incident to two quadrilateral faces $f_{1}$ and $f_{2}$ in $\overline{G}$. For
$i\in\\{1,2\\}$, let $d_{i}$ be the diagonal of $f_{i}$ with $b$ as one
endpoint. Let $\theta_{e}$ be the angle formed by $d_{1}$ and $d_{2}$
containing the edge $e$. Defined the flow from $b$ to $w$ to be
$\frac{\theta_{e}}{2\pi}$. Then it is straightforward to check the following

(6.1) $\displaystyle\sum_{e:e\sim
w}\frac{\theta_{e}}{2\pi}=1;\qquad\sum_{e:e\sim b}\frac{\theta_{e}}{2\pi}=1.$

Let $M$ be a perfect matching on $\overline{G}$. Define a preliminary height
function $\overline{h}_{M}$ associate to $M$ on faces of $\overline{G}$ as
follows. Let $f_{1}$ and $f_{2}$ be two faces of $G$ sharing an edge
$e=(w,b)$.

  * •

If $e\in M$, and

    * –

moving from $f_{1}$ to $f_{2}$ along the dual edge $e^{+}$, the black vertex
$b$ is on the left, then
$\overline{h}_{M}(f_{1})-\overline{h}_{M}(f_{2})=1-\frac{\theta_{e}}{2\pi}$;

    * –

moving from $f_{1}$ to $f_{2}$ along the dual edge $e^{+}$, the black vertex
$b$ is on the right, then
$\overline{h}_{M}(f_{1})-\overline{h}_{M}(f_{2})=-1+\frac{\theta_{e}}{2\pi}$;

  * •

If $e\notin M$, and

  * •

moving from $f_{1}$ to $f_{2}$ along the dual edge $e^{+}$, the black vertex
$b$ is on the left, then
$\overline{h}_{M}(f_{1})-\overline{h}_{M}(f_{2})=-\frac{\theta_{e}}{2\pi}$;

  * •

moving from $f_{1}$ to $f_{2}$ along the dual edge $e^{+}$, the black vertex
$b$ is on the right, then
$\overline{h}_{M}(f_{1})-\overline{h}_{M}(f_{2})=\frac{\theta_{e}}{2\pi}$;

The condition (6.1) guarantees that the preliminary height change around any
vertex is 0. Therefore $\overline{h}_{M}$ is well-defined up to an additive
constant. One can make $\overline{h}_{M}$ uniquely defined by fixing the
height at one specific face to be 0.

Let $M_{0}$ be a fixed perfect matching on $\overline{G}$. Define the height
function $h_{M}$ associated to $M$ as a function from faces of $\overline{G}$
to the set of integers ${\mathbb{Z}}$ by

$\displaystyle h_{M}(f)=\overline{h}_{M}(f)-\overline{h}_{M_{0}}(f).$

######  Definition 6.2.

Suppose Assumption 2.4 holds. Let $M_{0}$ be the fixed perfect matching on
$\overline{G}$ as in Definition 6.1.

  1. (a)

For each edge $e$ of $\overline{G}$, let $f_{1},f_{2}$ be the two faces of
$\overline{G}$ incident to $e$, such that moving along $e^{+}$ from $f_{1}$ to
$f_{2}$, the black vertex of $e$ is no the left.

Define a potential $V_{e}:{\mathbb{Z}}\rightarrow[0,\infty)$ as follows:

$\displaystyle V_{e}(\eta)=\begin{cases}0&\mathrm{if}\ \eta=0\\\
0&\mathrm{if}\ h_{M_{0}}(f_{1})>h_{M_{0}}(f_{2})\ \mathrm{and}\ \eta=-1\\\
0&\mathrm{if}\ h_{M_{0}}(f_{1})<h_{M_{0}}(f_{2})\ \mathrm{and}\ \eta=1\\\
\infty&\mathrm{otherwise}.\end{cases}$

Let $F(\overline{G})$ be the set of all the faces of $\overline{G}$. Let
$\Omega$ be the set of functions from $F(\overline{G})$ to ${\mathbb{Z}}$; and
let $\mathcal{F}$ be the Borel $\sigma$-algebra of the product topology on
$\Omega$. Let $\Lambda$ be an arbitrary finite subset of $F(\overline{G})$. A
gradient Gibbs measure $\mu$ on $\Omega$ is a probability measure on $\Omega$
satisfying the following conditions. For each $\phi\in\Omega$, let
$\phi|_{\Lambda}$ (resp. $\phi|_{F(\overline{G})\setminus\Lambda}$) be the
restriction of $\phi$ onto $\Lambda$ (resp.
$F(\overline{G})\setminus\Lambda$).

$\displaystyle\mu(\phi_{\Lambda}|\phi_{F(\overline{G})\setminus{\Lambda}})\propto\exp(-H_{\Lambda}(\phi))$

where

$\displaystyle H_{\Lambda}(\phi):$
$\displaystyle=\sum_{e^{+}=(f_{1},f_{2}):f_{1},f_{2}\in\Lambda}V_{e}(\phi(f_{1})-\phi(f_{2}))+\sum_{e^{+}=(f_{1},f_{2}):f_{1}\in\Lambda,f_{2}\notin\Lambda}V_{e}(\phi(f_{1})-\phi(f_{2}))$
$\displaystyle+\sum_{e^{+}=(f_{1},f_{2}):f_{1}\notin\Lambda,f_{2}\in\Lambda}V_{e}(\phi(f_{1})-\phi(f_{2}))$

and $f_{1},f_{2}$ satisfies (a).

Then it is straightforward to verify the following lemma

######  Lemma 6.3.

Given the 1-1 correspondence between perfect matchings on $\overline{G}$ and
its height function, any gradient Gibbs measure satisfying on Definition 6.2
gives a Gibbs measure on perfect matchings of $\overline{G}$ with edge weights
given by

(6.2) $\displaystyle\nu(e)=\nu(e^{+})=1;$

for any edge $e$ of $G$.

######  Lemma 6.4.

Let $G=(V,E)$ be an infinite, connected graph. Let $\mathcal{Q}$ be the
collection of all gradient Gibbs measures on $\Omega$ as defined by Definition
6.2. A Gibbs measure is called extreme in $\mathcal{Q}$ if it is an extreme
element in the convex set $\mathcal{Q}$. Let $\mathrm{ex}\mathcal{Q}$ be the
set of all the extremal Gibbs measures in $\mathcal{Q}$. For each
$A\in\mathcal{F}$, let $e_{A}:\mu\mapsto\mu(A)$ be the evaluation map. Denote
by $e(\mathrm{ex}\mathcal{Q})$ the smallest $\sigma$-algebra on
$\mathrm{ex}(\mathcal{Q})$ with respect to which each $e_{A}$ is measurable.
Then

  1. (1)

For each $\nu\in{\mathcal{Q}}$, $\nu\in ex{\mathcal{Q}}$ if and only if $\nu$
is tail-trivial;

  2. (2)

For each $\mu\in\mathcal{Q}$, there exists a unique weight $w_{\mu}$, which is
a probability measure on $(\mathrm{ex}\mathcal{Q},e(\mathrm{ex}\mathcal{Q}))$,
such that for each $A\in\mathcal{F}$,

(6.3) $\displaystyle\mu(A)=\int_{\mathrm{ex}\mathcal{Q}}\nu(A)w_{\mu}(d\nu)$

The mapping $\mu\mapsto w_{\mu}$ is a bijection between $\mathcal{Q}$ and
$\mathcal{P}(\mathrm{ex}\mathcal{Q},e(\mathrm{ex}\mathcal{Q}))$ (the set of
all the probability measures on
$(\mathrm{ex}\mathcal{Q},e(\mathrm{ex}\mathcal{Q}))$).

###### Proof.

Part (1) of the lemma follows from Theorem 7.7(1) in [18]. Part (2) of the
lemma follows from Lemma 3.2.4 of [62]; see also Theorem 7.26 of [18]. ∎

###  6.2. General boundary conditions.

######  Proposition 6.5.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4, 5.5
hold. Let $\mu$ be the probability measure for dimer coverings on
$\overline{G}$ obtained as in Lemma 5.6. Assume the edge weights satisfy
(6.2). Then the measure $\mu$ is tail-trivial.

In order to prove Proposition 6.5, we first review and prove a few definitions
and lemmas. Recall the flows are defined in Definition 3.11.

######  Lemma 6.6.

(Exercise 9.2 in [58])Let $H_{n}$ be increasing closed subspaces of a Hilbert
space $H$ and $P_{n}$ be the orthogonal projection on $H_{n}$. Let $P$ be the
orthogonal projection on the closure of $\cup_{n}H_{n}$. Then for all $u\in
H$, we have $\|P_{n}u-Pu\|\rightarrow 0$ as $n\rightarrow\infty$.

