use the edge $-f$. Then $\left|\sum_{x\in
V(G_{j}):x\sim
v}\Delta(v,x)\right|\theta_{j,N,v}=-\triangledown\mathbf{1}_{v}$ if $v\neq
b_{0,j}$ and $\theta_{j,N,b_{0,j}}=0$. Hence $\theta_{j,N,v}\in\bigstar_{j}$.
Note that

$\displaystyle J_{e,N,j}=\sum_{v\in
V_{j}}[G_{j,N,G}(\underline{e},v)\theta_{j,N,v}-G_{j,N,G}(\overline{e},v)\theta_{j,N,v}];$

then $J_{e,N,j}\in\bigstar_{j}$; therefore

$\displaystyle J_{e,N,j}=P_{\bigstar_{j}}\chi_{e}.$

By Proposition 3.18, we obtain

$\displaystyle\lim_{j\rightarrow\infty}P_{\bigstar_{j}}\chi_{e}=I_{e,\emptyset};$

and the convergence is in the norm $\|\cdot\|_{R}$. Then we obtain that for
any two adjacent vertices $b,b^{\prime}\in V(G_{j})$ with $e=(b,b^{\prime})$

$\displaystyle\lim_{j\rightarrow\infty}\left[\overline{D}_{j}^{-1}(b,w)-\overline{D}_{j}^{-1}(b^{\prime},w)\right]=\frac{1}{\zeta}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}\frac{\nu([bb^{\prime}]^{+})}{\nu(bb^{\prime})}P_{\triangledown\mathbf{HD}}\chi_{b_{3}b_{4}}(e)$
$\displaystyle+\frac{1}{\zeta}\left(\frac{G_{G}(b,b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(b,b_{4})}{|\Delta(b_{4},b_{4})|}-\frac{G_{G}(b^{\prime},b_{3})}{|\Delta(b_{3},b_{3})|}+\frac{G_{G}(b^{\prime},b_{4})}{|\Delta(b_{4},b_{4})|}\right)\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}};$

where $b_{3},b_{4},\zeta$ are given as in Lemma 4.21(2), $G_{G}$ is the
Green’s function on $G$ as defined in Definition 4.13. Then (2a) follows.

It remains to prove (2b). The proof is inspired by Theorem 1.1 in [3]. Let

$\displaystyle
R_{n}:=\lim_{j\rightarrow\infty}\overline{D}_{j}^{-1}(Z_{n},w)-\frac{1}{\zeta}\left(\frac{G_{G}(Z_{n},b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(Z_{n},b_{4})}{|\Delta(b_{4},b_{4})|}+H_{b_{3}b_{4}}(Z_{n})\right)\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}$
$\displaystyle-\lim_{j\rightarrow\infty}\overline{D}_{j}^{-1}(Z_{n-1},w)+\frac{1}{\zeta}\left(\frac{G_{G}(Z_{n-1},b_{3})}{|\Delta(b_{3},b_{3})|}-\frac{G_{G}(Z_{n-1},b_{4})}{|\Delta(b_{4},b_{4})|}+H_{b_{3}b_{4}}(Z_{n-1})\right)\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}$

We shall prove that

(4.15) $\displaystyle\lim_{n\rightarrow\infty}\sum_{k=1}^{n}R_{k}=0.$

Then the lemma follows.

Now we prove (4.15). We have for any $\epsilon>0$

$\displaystyle\mathbb{P}\left(\left|\sum_{k=1}^{n}R_{k}\right|>\epsilon\right)\leq\frac{\mathbb{E}\left|\sum_{k=1}^{n}R_{k}\right|^{2}}{\epsilon^{2}}$

Note that

$\displaystyle\mathbb{E}\left|\sum_{k=1}^{n}R_{k}\right|^{2}=\sum_{k=1}^{n}\mathbb{E}[R_{k}]^{2}+2\sum_{1\leq
r<s\leq n}\mathbb{E}R_{r}R_{s}$

For any $1\leq r<s\leq n$, let $\mathcal{F}_{r}$ be the sigma algebra
generated by $Z_{0},\ldots,Z_{r}$

$\displaystyle\mathbb{E}R_{r}R_{s}=\mathbb{E}\left[\mathbb{E}\left\\{R_{r}R_{s}|\mathcal{F}_{r}\right\\}\right]=\mathbb{E}R_{r}\mathbb{E}\left\\{R_{s}|\mathcal{F}_{r}\right\\}=0$

where the last identity follows from the fact that

$\displaystyle\mathbb{E}\left\\{R_{s}|\mathcal{F}_{s-1}\right\\}=0.$

and the harmonicity of $R_{n}$ with respect to the variable $Z_{n}$. Hence we
have

$\displaystyle\mathbb{E}\left|\sum_{k=1}^{n}R_{k}\right|^{2}=\sum_{k=1}^{n}\mathbb{E}[R_{k}]^{2}$

Let $Z_{0}=x$ be an arbitrary vertex in $V$. Moreover

$\displaystyle\sum_{k=1}^{\infty}\mathbb{E}[R_{k}]^{2}=\sum_{y\in
V}G_{G}(x,y)\mathbb{E}[R_{1}|Z_{0}=y]^{2}$

Let $\tau_{x}$ be the first hitting time of the vertex $x$ by $Z_{n}$. Note
that

$\displaystyle\frac{G_{G}(x,y)}{|\Delta(y,y)|}=\frac{G_{G}(y,x)}{|\Delta(x,x)|}=\frac{\mathbb{P}(\tau_{x}<\infty|Z_{0}=y)G_{G}(x,x)}{|\Delta(x,x)|}\leq\frac{G_{G}(x,x)}{|\Delta(x,x)|}$

Then we have

$\displaystyle\sum_{k=1}^{\infty}\mathbb{E}[R_{k}]^{2}\leq\frac{G_{G}(x,x)}{|\Delta(x,x)|}\sum_{y\in
V}\mathbb{E}[R_{1}|Z_{0}=y]^{2}=0,$

where the last identity follows from the fact that the left hand side of (7.4)
converges to the right hand side of (7.4) in the $\|\cdot\|_{R}$ norm. Then we
obtain for any $\epsilon>0$,
$\mathbb{P}\left(\left|\sum_{k=1}^{\infty}R_{k}\right|>\epsilon\right)=0$,
then (4.15) follows. ∎

###  4.8. Decay of the Dirichlet Green’s Function

######  Lemma 4.23.

(Proposition 6.6 in [58])Let $G=(V,E)$ be an infinite connected, non-amenable
graph with spectral radius $\rho<1$. For simple random walk on $G$,

$\displaystyle p_{n}(x,y)\leq\sqrt{\frac{d(y)}{d(x)}}\rho^{n}$

where $d(y)$ (resp. $d(x)$) is the degree of $y$ (resp. $x$).

######  Lemma 4.24.

Let $G=(V,E)$ be an infinite, connected, bounded degree, 3-connected,
nonamenable, one-ended, properly-embedded planar graph with spectral radius
$\rho(G)$. Assume the edge weights satisfy

(4.16) $\displaystyle\nu(e)=\nu(e^{+})=1;\ \forall e\in E.$

Let $G_{j}$ be a finite subgraph of $G$ consisting of faces of $G$ bounded by
a simple closed curve consisting of edges. Let $D$ be the maximal vertex
degree of $G$.

(4.17) $\displaystyle G_{j,d}(x,y)\leq
G(x,y)\leq\sqrt{\frac{D}{3}}\frac{[\rho(G)]^{d_{G}(x,y)}}{1-\rho(G)}.,\
\forall x,y\in V(G)$

where $G_{j,d}(\cdot,\cdot)$ is the Dirichlet Green’s function for $G_{j}$,
and $G(\cdot,\cdot)$ is the discrete Green’s function for the infinite graph
$G$.

###### Proof.

Note that

$\displaystyle G_{j,d}(u,v)\leq
G(u,v)=\sum_{n=d_{G}(u,v)}^{\infty}p_{n}(u,v)\leq\sum_{n=d_{G}(u,v)}^{\infty}\sqrt{\frac{d(v)}{d(u)}}[\rho(G)]^{n}\leq\sqrt{\frac{D}{3}}\frac{[\rho(G)]^{d_{G}(u,v)}}{1-\rho(G)}.$

Then (4.17) follows. ∎

Here we obtain the exponential decay of $G(u,v)$ with respect to the distance
of $u$ and $v$ from the nonamenability of the underlying graph. The decay rate
of the Green’s function $G(u,v)$ with respect to the distance of $u$ and $v$
is known to be related to the growth rate of the graph; see e.g. [30, 45].