######  Lemma 6.7.

(Thomson’s Principle; see page 35 of [58]). Let $G$ be a finite network and
$A$ and $Z$ be two disjoint subsets of its vertices. Let $\theta$ be a flow
from A to Z and $i=P_{\bigstar}\theta$ be the current flow from $A$ to $Z$
with $\mathrm{div}i=\mathrm{div}\theta$. Then
$\mathcal{E}(\theta)>\mathcal{E}(i)$ unless $\theta=i$.

######  Lemma 6.8.

(Proposition 2.12 in [58])Let G be a transient connected network and $G_{n}$
be finite induced subnetworks that contain a vertex $a$ and that exhaust $G$.
Identify the vertices outside $G_{n}$ to $z_{n}$, forming $G^{W}_{n}$. Let
$i_{n}$ be the unit current flow in $G^{W}_{n}$ from $a$ to $z_{n}$. Then
$\\{i_{n}\\}$ has a pointwise limit $i$ on $G$, which is the unique unit flow
on $G$ from $a$ to $\infty$ of minimum energy. Let $v_{n}$ be the voltages on
$G^{W}_{n}$ corresponding to $i_{n}$ and with $v_{n}(z_{n}):=0$. Then
$v:=lim_{n\rightarrow\infty}v_{n}$ exists on G and has the following
properties. Start a random walk at $a$. For all vertices $x$, the expected
number of visits to $x$ is $G(a,x)=c(x)v(x)$. For all edges $e$, the expected
signed number of crossings of e is $i(e)$.

######  Lemma 6.9.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4 and 5.5
hold. Assume the edge weights satisfy (6.2). Assume $\mathcal{A}_{1}$ in Lemma
5.6 satisfies condition (a) of Lemma 3.14 with $\mathcal{A}$ replaced by
$\mathcal{A}_{1}$. For each black vertex $b$ of $\overline{G}$, if $b$ is a
vertex of $G$,

$\displaystyle
i_{b}(e)=\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\zeta_{e}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w_{e});\
\forall(b,e)\in(V,\vec{E});$

is a finite energy unit flow from $b$ to $\infty$ on $G$, where $w_{e}$ is the
white vertex corresponding to the edge $e$, and

(6.4)
$\displaystyle\zeta_{e}:=\frac{\overline{e}-\underline{e}}{|\overline{e}-\underline{e}|}$

is the unit vector pointing from $\underline{e}$ to $\overline{e}$.

###### Proof.

Let $b\in V$ be arbitrary. By condition (a) of Lemma 3.14 and Corollary 3.13,
there exists an open set
$\mathcal{O}_{\mathcal{A}_{1}}\supset\mathcal{A}_{1}$, such that $G\setminus
U_{\mathcal{A}_{1}}$ is transient for every open set $U_{\mathcal{A}_{1}}$
satisfying $\mathcal{A}_{1}\subset
U_{\mathcal{A}_{1}}\subset\mathcal{O}_{\mathcal{A}_{1}}$. We may choose
$U_{\mathcal{A}_{1}}$ such that $b\in G\setminus U_{\mathcal{A}_{1}}$. By
Lemma 3.12, there exists a unit flow $q_{b}$ on $G\setminus
U_{\mathcal{A}_{1}}$ of finite energy from every vertex $b$ of $G\setminus
U_{\mathcal{A}_{1}}$ to $\infty$.

Let $G_{j}$ be the graph as Assumption 3.15 with $\mathcal{A}$ replaced by
$\mathcal{A}_{0}$. We may choose $j$ sufficiently large so that $G\setminus
U_{\mathcal{A}_{1}}$ is a subgraph of $G_{j}$. Then $q_{b}$ is a unit flow on
$G_{j}$ from $b$ to $\infty$. Let $G_{j,s}^{\bullet}$ be the subgraph of
$G_{j}$ as constructed in Assumption 3.15, and let $\bigstar_{j,s}$ be the
$\bigstar$ subspace of antisymmetric functions on directed edges of
$G_{j,s}^{\bullet}$. Define

$\displaystyle q_{b,j,s}=P_{\bigstar_{j,s}}q_{b}$

Then we have

$\displaystyle\bigstar_{j,s}\subset\bigstar_{j,s+1},\ \mathrm{and}\
\cup_{s}\bigstar_{j,s}=\bigstar_{j}$

where $\bigstar_{j}$ is the $\bigstar$ subspace of anti-symmetric functions on
directed edges of $G_{j}$. By Lemma 6.6,

$\displaystyle\lim_{s\rightarrow\infty}\|q_{b,j,s}-P_{\bigstar_{j}}q_{b}\|=0.$

Moreover,

$\displaystyle\bigstar_{j}\subset\bigstar_{j+1};\ \mathrm{and}\
\cup_{j}\bigstar_{j}=\diamondsuit_{\mathcal{A}_{0}}^{\perp}$

where $\diamondsuit_{\mathcal{A}_{0}}$ is the $\diamondsuit$ subspace of anti-
symmetric functions on directed edges of $G$ with $\mathcal{A}_{0}$ contracted
to one vertex. Again by Lemma 6.6 we infer that

$\displaystyle\lim_{j\rightarrow\infty}\|P_{\bigstar_{j}}q_{b}-P_{\diamondsuit_{\mathcal{A}_{0}}^{\perp}}q_{b}\|=0;$

We claim that $P_{\diamondsuit_{\mathcal{A}_{0}}^{\perp}}q_{b}$ is a unit
current flow from $b$ to $\infty$, which gives us exactly $i_{b}$. To see why
that is true, note that $P_{\bigstar_{j,s}}q_{b}$ is a unit current flow on
$G_{j,s}^{\bullet}$ with divergence $0$ everywhere other than $b$ and
$z_{j,s}$. Its voltage function is harmonic everywhere other than $b$ and
$U_{j,s}$, and equals a constant in $U_{j,s}$, hence it must be a constant
multiple of the Greens function of the random walk on $G_{j,s}$ absorbed at
$U_{j,s}$, plus an arbitrary constant. Taking limit by letting
$s\rightarrow\infty$, and then $j\rightarrow\infty$, gives us exactly $i_{b}$.

Hence $i_{b}$ has finite energy because $q_{b}$ has finite energy, and the
energy of $P_{\diamondsuit_{\mathcal{A}_{0}}^{\perp}}q_{b}$ is less than that
of $q_{b}$. ∎

Proof of Proposition 6.5. Let $F$ be a nonempty set of edges of
$\overline{G}$, and let $K$ be a nonempty set of white vertices of
$\overline{G}$, such that the vertex set of $F$ and $K$ are disjoint. Let
$\overline{E}_{K}$ be all the incident edges of $K$ in $\overline{G}$.

Let $A_{1}\in\mathcal{F}(F)$ and $A_{2}\in\mathcal{F}(\overline{E}_{K})$.

We shall show that

(6.5) $\displaystyle|\mu(A_{1}\cap A_{2})-\mu(A_{1})\mu(A_{2})|\leq
4^{2|F|}|F|\left[\sum_{b\in B(F)\cap
V}\|P_{\langle\chi_{F_{1}^{c}}\rangle}i_{b}\|^{2}+\sum_{b\in B(F)\cap
V^{+}}\|P_{\langle\chi_{F_{2}^{c}}\rangle}i_{b}\|^{2}\right]$

where $\tilde{F}_{1}$ (resp. $\tilde{F}_{2}$) consists of all the edges of $G$
(resp. $G^{+}$) with a half edge in $F_{1}$ (resp. $F_{2}$); then the tail-
triviality of the measure $\mu$ follows.

Let $M$ be a random perfect matching. We have

$\displaystyle\mu[F\subset
M|M\cap\overline{E}_{K}]=\left|\det\left[\overline{D}^{K}\right]^{-1}_{F}\right|$

where $\overline{D}^{K}$ is the weighted adjacency matrix obtained from
$\overline{D}$ by removing all the vertices in $M\cap\overline{E}_{K}$ and
their incident edges, and $[\overline{D}^{K}]^{-1}_{F}$ is the square
submatrix of $[\overline{D}^{K}]^{-1}$ with rows indexed by black vertices of
edges in $F$ and columns indexed by white vertices in $F$. It is
straightforward to see that if two edges in $F$ share a white vertex, then
$\det[\overline{D}^{K}]^{-1}_{F}=0$ since there are two identical columns in
$[\overline{D}^{K}]^{-1}_{F}$; and the probability that all these edges appear
in a perfect matching is indeed 0.

For each black vertex $w$ of $\overline{G}$, let $e_{w}$ be the edge
corresponding to $w$ in $E$ and $e_{w}^{+}$ be the edge corresponding to $w$
in $E^{+}$. We shall construct a new graph $\tilde{G}_{j,s}^{K}$ as a subgraph
of $G_{j,s}^{\bullet}$ in the following way

  * •

Consider Assumption 3.15. For each edge $(b,w)\in M\cap\overline{E}_{K}$ and
$b\in V$, remove the edge $e_{w}$ from $G_{j}$ and identify $b$ with all the
vertices in $U_{j,s}$.

  * •

For each edge $(b,w)\in\overline{E}_{K}\cap M$ and $b\in V^{+}$, remove the
edge $e_{w}$ from $G_{j,s}$.

The resulting graph obtained from $G_{j,s}^{\bullet}$ is
$\tilde{G}_{j,s}^{K}$.

Similarly, we construct a new graph $\tilde{G}_{j,s,+}^{K}$ in the following
way

  * •

Consider Assumptions 3.15 and 5.5, let $G_{j,s}^{+,\bullet}$ be the graph
obtained from $G_{j,s}^{+}$ by contracting all the dual vertices (or
equivalently, faces) in $G\setminus G_{j}$ to the same vertex $z_{j,s}^{+}$ as
well as dual edges joining two such dual vertices.

  * •

For each edge $(b,w)\in\overline{E}_{K}\cap M$ and $b\in V$, remove the edge
$e_{w}^{+}$ from $G_{j,s}^{+,\bullet}$.

  * •

For each edge $(b,w)\in\overline{E}_{K}\cap M$ and $b\in V^{+}$, remove the
edge $e_{w}^{+}$ from $G_{j,s}^{+,\bullet}$ and identify the vertex $b$ with
all the dual vertices in $G\setminus G_{j}$.

The resulting graph obtained from $G_{j,s}^{+,\bullet}$ is
$\tilde{G}_{j,s,+}^{K}$.

Let $Q_{\tilde{G}_{j,s}^{K}}$ (resp. $Q_{\tilde{G}_{j,s+}^{K}}$) be the
orthogonal projection onto the $\bigstar$ subspace of $\tilde{G}_{j,s}^{K}$
(resp. $\tilde{G}_{j,s,+}^{K}$).

For a black vertex $b$ of edges in $F$, and a white vertex $w$ of edges in
$F$, the following cases may occur

  * •

$b$ is a vertex of $V$; then

$\displaystyle[\overline{D}^{K}]^{-1}(b,w)=\frac{1}{\zeta_{e_{w}}}\langle
Q_{\tilde{G}_{j,s}}^{K}i_{b},\chi_{e_{w}}\rangle$

where the direction of $e_{w}$ can be any of the two possible ones.

  * •

$b$ is a vertex of $V^{+}$; then

$\displaystyle[\overline{D}^{K}]^{-1}(b,w)=\frac{1}{\zeta_{e_{w}^{+}}}\langle
Q_{\tilde{G}_{j,s,+}}^{K}i_{b},\chi_{e_{w}^{+}}\rangle$

where the direction of $e_{w}^{+}$ can be any of the two possible ones.

Let

$\displaystyle
Q_{j,s,K}i_{b}:=\begin{cases}Q_{\tilde{G}_{j,s}}^{K}i_{b}&\mathrm{if}\ b\in
V\\\ Q_{\tilde{G}_{j,s,+}}^{K}i_{b}&\mathrm{if}\ b\in V^{+}\end{cases}$

Then

(6.6) $\displaystyle\mathrm{Var}\left(\mu[F\subset
M|M\cap\overline{E}_{K}]\right)$
$\displaystyle=\mathrm{Var}\left(\left|\det\left[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle\right]_{b\in B(F),w\in
W(F)}\right|\right)$ $\displaystyle\leq\mathrm{Var}\left(\det\left[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle\right]_{b\in B(F),w\in W(F)}\right)$

where the $\mathrm{Var}$ is with respect to the distribution of
$M\cap\overline{E}_{K}$, $B(F)$ is the set of all the black vertices of edges
in $F$, $W(F)$ is the set of all the white vertices of edges in $F$; where
$\zeta_{e_{w}}$ is defined as in (6.4) and

$\displaystyle\chi_{e_{w,b}}=\begin{cases}\chi_{e_{w}}&\mathrm{if}\ b\in V\\\
\chi_{e_{w}^{+}}&\mathrm{if}\ b\in V^{+}\end{cases}$

We make the following claim

######  Claim 6.10.

$\displaystyle\mathrm{Var}(\det[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in B(F),w\in W(F)}\leq
2|F|\left[\sum_{b\in B(F)\cap V}\|P_{K_{1}}i_{b}\|_{R}^{2}+\sum_{b\in B(F)\cap
V^{+}}\|P_{K_{2}}i_{b}\|_{R}^{2}\right]$

Claim 6.10 will be proved later.

Now we apply Claim 6.10 to prove Proposition (6.5). As discussed before, (6.5)
will imply the tail-triviality of the measure $\mu$. Given
$A_{1}\in\mathcal{F}(F)$, $A_{1}$ is the union at most $2^{|F|}$ events, each
of which specifies the states of each edge in the perfect matching; each such
event is the union or difference of at most $2^{|F|}$ events, each of which is
the event that a certain subset of edges of $F$ is present in the perfect
matching. Therefore if we write

$\displaystyle a:=4^{2|F|+1}|F|\left[\sum_{b\in B(F)\cap
V}\|P_{K_{1}}i_{b}\|_{R}^{2}+\sum_{b\in B(F)\cap
V^{+}}\|P_{K_{2}}i_{b}\|_{R}^{2}\right]$

Then

$\displaystyle\mathrm{Var}(\mu(A_{1})|M\cap\overline{E}_{K})\leq a$

By convexity of quadratic functions we have

$\displaystyle|\mu[A_{1}|A_{2}]-\mu(A_{1})|^{2}\mu(A_{2})\leq a$

and therefore

$\displaystyle|\mu[A_{1}|A_{2}]-\mu(A_{1})|^{2}\leq a\mu(A_{2})\leq a.$

$\hfill\Box$

We recall a lemma before proving Claim 6.10.

######  Lemma 6.11.

(Lemma 8.7 in [5])Let $\mathbb{P}$ be any probability measure on the set of
$k\times k$ real matrices. For a matrix $A$, write its rows as $A_{i}$ and its
entries as $A_{i,j}$ ($i=1,2,\ldots,k$), if $\mathbb{E}\det A=\det\mathbb{E}A$
and each $\|A_{i}\|_{2}\leq 1$, then

$\displaystyle\mathrm{Var}\det A\leq k\sum_{i,j=1}^{k}\mathrm{Var}A_{i,j}.$

Proof of Claim 6.10. Now we fix the direction of each $e_{w,b}$ to be from $w$
to $b$. Note that

(6.7) $\displaystyle\mathbb{E}\det[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in B(F),w\in
W(F)}=\mathbb{E}\frac{(-1)^{\sum_{b\in B(F)}\sum_{w\in
W(F)}|b|+|w|}\det\overline{D}^{K}_{\setminus\\{B(F),W(F)\\}}}{\det\overline{D}^{K}}$
$\displaystyle=\mathbb{E}\mu[F\subset M|M\cap\overline{E}_{K}]=\mu[F\subset
M]=\frac{(-1)^{\sum_{b\in B(F)}\sum_{w\in
W(F)}|b|+|w|}\det\overline{D}_{\tilde{G}_{j,s}\setminus\\{B(F),W(F)\\}}}{\det\overline{D}_{\tilde{G}_{j,s}}}$

where

  * •

$\overline{D}^{K}_{\setminus\\{B(F),W(F)\\}}$ is the submatrix of
$\overline{D}^{K}$ by removing rows and columns indexed by $B(F)$ and $W(F)$,
respectively.

Now we show that

(6.8) $\displaystyle\frac{(-1)^{\sum_{b\in B(F)}\sum_{w\in
W(F)}|b|+|w|}\det\overline{D}_{\tilde{G}_{j,s}\setminus\\{B(F),W(F)\\}}}{\det\overline{D}_{\tilde{G}_{j,s}}}=\det\mathbb{E}[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in B(F),w\in W(F)}$

Since

$\displaystyle\mathbb{E}[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]=\sum_{W^{\prime}:W^{\prime}=W(M\cap\overline{E}_{K})}\frac{(-1)^{|w|+|b|+\sum_{w^{\prime}\in
W^{\prime}}\sum_{b^{\prime}\in
K}|w^{\prime}|+|b^{\prime}|}\det{\overline{D}}_{\tilde{G}_{j,s}\setminus[\\{b,w\\}\cup
W^{\prime}\cup K]}}{\det\overline{D}_{\tilde{G}_{j,s}}}$
$\displaystyle=\frac{(-1)^{|w|}\det{\overline{D}}_{\tilde{G}_{j,s}\setminus[\\{b,w\\}\cup
W^{\prime}\cup K]}}{\det\overline{D}_{\tilde{G}_{j,s}}}$

  * •

$W(M\cap\overline{E}_{K})$ is the set of white vertices of edges in
$M\cap\overline{E}_{K}$ (recall that the set of black vertices of edges in
$M\cap\overline{E}_{K}$ is $K$).