Recall also the following general results proved in [58].

######  Lemma 4.25.

(Corollary 6.32 in [58]) Let $p(\cdot,\cdot)$ be the transition probabilities
of an irreducible Markov chain on a countable state space $V$. Assume the
chain has an infinite stationary measure $\pi$. Let $\pi_{\min}:=\inf_{x\in
v}\pi(x)$. For $x,y\in V$, let

$\displaystyle Q(x,y):=\pi(x)p(x,y)$

For $S,A\subset V$, define

$\displaystyle Q(S,A):=\sum_{s\in S,a\in A}Q(s,a);\qquad\mathrm{and}\
|\partial_{E}S|_{Q}:=Q(S,S^{c})$

The edge expansion of a finite set $S\subset V$ is

$\displaystyle\Phi_{S}:=\frac{|\partial_{E}S|_{Q}}{\pi(S)}.$

The expansion profile of the chain is defined for $u>0$ by

(4.18) $\displaystyle\Phi(u):=\inf\\{\Phi_{S}:0<\pi(S)\leq u\\},$

where we take the convention that the inf of an empty set is $\infty$. Suppose
the Markov chain $(V,P)$ is reversible. Then

  1. (1)

If $\Phi(u)\geq\varphi_{0}$ for some $\varphi_{0}>0$, then

$\displaystyle\frac{p_{n}(x,y)}{\pi(y)}\leq\frac{4}{\pi_{\min}}\mathrm{exp}\left(-\frac{\varphi_{0}^{2}(n-1)}{16}\right)\qquad\forall
x,y\in V,\ \mathrm{and}\ n\geq 1$

  2. (2)

Let $d>0$. If $\Phi(u)\geq cu^{-\frac{1}{d}}$ for some $c>0$ and all $u>0$,
then

$\displaystyle\frac{p_{n}(x,y)}{\pi(y)}\leq
C^{\prime}n^{-\frac{d}{2}};\qquad\forall x,y\in V,\ \mathrm{and}\ n\geq 1$

where $C^{\prime}=C^{\prime}(d,c)$.

  3. (3)

If $\Phi(u)\geq\frac{c}{\log(bu)}$ for some $b,c>0$ and all $u>0$, then

$\displaystyle\frac{p_{n}(x,y)}{\pi(y)}\leq
C_{1}\exp(-C_{2}n^{\frac{1}{3}});\qquad\forall x,y\in V,\ \mathrm{and}\ n\geq
1$

######  Lemma 4.26.

Let $G=(V,E)$ be an infinite, connected, bounded degree, 3-connected,
nonamenable, one-ended, properly-embedded planar graph. Consider the weighted
random walk on defined by (4.6) with $G^{+}$ replaced by $G$. Assume

(4.19) $\displaystyle\Phi(u)\geq cu^{-\frac{1}{d}}$

for some $d>2$. Then the discrete Green’s function on the infinite graph $G$
satisfies

$\displaystyle\frac{G(u,v)}{|\Delta(v,v)|}\leq
C^{\prime\prime}[d_{G}(u,v)]^{-\frac{d-2}{2}}$

where $C^{\prime\prime}=C^{\prime\prime}(d,c)$.

###### Proof.

The random walk defined by (4.6) on $d$ is reversible and has an infinite
stationary measure $\pi(x)=|\Delta(x,x)|$.

By Lemma 4.25(2), we have

$\displaystyle\frac{G(u,v)}{|\Delta(v,v)|}=\sum_{n=d_{G}(u,v)}^{\infty}\frac{p_{n}(u,v)}{|\Delta(v,v)|}\leq
C^{\prime}\sum_{n=d_{G}(u,v)}^{\infty}n^{-\frac{d}{2}}$

Then the lemma follows. ∎

##  5\. General Boundary Conditions

In this section, we discuss the infinite volume Gibbs measure of perfect
matchings on $\overline{G}$ obtained by using sequences of finite graphs
exhausting the infinite graph with boundary conditions other than the
Temperley boundary conditions. We then prove that using a sequence of finite
graphs with two convex corners and no concave corners, such that the two
boundaries portions divided by the two convex corners converge to two closed
subsets $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ of $\partial{\mathbb{H}}^{2}$,
the limits of the entries of the inverse of the weighted adjacency matrix
converge to the difference of the Dirichlet Green’s function plus explicit
harmonic Dirichlet functions; see Lemma 5.6.