We have (6.8) follows.

It follows from (6.7), (6.8) that

$\displaystyle\mathbb{E}\det[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in B(F),w\in
W(F)}=\det\mathbb{E}[\langle Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in
B(F),w\in W(F)}$

One may also interpret (6.7) as follows. By Temperley’s bijection, perfect
matching configurations are in 1-1 correspondence with directed spanning trees
(recall that the dual tree configuration is uniquely determined by the
spanning tree configuration on the primal graph). The probabilistic
interpretation of $\langle Q_{j,s,K}i_{b},\chi_{e_{w},b}\rangle$ is,
conditional on the directed tree configurations on edges and dual edges
corresponding to white vertices in $F$, the probability that the branch of the
tree staring at $b$ passes through $e_{w}$ minus the probability that it
passes through $-e_{w}$. Then the last identity of (6.7) follows.

Then by Lemma 6.11, we have

(6.9) $\displaystyle\mathrm{Var}(\det[\langle
Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle]_{b\in B(F),w\in W(F)}\leq|F|\sum_{b\in
B(F),w\in W(F)}\mathrm{Var}\langle Q_{j,s,K}i_{b},\chi_{e_{w,b}}\rangle$
$\displaystyle=|F|\left[\sum_{b\in B(F)\cap V,w\in W(F)}\mathrm{Var}\langle
Q_{j,s,K}i_{b},\chi_{e_{w}}\rangle+\sum_{b\in B(F)\cap V^{+},w\in
W(F)}\mathrm{Var}\langle Q_{j,s,K}i_{b},\chi_{e_{w}^{+}}\rangle\right]$

Recall that the graph $G_{j,s}^{\bullet}$ in Assumption 3.15 has a unique
identified boundary vertex $z_{j,s}$ with Dirichlet boundary condition; let
$\bigstar_{G_{j,s}^{\bullet}}$ be the $\bigstar$ space of anti-symmetric
functions on directed edges of $\bigstar_{G_{j,s}^{\bullet}}$. Let

$\displaystyle K_{1}:=\\{e_{w}:w\in K\\};\qquad K_{2}:=\\{e_{w}^{+}:w\in
K\\}.$

For each possible $S=\overline{E}_{K}\cap M$, let $L_{S}$ consist of all the
shortest paths in $G_{j,s}^{\bullet}$ joining $z_{j,s}$ and a vertex in $S\cap
V$ without using edges in $K_{1}$.

Then

(6.10)
$\displaystyle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}i_{b}=\sum_{S\subset\overline{E}_{K}}\mathbb{P}[M\cap\overline{E}_{K}=S]P_{[\bigstar_{G_{j,s}^{\bullet}}+\langle\chi_{K_{1}}\rangle]\cap\langle\chi_{K_{1}}\rangle^{\perp}\cap\langle
L_{S}\rangle^{\perp}}i_{b}$

where $\langle\chi_{K_{1}}\rangle$ is the subspace of antisymmetric functions
generated by $\\{\chi_{e}\\}_{e\in K_{1}}$.

We shall need the following claims

######  Claim 6.12.

  1. (1)

$\displaystyle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}=P_{K_{1}}^{\perp}P_{\bigstar_{G_{j,s}^{\bullet}}}P_{K_{1}}^{\perp};\forall
b\in B(F)\cap V,\ b\notin K.$

  2. (2)

$\displaystyle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}=P_{K_{2}}^{\perp}P_{\bigstar_{G_{j,s}^{+,\bullet}}}P_{K_{2}}^{\perp};\forall
b\in B(F)\cap V^{+},\ b\notin K.$

######  Claim 6.13.

For any $\xi\in l_{-}(E(G_{j,s}^{\bullet}))$, we have

$\displaystyle\mathrm{Var}(Q_{j,s,K}\xi):=\mathbb{E}\|Q_{j,s,K}\xi-\mathbb{E}Q_{j,s,K}\xi\|_{R}^{2}=\|P_{K_{1}}P_{\bigstar_{G_{j,s}^{\bullet}}}P_{K_{1}}^{\perp}\xi\|_{R}^{2}.$

For any $\eta\in l_{-}(E(G_{j,s}^{+,\bullet}))$, we have

$\displaystyle\mathrm{Var}(Q_{j,s,K}\eta):=\mathbb{E}\|Q_{j,s,K}\eta-\mathbb{E}Q_{j,s,K}\eta\|_{R}^{2}=\|P_{K_{2}}P_{\bigstar_{G_{j,s}^{+,\bullet}}}P_{K_{2}}^{\perp}\eta\|_{R}^{2}.$

Claims 6.12 and 6.13 will be proved later.

We infer that

(6.11) $\displaystyle\sum_{w\in W(F)}\mathrm{Var}\langle
Q_{j,s,K}i_{b},\chi_{e_{w}}\rangle\leq\mathbb{E}\|Q_{j,s,K}i_{b}-\mathbb{E}Q_{j,s,K}i_{b}\|^{2}_{R}\leq\|P_{K_{1}}P_{\bigstar_{G_{j,s}^{\bullet}}}P_{K_{1}}^{\perp}i_{b}\|_{R}^{2}$
$\displaystyle=\|P_{K_{1}}P_{\bigstar_{G_{j,s}^{\bullet}}}(1-P_{K_{1}})i_{b}\|_{R}^{2}\leq
2\|P_{K_{1}}i_{b}\|^{2}$

Similarly we can show that if $b\in V^{+}$

(6.12) $\displaystyle\sum_{w\in W(F)}\mathrm{Var}\langle
Q_{j,s,K}i_{b},\chi_{e_{w}}\rangle\leq 2\|P_{K_{2}}i_{b}\|_{R}^{2}.$

Then the Claim 6.10 follows from (6.11), (6.12) and (6.9). $\hfill\Box$

Proof of Claim 6.12. We only prove part (1) of Claim 6.12, part (2) can be
proved using the same arguments.

Let $T$ be a random spanning tree of $G_{j,s}^{\bullet}$ and let
$e=(u,v)\in\vec{E}(G_{j,s}^{\bullet})$. Let $L_{u,v}$ be the oriented path in
$T$ starting from $u$ and ending in $v$. Let

$\displaystyle\zeta_{T}^{e}=\sum_{e_{i}\in L_{u,v}}\chi_{e_{i}}$

Then for any $f\in\vec{E}(G_{j,s}^{\bullet})$

(6.13) $\displaystyle
P_{\bigstar_{G_{j,s}^{\bullet}}}\chi_{e}(f)=\mathbb{E}\zeta_{T}^{e}(f)=\left[-\frac{G_{G_{j,s}^{\bullet},N}(u,\overline{f})}{\left|\sum_{x\in
V(G_{j,s}^{\bullet}),x\sim\overline{f}}\Delta(\overline{f},x)\right|}+\frac{G_{G_{j,s}^{\bullet},N}(u,\underline{f})}{\left|\sum_{x\in
V(G_{j,s}^{\bullet}),x\sim\underline{f}}\Delta(\underline{f},x)\right|}\right.$
$\displaystyle\left.+\frac{G_{G_{j,s}^{\bullet},N}(v,\overline{f})}{\left|\sum_{x\in
V(G_{j,s}^{\bullet}),x\sim\overline{f}}\Delta(\overline{f},x)\right|}-\frac{G_{G_{j,s}^{\bullet},N}(v,\underline{f})}{\left|\sum_{x\in
V(G_{j,s}^{\bullet}),x\sim\underline{f}}\Delta(\underline{f},x)\right|}\right]$

where $G_{G_{j,s}^{\bullet},N}$ is the Green’s function for the random walk on
$G_{j,s}^{\bullet}$ absorbed at $z_{j,s}$.

Let $M$ be the random perfect matching obtained from $T$ via the Temperley
bijection. More precisely, let $T^{+}$ be the dual spanning tree on
$G_{j,s}^{+,\bullet}$ such that a dual edge $e^{+}\in T^{+}$ if and only if
its corresponding primal edge $e\notin T$. For each $e\in T$ (resp. $e^{+}\in
T^{+}$), let $L_{e}$ ($L_{e^{+}}$) be the unique branch of $T$ (resp. $T^{+}$)
starting from $e$ (resp. $T^{+}$) and ending in $z_{j,s}$ (resp.
$z_{j,s}^{+}$), orient $e=(u,v)$ (resp. $e^{+}=(x,y)$) from $u$ (resp. $x$) to
$v$ (resp. $y$) such that $u$ (resp. $x$) is the first vertex along $L_{e}$
($L_{e^{+}}$). Then an edge $s=(b,w)$ of $\overline{G}$ is present in the
perfect matching $M$ if and only if one of the following two conditions holds

  * •

$s$ is a half edge of a directed edge $e\in T$ such that $e$ starts at $b$; or

  * •

$s$ is a half edge of a directed edge $e^{+}\in T^{+}$ such that $e^{+}$
starts at $b$.

Similarly to (6.13) we have if $b\in V$ and $e\notin K_{1}$,

$\displaystyle
P_{[\bigstar_{G_{j,s}^{\bullet}}+\langle\chi_{K_{1}}\rangle]\cap\langle\chi_{K_{1}}\rangle^{\perp}\cap\langle
L_{S}\rangle^{\perp}}\chi_{e}=P_{K_{1}}^{\perp}\mathbb{E}[\zeta_{T}^{e}|M\cap\overline{E}_{K}=S]$

To prove Claim 6.12(1), it suffices to show that for any
$e,h\in\vec{E}(G_{j,s}^{\bullet})$,

(6.14)
$\displaystyle\langle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}\chi_{e},\chi_{h}\rangle=\langle
P_{K_{1}}^{\perp}P_{\bigstar_{G_{j,s}^{\bullet}}}P_{K_{1}}^{\perp}\chi_{e},\chi_{h}\rangle.$

If $h\in K_{1}$ or $e\in K_{1}$, by (6.10) we see both the left hand side and
the right hand side of (6.14) are 0. Assume $h\notin K_{1}$ and $e\notin
K_{1}$; then to prove (6.14) it suffices to show that

(6.15)
$\displaystyle\langle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}\chi_{e},\chi_{h}\rangle=\langle
P_{\bigstar_{G_{j,s}^{\bullet}}}\chi_{e},\chi_{h}\rangle$

Note that

$\displaystyle\langle
P_{\bigstar_{G_{j,s}^{\bullet}}}\chi_{e},\chi_{h}\rangle$
$\displaystyle=\langle\mathbb{E}\zeta_{T}^{e},\chi_{h}\rangle=\langle\mathbb{E}\zeta_{T}^{e},P_{K_{1}}^{\perp}\chi_{h}\rangle=\langle
P_{K_{1}}^{\perp}\mathbb{E}\zeta_{T}^{e},\chi_{h}\rangle$
$\displaystyle=\sum_{S\subset\overline{E}_{K}}\mathbb{P}(M\cap\overline{E}_{K}=S)\langle
P_{K_{1}}^{\perp}\mathbb{E}[\zeta_{T}^{e}|M\cap\overline{E}_{K}=S],\chi_{h}\rangle$
$\displaystyle=\sum_{S\subset\overline{E}_{K}}\mathbb{P}(M\cap\overline{E}_{K}=S)\langle
P_{[\bigstar_{G_{j,s}^{\bullet}}+\langle\chi_{K_{1}}\rangle]\cap\langle\chi_{K_{1}}\rangle^{\perp}\cap\langle
L_{S}\rangle^{\perp}}\chi_{e},\chi_{h}\rangle$
$\displaystyle=\langle\mathbb{E}Q_{\tilde{G}_{j,s}^{K}}\chi_{e},\chi_{h}\rangle.$

Then Claim 6.12(1) follows. $\hfill\Box$

Proof of Claim 6.13. Same arguments as the proof of Lemma 8.6 in [5].
$\hfill\Box$

######  Remark 6.14.

Note that although the infinite-volume Gibbs measure obtained in Theorem 5.6
is extremal, it is in general not automorphism-invariant with respect to the
whole automorphism group of $\overline{G}$. For example let $G$ be the
degree-7 transitive triangulation of the hyperbolic plane. Let $b_{1}$ be a
vertex of $G$ and let $b_{2}$ be a vertex of $G^{+}$. Let
$\\{e_{i}\\}_{i\in\\{1,2,\ldots,7\\}}$ be all the edges incident to $b_{1}$
and let $\\{f_{j}\\}_{j\in\\{1,2,3\\}}$ be all the edges incident to $b_{2}$.
If $\eta$ is an automorphism-invariant measure for perfect matchings on
$\overline{G}$ with respect to the whole automorphism group, then

$\displaystyle\eta(e_{i}\in M)=\frac{1}{7};\qquad\forall
i\in\\{1,2,\ldots,7\\}$ $\displaystyle\eta(f_{j}\in
M)=\frac{1}{3};\qquad\forall j\in\\{1,2,3\\}$

then at each white vertex, the sum of probabilities of the 4 incident edges
present in a perfect matching is
$\frac{1}{3}+\frac{1}{3}+\frac{1}{7}+\frac{1}{7}<1$, which is not possible.
However, from Theorem 5.6, one sees that this infinite-volume Gibbs measure is
invariant with respect to automorphisms of $G$ that preserving
$\mathcal{A}_{0}$. But the automorphisms of $G$ that preserving
$\mathcal{A}_{0}$ do not have a finite orbit.

###  6.3. Temperley boundary conditions

######  Assumption 6.15.

Suppose the edge weights satisfy (6.2). Let $G_{j}$ be a finite subgraph of
$G$ as in section 4.2. Let $b_{0,j}$ be the removed vertex to obtain
$\mathcal{G}_{j}$ from $\overline{G}_{j}$. For $b\in V(G_{j})$, add an extra
edge $e_{bb_{0,j}}$ with end points $b$ and $b_{0,j}$ in $G_{j}$ and let
$G_{j,e_{bb_{0,j}}}$ be the new graph $G_{j}\cup\\{e_{bb_{0,j}}\\}$. Define a
flow on the new graph by

$\displaystyle\theta_{j}:=P_{\bigstar_{G_{j,e_{bb_{0,j}}}}}\chi_{(b,b_{0,j})}$

where $(b,b_{0,j})$ is the directed edge from $b$ to $b_{0,j}$, and
$\bigstar_{G_{j,e_{bb_{0,j}}}}$ is the $\bigstar$ subspace of the graph
$G_{j,e_{bb_{0,j}}}$. Assume $\lim_{j\rightarrow\infty}b_{0,j}=b_{0}$. Assume
as $j\rightarrow\infty$, $\theta_{j}$ converges edgewise to a flow $\theta$
such that

$\displaystyle\langle\theta,\chi(b,b_{0})\rangle\neq 1$

######  Lemma 6.16.

Suppose Assumption 6.15 holds. Then

(6.16)
$\displaystyle\frac{P_{(b,b_{0})}^{\perp}\theta}{1-\langle\theta,\chi(b,b_{0})\rangle}$

is a finite energy unit current flow from $b$ to $b_{0}$ in $G$.

###### Proof.

Note that $\chi_{(b,b_{0,j})}$ is a unit flow from $b$ to $b_{0,j}$ on
$G_{j,e_{bb_{0,j}}}$. Since the orthogonal projection does not change the
divergence, we infer that $\theta_{j}$ is a unit current flow from $b$ to
$b_{0,j}$ on $G_{j,e_{bb_{0,j}}}$. The edgewise limit $\theta$ is a unit
current flow from $b$ to $b_{0}$ in $G_{e_{bb_{0}}}$, where $G_{e_{bb_{0}}}$
is the graph obtained from $G$ by adding an extra edge joining $b$ and
$b_{0}$. $P_{(b,b_{0})}^{\perp}\theta$ is a current flow on $G$ with
divergence $1-\langle\theta,\chi(b,b_{0})\rangle$ at $b$ and divergence
$-1+\langle\theta,\chi(b,b_{0})\rangle$ at $b_{0}$. When
$1-\langle\theta,\chi(b,b_{0})\rangle\neq 0$, (6.16) is a unit current flow on
$G$ with Dirichlet energy bounded above by
$\frac{1}{[1-\langle\theta,\chi(b,b_{0})\rangle]^{2}}$. Then the lemma
follows. ∎

######  Lemma 6.17.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4 and 5.5
hold. Assume the edge weights satisfy (6.2). Consider the infinite volume
Gibbs measure obtained in Theorem 4.22. For each black vertex $b$ of
$\overline{G}\cap V^{+}$,

$\displaystyle
i_{b}(e):=\lim_{j\rightarrow\infty}\zeta_{e}\overline{D}_{\overline{G}_{j}}^{-1}(b,w_{e});\
\forall(b,e)\in(V^{+},\vec{E});$

is a finite energy unit flow from $b$ to $\infty$ on $G$, where $w_{e}$ is the
white vertex corresponding to the edge $e$, and $\zeta_{e}$ is given as in
(6.4).