######  Assumption 5.1.

Suppose Assumption 2.4 holds. Consider a sequence of finite, simply-connected
subgraphs of $\overline{G}$ given by

$\displaystyle\tilde{G}_{1},\tilde{G}_{2},\ldots,\tilde{G}_{n},\ldots$

satisfying all the following conditions

  1. (1)

Each $\tilde{G}_{i}$ is induced by a finite subset of vertices $\tilde{V}_{i}$
of $\overline{G}$; and

  2. (2)

For any two vertices $u,v\in V(\tilde{G}_{i})\cap V$, such that $(u,v)\in E$
is an edge of $G$, the white vertex of $\overline{G}$ corresponding to $(u,v)$
is also a vertex of $\tilde{G}_{i}$; and

  3. (3)

For any two vertices $u,v\in V(\tilde{G}_{i})\cap V^{+}$, such that $(u,v)\in
E^{+}$ is an edge of $G^{+}$, the white vertex of $\overline{G}$ corresponding
to $(u,v)$ is also a vertex of $\tilde{G}_{i}$; and

  4. (4)

For each $i\geq 1$, $\tilde{G}_{i}\subset\tilde{G}_{i+1}$; and

  5. (5)

$\cup_{i=1}^{\infty}\tilde{G}_{i}=\overline{G}$; and

  6. (6)

Each $\tilde{G}_{i}$ admits a perfect matching.

######  Definition 5.2.

Let $\tilde{G}_{j}$ be a finite subgraph of $\overline{G}$. We call a white
vertex $w$ of $\tilde{G}_{j}$ a concave corner if there exists a face of
$\overline{G}$ consisting of vertices $w,b_{1},w^{\prime},b_{2}$ such that
$w,b_{1},b_{2}$ are vertices of $\tilde{G}_{j}$, but $w^{\prime}$ is not a
vertex of $\tilde{G}_{j}$.

We call a white vertex $w$ a convex corner if there is exactly one incident
face of $w$ in $\tilde{G}_{j}$.

######  Lemma 5.3.

Let $\\{\tilde{G}_{j}\\}$ be a finite subgraph of $\overline{G}$ satisfying
Assumption 5.1(1)(2)(3)(6). Let $K_{j}$ be the number of concave white corners
in $\tilde{G}_{j}$. Let $w$ be a white vertex of $\overline{G}$ and let $b\in
V^{+}$. Let

$\displaystyle\overline{D}_{\tilde{G}_{j}}:=\left.\overline{D}\right|_{W(\tilde{G}_{j}),B(\tilde{G}_{j})}$

i.e. $\overline{D}_{\tilde{G}_{j}}$ is the submatrix of $\overline{D}$ with
rows restricted to white vertices of $\tilde{G}_{j}$, and columns restricted
to black vertices of $\tilde{G}_{j}$.

If $K_{j}=0$, then

  * •

If $b$ is a vertex of $G$ and $v\in V(\tilde{G}_{j})\cap V$, then

$\displaystyle\overline{D}^{-1}_{\tilde{G}_{j}}(b,w)=\frac{1}{\zeta}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}\left[\Delta_{{G}_{1,j}}^{-1}(b,b_{3})-\Delta_{G_{1,j}}^{-1}(b,b_{4})\right]$

where $b_{3}$, $b_{4}$, $\zeta$ are given as in Lemma 4.21(2).

  * •

If $b$ is a vertex of $\tilde{G}_{j}^{+}$, then

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{-1}(b,w)=\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu([b_{1}b_{2}]^{+})}}\left[\Delta_{G_{0,j}}^{-1}(b,b_{1})-\Delta_{G_{0,j}}^{-1}(b,b_{2})\right]$

where $b_{1}$, $b_{2}$, $\xi$ are given as in Lemma 4.21(1).

###### Proof.

For each pair of black vertices $b_{1},b_{2}$ of $\tilde{G}_{j}$, the
following cases might occur:

  1. (1)

If $b_{1},b_{2}$ are adjacent vertices in $G$ or if $b_{1},b_{2}$ are adjacent
vertices in $G^{+}$

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{2})=\frac{\nu(b_{1}b_{2})}{\nu((b_{1}b_{2})^{+})}$

  2. (2)

If $d_{\overline{G}}(b_{1},b_{2})>2$, then

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{2})=0.$

  3. (3)

If $d_{\overline{G}}(b_{1},b_{2})=2$, $b_{1}$ is a vertex of $G$, $b_{2}$ is a
vertex of $G^{+}$, in the quadrilateral face $b_{1},w,b_{2},w^{\prime}$ of
$\overline{G}$, both $w$ and $w^{\prime}$ are vertices of $\tilde{G}_{j}$,
then

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{2})=\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{2},b_{1})=0.$

  4. (4)

If $d_{\overline{G}}(b_{1},b_{2})=2$, $b_{1}$ is a vertex of $G$, $b_{2}$ is a
vertex of $G^{+}$, in the quadrilateral face $b_{1},w,b_{2},w^{\prime}$ of
$\overline{G}$, exactly one of $w$ and $w^{\prime}$ is a vertex of
$\tilde{G}_{j}$, then

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{2})=\mathbf{i}\
\mathrm{and}\
\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{2},b_{1})=-\mathbf{i}$

or

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{2})=-\mathbf{i}\
\mathrm{and}\
\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{2},b_{1})=\mathbf{i}$

  5. (5)

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}(b_{1},b_{1})=-\sum_{w\in
V(\tilde{G}_{j}):w\sim b_{i}}\frac{\nu(e_{wb})}{\nu(e_{wb}^{+})};$

where $e_{wb}$ is the edge in $G$ or $G^{+}$ with $wb$ as a half edge.

Then we obtain

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}=\left(\begin{array}[]{cc}\Delta_{G_{1,j}}&\mathbf{i}A\\\
-\mathbf{i}A^{t}&\Delta_{G_{0,j}}\end{array}\right)$

where $A$ is a matrix with entries $+1$ or $-1$, rows index by vertices of
$G_{j,1}$ and columns indexed by vertices of $G_{j,0}$; and $A^{t}$ is the
transpose of $A$. Note that the number of non-zero entries in $A$ is exactly
$K_{j}$. Since

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{*}\overline{D}_{\tilde{G}_{j}}\overline{D}_{\tilde{G}_{j}}^{-1}=\overline{D}_{\tilde{G}_{j}}^{*};$

We obtain that

$\displaystyle\overline{D}_{\tilde{G}_{j}}^{-1}$
$\displaystyle=\left(\begin{array}[]{cc}\Delta_{G_{1,j}}&\mathbf{i}A\\\
-\mathbf{i}A^{t}&\Delta_{G_{0,j}}\end{array}\right)^{-1}\overline{D}_{G_{0,j}}^{*}$
$\displaystyle=\left(I+\left(\begin{array}[]{cc}0&\mathbf{i}\Delta_{G_{1,j}}^{-1}A\\\
\mathbf{i}\Delta_{G_{0,j}}^{-1}A&0\end{array}\right)\right)^{-1}\left(\begin{array}[]{cc}\Delta_{G_{1,j}}^{-1}&0\\\
0&\Delta_{G_{0,j}}^{-1}\end{array}\right)\overline{D}_{\tilde{G}_{j}}^{*}$

Note that

  * •

If $b$ is a vertex of $G$ and $v\in V(\tilde{G}_{j})\cap V$, then

$\displaystyle\left(\begin{array}[]{cc}\Delta_{G_{1,j}}^{-1}&0\\\
0&\Delta_{G_{0,j}}^{-1}\end{array}\right)\overline{D}_{\tilde{G}_{j}}^{*}(b,w)=\frac{1}{\zeta}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}\left[\Delta_{{G}_{1,j}}^{-1}(b,b_{3})-\Delta_{G_{1,j}}^{-1}(b,b_{4})\right]$

where $b_{3}$, $b_{4}$, $\zeta$ are given as in Lemma 4.21(2).

  * •

If $b$ is a vertex of $\tilde{G}_{j}^{+}$, then

$\displaystyle\left(\begin{array}[]{cc}\Delta_{G_{1,j}}^{-1}&0\\\
0&\Delta_{G_{0,j}}^{-1}\end{array}\right)\overline{D}_{\tilde{G}_{j}}^{*}(b,w)=\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu([b_{1}b_{2}]^{+})}}\left[\Delta_{G_{0,j}}^{-1}(b,b_{1})-\Delta_{G_{0,j}}^{-1}(b,b_{2})\right]$

where $b_{1}$, $b_{2}$, $\xi$ are given as in Lemma 4.21(1).