###### Proof.

Same arguments as in the proof of Lemma 6.9. ∎

######  Proposition 6.18.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4, 6.15
hold. Let $\mu$ be the probability measure for dimer coverings on
$\overline{G}$ obtained as in Lemma 4.22. Assume the edge weights satisfy
(6.2). Then the measure $\mu$ is tail-trivial.

###### Proof.

Let $F,K$, $K_{1}$, $K_{2}$ be given as in the proof of Proposition 6.5.

Compared to Proposition 6.5 in this case we may consider
$\mathcal{A}_{0}=\\{b_{0}\\}\in\partial{\mathbb{H}}^{2}$. In this case we no
longer have condition (a) in Lemma 3.14 holds with $\mathcal{A}$ replaced by
$\mathcal{A}_{1}$. Therefore we cannot prove the $i_{b}$ for $b\in V$ in this
case has finite energy as in Lemma 6.9. However, we can still follow the
computations as in the proof of Proposition 6.5 and obtain (6.9). Following
the same arguments as in the proof of Claim 6.10 and applying Lemma 6.17, we
obtain (6.12). It remains to give an upper bound for the first term in the
right hand side of (7.4) in the parentheses. Let

$\displaystyle
i_{b,j}(e):=\zeta_{e}\overline{D}_{\overline{G}_{j}}^{-1}(b,w_{e})$

By Claim 6.12 we obtain

$\displaystyle\sum_{b\in B(F)\cap V,w\in W(F)}\mathrm{Var}\langle
Q_{j,s,K}i_{b,j},\chi_{e_{w}}\rangle=\sum_{b\in B(F)\cap V,w\in
W(F)}\mathbb{E}[\langle Q_{j,s,K}i_{b,j},\chi_{e_{w}}\rangle-\mathbb{E}\langle
Q_{j,s,K}i_{b,j},\chi_{e_{w}}\rangle]^{2}$ $\displaystyle=\sum_{b\in B(F)\cap
V,w\in W(F)}\mathbb{E}[\langle
i_{b,j},Q_{j,s,K}\chi_{e_{w}}\rangle-\mathbb{E}\langle
i_{b,j},Q_{j,s,K}\chi_{e_{w}}\rangle]^{2}$ $\displaystyle=\sum_{b\in B(F)\cap
V,w\in W(F)}\mathbb{E}[\langle
i_{b,j},Q_{j,s,K}\chi_{e_{w}}-\mathbb{E}Q_{j,s,K}\chi_{e_{w}}\rangle]^{2}\leq\|i_{b,j}\|_{R}^{2}\mathbb{E}\|Q_{j,s,K}\chi_{e_{w}}-\mathbb{E}Q_{j,s,K}\chi_{e_{w}}\|^{2}$

When Assumption 6.15 holds, we have

$\displaystyle\lim_{j\rightarrow\infty}\|i_{b,j}\|_{R}^{2}<\infty$

and by Claim 6.13 we have

$\displaystyle\mathbb{E}\|Q_{j,s,K}\chi_{e_{w}}-\mathbb{E}Q_{j,s,K}\chi_{e_{w}}\|^{2}=\|P_{K_{1}}P_{\bigstar
G_{j}}P_{K_{1}}^{\perp}\chi_{e_{w}}\|^{2}$

When $e_{w}\notin K_{1}$ we have

$\displaystyle\lim_{j\rightarrow\infty}\|P_{K_{1}}P_{\bigstar
G_{j}}P_{K_{1}}^{\perp}\chi_{e_{w}}\|^{2}\|=\|P_{K_{1}}P_{\bigstar
G_{j}}\chi_{e_{w}}\|^{2}$

which goes to 0 as the distance of $K_{1}$ and $e_{w}$ goes to infinity. Then
the proposition follows. ∎

######  Example 6.19.

We shall give examples where Assumption 6.15 holds and where Assumption 6.15
does not hold.

First consider the graph $G$ with vertex set ${\mathbb{N}}$ and edges joining
two nonnegative integers with distance 1. Let $G_{j}$ be the graph with vertex
set $\\{0,1,\ldots,j\\}$. Let $b=0$ and $b_{0,j}=j$. Then

$\displaystyle\theta_{j}(i-1,i)=\frac{1}{j},\qquad\forall 1\leq i\leq j;$
$\displaystyle\theta_{j}(0,j)=\frac{j-1}{j}$

We have

$\displaystyle\lim_{j\rightarrow\infty}\theta_{j}(0,j)=1.$

Assumption 6.15 does not hold in this case.

Now consider $G_{j}$ to be constructed from two depth $j$ trees $T_{1}$ and
$T_{2}$ as follows

  * •

The root of $T_{1}$ is $b$;

  * •

The root of $T_{2}$ is $b_{0,j}$;

  * •

Every vertex in $T_{1}$ and $T_{2}$ other than the leaves has two children;
such that both $T_{1}$ and $T_{2}$ have exactly $2^{j}$ leaves;

  * •

Identify each leave with $T_{1}$ with a unique leave of $T_{2}$ gives $G_{j}$.

In this case we have

$\displaystyle\theta_{j}(b,b_{0,j})=\frac{4-2^{2-j}}{6-2^{2-j}}$

If $x$ is a depth $k-1$ vertex of $T_{1}$ while $y$ is a depth $k$ vertex of
$T_{1}$ (or if $x$ is a depth $k$ vertex of $T_{2}$ and $y$ is a depth $k-1$
vertex of $T_{2}$), we have

$\displaystyle\theta_{j}(x,y)=\frac{2^{1-k}}{6-2^{2-j}};\qquad\forall 1\leq
k\leq j$

In this case we have

$\displaystyle\lim_{j\rightarrow\infty}\theta_{j}(b,b_{0,j})=\frac{2}{3}\neq
1;$

Assumption 6.15 holds in this case.

In Example 6.19, we have an nonamenable graph where Assumption 6.15 holds.
However, as we shall see in the next proposition, nonamenability is not a
necessary condition for Assumption 6.15.

######  Proposition 6.20.

Let $G_{j}$ be finite subgraphs of $G$ as in section 4.2. Let $b_{0,j}$ be the
removed vertex of $G_{j}$ to obtain $\mathcal{G}_{j}$ from $\overline{G}_{j}$.
Assume for each vertex $b$ of $G$ and all sufficiently large $j$, there exists
a subtree $T_{j,b}$ of an infinite transient tree $T$ rooted at $b$ and a
subtree $T_{j,b_{0,j}}$ of $T$ rooted at $b_{0,j}$ such that

  * •

$T_{j,b}$ and $T_{j,b_{0,j}}$ are edge-disjoint subgraphs of $G$; and

  * •

there exists an isomorphism $\sigma:T_{j,b}\rightarrow T_{j,b_{0,j}}$, such
that $\sigma(b)=b_{0,j}$; and if $v$ is a leave of $T_{j,b}$ then
$\sigma(v)=v$; and

  * •

$\lim_{j\rightarrow\infty}T_{j,b}$ is isomorphic to $T$.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4, 6.15
hold. Let $\mu$ be the probability measure for dimer coverings on
$\overline{G}$ obtained as in Lemma 4.22. Assume the edge weights satisfy
(6.2). Then the measure $\mu$ is tail-trivial.

###### Proof.

Since $T$ is an infinite transient tree, there exists a finite energy unit
flow $\theta$ from the root of $T$ to $\infty$. Let $\theta|_{T_{j,b}}$ be the
restriction of the flow to the subtree $T_{j,b}$. Then
$\left[\theta|_{T_{j,b}}\right]\cup\left[-\theta|_{T_{j,b_{0,j}}}\right]$
gives a unit flow on $G_{j}$ from $b$ to $b_{0,j}$. As $j\rightarrow\infty$
this gives a finite energy unit flow from $b$ to $b_{0}$; where
$b_{0}=\lim_{j\rightarrow\infty}b_{0,j}$. Then

$\displaystyle
P_{\bigstar}\lim_{j\rightarrow\infty}\left[\theta|_{T_{j,b}}\right]\cup\left[-\theta|_{T_{j,b_{0,j}}}\right]$

gives a finite energy unit current flow from $b$ to $b_{0}$. Following the
same arguments as in the proof of Proposition 6.18, we infer that the measure
$\mu$ is tail-trivial. ∎

##  7\. Double Dimer Contours

In this section, we prove that for the infinite-volume Gibbs measure given in
Theorem 5.6 for uniformly weighted perfect matchings on $\overline{G}$, if $G$
satisfies certian 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. This 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. See Theorems 7.5 and
7.3 for precise statements. The proof depends on the fact that the Dirichlet
Green’s function on such graphs has certain decay rate with respect to the
distance of two points due to isoperimetric inequalities.