If $K_{j}=0$ we have $A=0$, then the lemma follows. ∎

######  Definition 5.4.

Let $\tilde{G}_{j}$ be a finite subgraph of $\overline{G}$. Assume
$\tilde{G}_{j}$ consist of faces of $\overline{G}$. Let $w$ be a white vertex
of $\tilde{G}_{j}$. We call $w$ a convex corner of $\tilde{G}_{j}$ if exactly
one incident face of $w$ in $\overline{G}$ is in $\tilde{G}_{j}$.

Figure 5.1. A graph with two convex white corners and no concave white
corners: the dual graph is represented by black lines; the primal graph is
represented by gray lines.

######  Assumption 5.5.

Suppose Assumption 3.15 holds with $\mathcal{A}$ replaced by
$\mathcal{A}_{0}$. Assume each $G_{j,s}=(V_{j,s},E_{j,s})$ is a finite
subgraph of $G$ such that $V_{j,s}$ consists all the vertices and edges in the
bounded component of ${\mathbb{R}}^{2}\setminus C_{j,s}$ (where $C_{j,s}$ is a
simple closed curve consisting of edges of $G$), as well as all the vertices
and edges along $C_{j,s}$; and $G_{j,s}$ is the induced subgraph of $V_{j,s}$.
Assume there exist two vertices $v_{1,j,s}$ and $v_{2,j,s}$ along $C_{j,s}$,
such that

$\displaystyle C_{j,s}\setminus\\{v_{1,j,s},v_{2,j,s}\\}=C_{0,j,s}\cup
C_{1,j,s}\ \mathrm{and}\ C_{0,j,s}\cap C_{1,j,s}=\emptyset.$

Assume

$\displaystyle\lim_{s\rightarrow\infty}v_{1,j,s}=z_{1};\ \mathrm{and}\
\lim_{s\rightarrow\infty}v_{2,j,s}=z_{2};\ \forall j\in{\mathbb{N}}.$

and

$\displaystyle\lim_{s\rightarrow\infty}C_{0,j,s}=\mathcal{A}_{0};$

where $\mathcal{A}_{0}$ is a closed interval along $\partial{\mathbb{H}}^{2}$,
and the convergence is in the sense of Hausdorff distance of two closed set.
Assume
$\mathcal{A}_{1}=\partial{\mathbb{H}}^{2}\setminus\mathcal{A}_{0}^{\circ}$;
where $\mathcal{A}_{0}^{\circ}$ is the interior of $\mathcal{A}_{0}$. Let
$G_{j,s}^{+}=(V_{j,s}^{+},E_{j,s}^{+})$ be a finite subgraph of $G^{+}$
constructed as follows:

  1. (1)

Let $\hat{G}_{j,s}^{+}$ be a subgraph of $G^{+}$ that contains $G_{j,s}$ as an
interior dual graph;

  2. (2)

Assume $\hat{G}_{j,s}^{+}$ is bounded by a simple closed curve $\hat{C}_{j,s}$
consisting of edges of $G^{+}$; i.e., the vertex set of $\hat{G}_{j,s}^{+}$
consists of all the vertices of $G^{+}$ in the bounded component of
${\mathbb{R}}^{2}\setminus\hat{C}_{j,s}$; as well as vertices along
$\hat{C}_{j,s}$, and $\hat{G}_{j,s}^{+}$ is the induced subgraph by its vertex
set. Let $v^{+}_{1,j,s}$ be a vertex along $\hat{C}_{j,s}$ sharing a face with
$v_{1,j,s}$ in $\overline{G}$; let $v^{+}_{2,j,s}$ be a vertex along
$\hat{C}_{j,s}$ sharing a face with $v_{2,j,s}$ in $\overline{G}$;

  3. (3)

Assume

$\displaystyle\hat{C}_{j,s}\setminus\\{v^{+}_{1,j,s},v^{+}_{2,j,s}\\}=\hat{C}_{0,j,s}\cup\hat{C}_{1,j,s}\
\mathrm{and}\ \hat{C}_{0,j,s}\cap\hat{C}_{1,j,s}=\emptyset.$

Assume vertices in $\hat{C}_{1,j,s}\cap V^{+}$ share no faces in
$\overline{G}$ with vertices in $C_{0,j,s}\cap V$; and vertices in
$\hat{C}_{0,j,s}\cap V^{+}$ share no faces in $\overline{G}$ with vertices in
$C_{1,j,s}\cap V$.

Let $G_{j,s}^{+}$ be the subgraph of $\hat{G}_{j,s}^{+}$ obtained by removing
all the vertices in $\hat{C}_{1,j,s}\cap V^{+}$ as well as their incident
edges in $G^{+}$.

Let $\tilde{G}_{j,s}$ be the bipartite graph obtained from the superposition
of $G_{j,s}$ and $\hat{G}_{j,s}^{+}$, and then removing all the vertices along
$\hat{C}_{j,s}$ between $v_{1,j,s}$ and $v_{2,j,s}$ surrounding $C_{1,j,s}$ as
well as their incident edges in $\overline{G}$. See Figure 5.1.

######  Theorem 5.6.

Suppose the Assumption 5.5 holds.

Then we have

  1. (1)

If $b$ is a vertex of $G$ and $v\in V(\tilde{G}_{j,s})\cap V$, then

(5.1)
$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}^{-1}_{\tilde{G}_{j,s}}(b,w)=\frac{1}{\zeta}\sqrt{\frac{\nu(b_{3}b_{4})}{\nu([b_{3}b_{4}]^{+})}}\left[\frac{G_{G}(b,b_{3})}{\Delta(b_{3},b_{3})}-\frac{G_{G}(b,b_{4})}{\Delta(b_{4},b_{4})}+H_{\mathcal{A}_{0},G,b_{3}b_{4}}(b)\right]$

where $b_{3}$, $b_{4}$, $\zeta$ are given as in Lemma 4.21(2), $G_{G}$ is the
Green’s function on $G$ as defined in Definition 4.13 and
$H_{\mathcal{A}_{0},G,b_{3}b_{4}}$ is a harmonic Dirichlet function on $G$
satisfying

    1. (A)

For almost every singly infinite path $\gamma:=(\gamma(0),\gamma(1),\ldots)$
consisting of vertices in $G$ satisfying

$\displaystyle\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0$

for some $z\in\mathcal{A}_{0}$, we have

$\displaystyle\lim_{n\rightarrow\infty}H_{\mathcal{A}_{0},G,b_{3}b_{4}}(\gamma(n))=0;$

and

    2. (B)

$\triangledown
H_{\mathcal{A}_{0},G,b_{3}b_{4}}(e)=P_{\mathcal{S}_{\mathcal{A}_{0}}(G)}\chi_{b_{3}b_{4}}(e)$;
i.e. the gradient of $H_{\mathcal{A}_{0},G,b_{3}b_{4}}$ is the orthogonal
projection of $\chi_{b_{3}b_{4}}$ on the subspace
$\mathcal{S}_{\mathcal{A}_{0}}(G)$ of the gradient of harmonic Dirichlet
functions on $G$ which are constant on $\mathcal{A}_{0}$.

  2. (2)

If $b$ is a vertex of $G^{+}$ and and $v\in V(\tilde{G}_{j,s})\cap V^{+}$,
then

$\displaystyle\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w)=\frac{1}{\xi}\sqrt{\frac{\nu(b_{1}b_{2})}{\nu([b_{1}b_{2}]^{+})}}\left[\frac{G_{G^{+}}(b,b_{1})}{\Delta(b_{1},b_{1})}-\frac{G_{G^{+}}(b,b_{2})}{\Delta(b_{2},b_{2})}+H_{\mathcal{A}_{1},G^{+},b_{1}b_{2}}(b)\right]$

where $b_{1}$, $b_{2}$, $\xi$ are given as in Lemma 4.21(1), $G_{G^{+}}$ is
the Green’s function on $G^{+}$ as defined in Definition 4.13 and
$H_{\mathcal{A}_{1},G^{+},b_{3}b_{4}}$ is a Dirichlet harmonic function on
$G^{+}$ satisfying

     * •

For almost every singly infinite path $\gamma:=(\gamma(0),\gamma(1),\ldots)$
consisting of vertices in $G^{+}$ satisfying