For any two dimer configurations $M_{1}$, $M_{2}$ on the same graph, the
symmetric difference $M_{1}\triangle M_{2}$ is a disjoint union of cycles and
doubly infinite self-avoiding paths.

######  Definition 7.1.

Let $\overline{G}$ be a bipartite, planar graph. Let $M_{1}$, $M_{2}$ be two
dimer configurations on $\overline{G}$.

  1. (1)

The height function $h_{M_{1},M_{2}}$ of double dimer configurations is a
function from faces of $\overline{G}$ to the set of all integers, such that
for any pair of faces $f_{1},f_{2}$ sharing an edge $e$,

    1. (a)

If $e\notin M_{1}\triangle M_{2}$,
$h_{M_{1},M_{2}}(f_{1})=h_{M_{1},M_{2}}(f_{2})$;

    2. (b)

If $e\in M_{1}\triangle M_{2}$,

      1. (i)

If $e\in M_{1}\setminus M_{2}$, and moving from $f_{1}$ to $f_{2}$, the white
vertex is on the left, then $h_{M_{1},M_{2}}(f_{1})-h_{M_{1},M_{2}}(f_{2})=1$;

      2. (ii)

If $e\in M_{2}\setminus M_{1}$, and moving from $f_{1}$ to $f_{2}$, the white
vertex is on the left, then
$h_{M_{1},M_{2}}(f_{1})-h_{M_{1},M_{2}}(f_{2})=-1$.

  2. (2)

We define a cluster in $M_{1}\triangle M_{2}$ to be a maximal connected
subgraph of $\overline{G}^{+}$ in which every vertex has the same height. A
finite (resp. infinite) cluster is a cluster consisting of finitely many
(resp. infinitely many) vertices.

Let $k\in{\mathbb{Z}}$. A level-$k$ cluster in $M_{1}\triangle M_{2}$ is a
cluster in which every vertex of $\overline{G}^{+}$ in the cluster has height
$k$.

One can see that the height function defined in Definition 7.1 satisfies

$\displaystyle h_{M_{1},M_{2}}=h_{M_{2}}-h_{M_{1}}$

where $h_{M}$ is defined as in Definition 6.1.

Recall the following definition:

######  Definition 7.2.

We say a graph $G$ satisfies an isoperimetric inequality of dimension $d$ if
there exists some constant $c>0$ such that

(7.1) $\displaystyle|\partial K|\geq c|K|^{\frac{d-1}{d}}\ \forall\
\mathrm{finite}\ K\subset V.$

where $\partial K:=\\{(x,y)\in E:x\in K,y\notin K\\}$ is the edge boundary of
$K$. The isoperimetric dimension of $G$ which we denote by $\mathrm{Dim}(G)$,
is defined as the supreme over all $d$ such that such that (7.1) holds.

###  7.1. General Boundary Conditions

######  Theorem 7.3.

Let $G,G^{+},\overline{G}$ be given as in Proposition 6.5. Let $\mu$ be the
probability measures for dimer coverings on $\overline{G}$ obtained as in
Theorem 5.6, or Proposition 6.18, or Proposition 6.20. Let $M_{1}$, $M_{2}$ be
two independent dimer configurations on $\overline{G}$ with distribution
$\mu$. Let $\mathcal{N}$ be the number of doubly-infinite self-avoiding paths
in $M_{1}\triangle M_{2}$. Then

(7.2) $\displaystyle\mu\times\mu(\mathcal{N}=0)=1.$

###### Proof.

For $k\in{\mathbb{Z}}$, let $\mathcal{H}_{k}$ be the event that there exists
an infinite level-$k$ cluster in $M_{1}\triangle M_{2}$. Since
$\mathcal{H}_{k}$ is measurable with respct to the tail-$\sigma$ field and
$\mu$ is tail-trivial by Lemma 6.5,

$\displaystyle\mu\times\mu(\mathcal{H}_{k})\in\\{0,1\\}.$

Let $m\geq 1$. Assume

(7.3) $\displaystyle\mu\times\mu(\mathcal{H}_{m})=1.$

Then interchanging the configuration $M_{1}$ and $M_{2}$ we obtain
$\mu\times\mu(\mathcal{H}_{-m})=1$. We infer that
$\mu\times\mu(\mathcal{H}_{m}\cap\mathcal{H}_{-m})=1$. Since when $m\geq 1$,
$\mathcal{H}_{m}$ and $\mathcal{H}_{-m}$ are distinct, each infinite level-$m$
cluster has a doubly infinite self-avoiding path in $M_{1}\triangle M_{2}$ as
its boundary. For any $r\geq m$, if we interchange the configurations of
$M_{1}$ and $M_{2}$ along all the doubly-infinite self-avoiding paths which
are boundaries of some infinite level-$r$ cluster for all $r\geq m$, we obtain
a configuration with no infinite level-$m$ cluster. Then
$\mu\times\mu(\mathcal{H}_{m})=0$; which contradicts (7.3). The contradiction
implies

(7.4) $\displaystyle\mu\times\mu(\mathcal{H}_{m})=0.\qquad\forall m\neq 0$

We infer from (7.4) that all the infinite clusters have level 0 a.s. However
if there exists a doubly infinite self-avoiding path in $M_{1}\triangle
M_{2}$, on the two sides of the path there are two infinite clusters of
different levels. The contradiction implies (7.2). ∎

######  Lemma 7.4.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumptions 2.4, 5.5
holds. Suppose Assumption 3.15 holds with $\mathcal{A}$ replaced by
$\mathcal{A}_{0}$. Assume the edge weights satisfy (6.2).

Let $\\{G_{j}\\}_{j=1}^{\infty}$ be given as in Assumption 3.15.

  1. (1)

Assume

(7.5) $\displaystyle\Phi(u;G_{j})>cu^{-\frac{1}{d}}$

for some $d>2$, where $\Phi(u;G_{j})$ is defined as in (4.18) for the graph
$G_{j}$, and $c$ is a constant independent of $j$.

Then for any $b\in V$, and any white vertex $w$ of $\overline{G}$, we have

$\displaystyle\left|\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w)\right|\leq
C_{1}[d_{G}(u,v)]^{-\frac{d-2}{2}}$

where $C_{1}=C_{1}(c,d)$ is a positive constant.

  2. (2)

Assume (7.5) holds for some $d>4$ with a constant $c>0$ independent of $j$.
Then
$\left|\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w)\right|$
is uniformly bounded for all $b\in V^{+}$ and white vertices $w$ of
$\overline{G}$.

###### Proof.

Let $G_{G_{j}}(\cdot,\cdot)$ be the discrete Green’s function on the infinite
graph $G$. If $b\in V$, we see from Theorem 5.6(1) that

$\displaystyle\left|\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w)\right|=\left|\lim_{j\rightarrow\infty}\frac{G_{G_{j}}(b,b_{3})}{\Delta(b_{3},b_{3})}-\frac{G_{G_{j}}(b,b_{3})}{\Delta(b_{3},b_{3})}\right|$

Then part (1) the lemma follows from Lemma 4.26.

Now we prove part (2) of the lemma. Let $b\in V^{+}$. Let $u,v\in V^{+}$ be
adjacent vertices in $G^{+}$ such that the white vertex $w$ corresponds to the
edge $(u,v)\in E^{+}$. Let $l_{v,b}$ be a shortest path in $G^{+}$ joining $v$
and $b$. More precisely, let

$\displaystyle l_{v,b}:=v_{0}(=v),v_{1},\ldots,v_{n}(=b)$

Then we have

(7.6)
$\displaystyle\overline{D}_{\tilde{G}_{j,s}}^{-1}(b,w)=\sum_{k=1}^{n}\left[\overline{D}_{\tilde{G}_{j,s}}^{-1}(v_{k},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(u_{k-1},w)\right]+\overline{D}_{\tilde{G}_{j,s}}^{-1}(v,w)$

For $1\leq k\leq n$, let $x_{k},y_{k}$ be the dual edge of $(v_{k-1},v_{k})$.
Assume $x_{k},v_{k-1},y_{k},v_{k}$ are adjacent to a white vertex of
$\overline{G}$ in clockwise order. When edge weights satisfy (6.2) we obtain

(7.7)
$\displaystyle[\overline{D}_{\tilde{G}_{j,s}}^{-1}(v_{k},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(v_{k-1},w)]=-\mathbf{i}[\overline{D}_{\tilde{G}_{j,s}}^{-1}(y_{k},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(x_{k},w)]$

Moreover,

(7.8)
$\displaystyle|\overline{D}_{\tilde{G}_{j,s}}^{-1}(v,w)|=\mathbb{P}((v,w)\in
M(\tilde{G}_{j,s}))\in[0,1]$

where $\mathbb{P}((v,w)\in M(\tilde{G}_{j,s}))$ is the probability that the
edge $(v,w)$ appear in a perfect matching of $\tilde{G}_{j,s}$. Then the
uniform boundedness of $\overline{D}_{\tilde{G}_{j,s}}(b,w)$ when $b\in V$
follows from (7.6), (7.7), (7.8) and part (1) of the lemma. ∎

######  Theorem 7.5.

Let $G=(V,E)$ be a 3-connected, transient, simple proper planar graph with
bounded vertex degree and locally finite dual. Suppose Assumption 2.4 holds.