$\displaystyle\lim_{n\rightarrow\infty}d_{m}(\gamma(n),z)=0$

for some $z\in\mathcal{A}_{1}$, we have

$\displaystyle\lim_{n\rightarrow\infty}H_{\mathcal{A}_{1},G,b_{3}b_{4}}(\gamma(n))=0;$

and

     * •

$\triangledown
H_{\mathcal{A}_{1},G^{+},b_{1}b_{2}}(e)=P_{\mathcal{S}_{\mathcal{A}_{1}}(G^{+})}\chi_{b_{1}b_{2}}(e)$;
i.e. the gradient of $H_{\mathcal{A}_{1},G^{+},b_{1}b_{2}}$ is the orthogonal
projection of $\chi_{b_{1}b_{2}}$ on the subspace
$\mathcal{S}_{\mathcal{A}_{1}}(G^{+})$ of the gradient of harmonic Dirichlet
functions on $G^{+}$ which are constants on $\mathcal{A}_{1}$.

  3. (3)

Suppose condition (a) in Lemma 3.14 holds with $\mathcal{A}$ replaced by
$\mathcal{A}_{0}$; and condition (a) in Lemma 3.14 holds with $\mathcal{A}$
replaced by $\mathcal{A}_{1}$ and $G$ replaced by $G^{+}$. Then the infinite
volume Gibbs measure obtained above is distinct from the infinite volume Gibbs
measures obtained from the Temperley boundary conditions as obtained in
Theorem 4.22.

###### Proof.

Note that under Assumption 5.5, the sequence of graphs $\\{G_{j,s}\\}$ satisfy
Assumption 3.15 with $\mathcal{A}$ replaced by $\mathcal{A}_{0}$, while the
sequence of graphs $\\{G_{j,s}^{+}\\}$ satisfy Assumption 3.16 with
$\mathcal{A}$ replaced by $\mathcal{A}_{1}$. We only prove part (1) here; part
(2) can be proved similarly.

Following similar arguments as in the proof of Lemma 4.22, one obtains for
each fixed $w$

$\displaystyle\zeta\sqrt{\frac{\nu([b_{3}b_{4}]^{+})}{\nu(b_{3}b_{4})}}\overline{D}^{-1}_{\tilde{G}_{j,s}}(\cdot,w)-F_{j,d,G}^{b_{3}}(\cdot)+F_{j,d,G}^{b_{4}}(\cdot)\rightarrow
H_{\mathcal{A}_{0},G,b_{3},b_{4}}(\cdot)$

with respect to $\sqrt{\mathcal{E}(\triangledown)}$. Then the convergence of
the left hand side to the right hand side for each $b\in V$ follows from Lemma
3.9 and condition (A) of $H_{\mathcal{A}_{0},G,b_{3},b_{4}}(\cdot)$.

Part (3) of the lemma follows from the explicit expressions of the infinite
volume Gibbs measures in Theorem 4.22, part (1)(2) and Theorem 3.14. ∎

##  6\. Extremity

In this section, we prove that the infinite-volume measure for dimer
configurations on $\overline{G}$ obtained as in Theorem 5.6 is extreme; see
Proposition 6.5. The idea is as follows. We first show that the extremity of
the infinite-volume measure is equivalent to the tail-triviality; then we show
that the infinite-volume measure is tail-trivial. The proof of the tail-
triviality of the infinite volume measure is inspired by the proof of Theorem
8.4 in [5]. One of the major differences is instead of considering uniform
spanning trees, we are considering perfect matchings. The corresponding anti-
symmetric function for the perfect matchings is a “discrete integral” version
of the anti-symmetric function for the spanning trees. Also it is
straightforward to see that the anti-symmetric function for spanning trees has
finite energy, the fact that the anti-symmetric function for perfect matchings
has finite energy is not so obvious; but we still manage to prove it by using
the transience property of the graph. Using similar technique, we prove that
the infinite-volume measure for dimer configurations on $\overline{G}$
obtained as in Theorem 4.22 is extreme under natural technical assumptions;
see Propositions 6.18 and 6.20.

###  6.1. Extremity and Tail-triviality

We first review a few definitions and lemmas.

######  Definition 6.1.

Suppose Assumption 2.4 holds. Consider the embedding of $\overline{G}$ into
the hyperbolic plane such that $(G,G^{+})$ are nerves of the double circle
packings. Define a unit flow from black vertices of $\overline{G}$ to white
vertices of $\overline{G}$ as follows. Let $e=(w,b)$ be an edge of
$\overline{G}$ with black vertex $b$ and white vertex $w$. The edge $e$ is