Let $\mathbb{P}$ be the probability measures for dimer coverings on
$\overline{G}$ obtained as in Theorem 5.6. Assume the edge weights satisfy
(6.2).

Suppose

  * •

Assumption 3.15 holds with $\mathcal{A}$ replaced by $\mathcal{A}_{0}$. Let
$\\{G_{j}\\}_{j=1}^{\infty}$ be given as in Assumption 3.15. Assume

$\displaystyle\Phi(u;G_{j})>c_{1}u^{-\frac{1}{d}}$

for some $d>8$, and $c_{1}$ is a constant independent of $j$.

Let $M_{1}$, $M_{2}$ be two independent dimer configurations, both have
distribution $\mathbb{P}$. Let $f_{0}$ be an arbitrary a fixed face of
$\overline{G}$.

Then all the following conclusions hold.

  1. (1)

For any face $f$, $\mathrm{Var}[h_{M_{1},M_{2}}(f)]^{2}<\infty$.

  2. (2)

${\mathbb{P}}\times{\mathbb{P}}$ a.s., each face of $\overline{G}$ is enclosed
by finitely many self-avoiding cycles in $M_{1}\triangle M_{2}$.

###### Proof.

We first prove Part (1) of the lemma. Let $\tilde{G}_{j,s}$ be a finite
subgraph of $G$ obtained as in Assumption 5.5. Let $M_{1}$ and $M_{2}$ be two
i.i.d. dimer configurations on $\tilde{G}_{j,s}$. Without loss of generality,
assume that for any face $g$ of $\overline{G}$ not in $\tilde{G}_{j,s}$,
$h_{M_{1},M_{2}}(g)=0$.

Let $g_{1}$ be a face of $\overline{G}$ not in $\tilde{G}_{j,s}$ but are
incident to an edge in $\tilde{G}_{j,s}$. Assume $g_{0}$ is a face of
$\overline{G}$ not in the superposition of $G_{j,s}$ and $\hat{G}_{j,s}^{+}$
but is incident to an edge in $\hat{C}_{1,j,s}$. Let $l_{fg_{0}}$ (resp.
$l_{fg_{1}}$) be a path in $\overline{G}^{+}$ joining $f$ and $g_{0}$ (resp.
$f$ and $g_{1}$) of $\overline{G}$, such that all the faces along $l_{fg_{0}}$
(resp. $l_{fg_{1}}$) between $f,g_{0}$ (resp. $f$ and $g_{1}$) are faces of
$\tilde{G}_{j,s}$. Assume

  * •

$l_{fg_{0}}\cup l_{fg_{1}}$ is a geodesic in $\overline{G}^{+}$ joining
$g_{0}$ and $g_{1}$; and

  * •

$\displaystyle l_{f,g_{0}}$
$\displaystyle=s_{0}=(f),s_{1},\ldots,s_{m}(=g_{0});$ $\displaystyle
l_{f,g_{1}}$ $\displaystyle=q_{0}=(f),q_{1},\ldots,q_{k}(=g_{1}).$

Let

$\displaystyle\alpha_{i}$ $\displaystyle=(s_{i-1}s_{i})^{+};\qquad\forall
1\leq i\leq m$ $\displaystyle\beta_{t}$
$\displaystyle=(q_{t-1},q_{t})^{+};\qquad\forall 1\leq t\leq k$

such that $\\{\alpha_{i}\\}_{i=1}^{m-1}$, $\\{\beta_{t}\\}_{t=1}^{k}$ are
edges of $\tilde{G}_{j,s}$, and $\alpha_{m}$ is an edge incident to a vertex
along $\hat{C}_{1,j,s}$.

  * •

Let $L_{0}$ (resp. $L_{1}$) be the path in $G^{+}$ (resp. $G$) consisting of
all the vertices of $G^{+}$ (resp. $G$) in a face of $\overline{G}$
corresponding to a vertex of $\overline{G}^{+}$ along $l_{f,g_{0}}$ (resp.
$l_{f,g_{1}}$). Assume

$\displaystyle L_{0}:$ $\displaystyle=b_{0,0}(\in
f),b_{0,1},\ldots,b_{0,m_{0}}(\in g_{0})$ $\displaystyle L_{1}:$
$\displaystyle=b_{1,0}(\in f),b_{1,1},\ldots,b_{1,k_{1}}(\in g_{1})$

where $b_{0,m_{0}}$ is a vertex along $\hat{C}_{1,j,s}$.

Let

$\displaystyle E_{j}:=\\{\alpha_{1},\ldots,\alpha_{m}\\};\qquad
F_{j}:=\\{\beta_{1},\ldots,\beta_{k}\\};$

We shall write

  * •

$E_{j}:=E_{j,1}\cup E_{j,2}$, and $E_{j,1}\cap E_{j,2}=\emptyset$, where for
$1\leq i\leq m$, $\alpha_{i}\in E_{j,1}$ if and only if moving along
$l_{f,g_{1}}$ from $f$ to $g_{1}$, when crossing $\alpha_{i}$ the white vertex
of $\alpha_{i}$ is on the left.

  * •

$F_{j}:=F_{j,1}\cup F_{j,2}$, and $F_{j,1}\cap F_{j,2}=\emptyset$, where for
$1\leq t\leq k$, $\beta_{t}\in F_{j,1}$ if and only if moving along
$l_{f,g_{2}}$ from $f$ to $g_{2}$, when crossing $\beta_{t}$ the white vertex
of $\beta_{t}$ is on the left.

Then

$\displaystyle\mathrm{Var}[h_{M_{1},M_{2}}(f)]^{2}={\mathbb{E}}[h_{M_{1},M_{2}}(f)-h_{M_{1},M_{2}}(g_{1})][h_{M_{1},M_{2}}(f)-h_{M_{1},M_{2}}(g_{2})]$
$\displaystyle=\mathbb{E}\left[\sum_{\alpha\in E_{j,1}}[\mathbf{1}_{\alpha\in
M_{1}}-\mathbf{1}_{\alpha\in M_{2}}]-\sum_{\alpha\in
E_{j,2}}[\mathbf{1}_{\alpha\in M_{1}}-\mathbf{1}_{\alpha\in
M_{2}}]\right]\left[\sum_{\beta\in F_{j,1}}[\mathbf{1}_{\beta\in
M_{1}}-\mathbf{1}_{\beta\in M_{2}}]-\sum_{\beta\in
F_{j,2}}[\mathbf{1}_{\beta\in M_{1}}-\mathbf{1}_{\beta\in M_{2}}]\right]$
$\displaystyle=2\sum_{\alpha\in E_{j,1}}\sum_{\beta\in
F_{j,1}}\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in M_{1}})-2\sum_{\alpha\in
E_{j,1}}\sum_{\beta\in F_{j,2}}\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in M_{1}})$ $\displaystyle+2\sum_{\alpha\in
E_{j,2}}\sum_{\beta\in F_{j,2}}\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in M_{1}})-2\sum_{\alpha\in
E_{j,2}}\sum_{\beta\in F_{j,1}}\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in M_{1}})$

where we used the fact that $M_{1}$ and $M_{2}$ are i.i.d. Moreover, note that
if $\alpha$ and $\beta$ are two distinct edges sharing a vertex,

$\displaystyle\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in M_{1}})=-\mathbb{P}(\beta\in
M_{1})\mathbb{P}(\alpha\in M_{1}).$

If $\alpha$ and $\beta$ are two distinct edges sharing no vertices; assume
$\alpha=(w_{1},b_{1})$ and $\beta=(w_{2},b_{2})$; then by Corollary 4.3(2) we
have

$\displaystyle\left|\mathbb{E}(\mathbf{1}_{\alpha\in
M_{1}}-\mathbb{E}\mathbf{1}_{\alpha\in M_{1}})(\mathbf{1}_{\beta\in
M_{1}}-\mathbb{E}\mathbf{1}_{\beta\in
M_{1}})\right|=\sqrt{\frac{\nu(\alpha)\nu(\beta)}{\nu(\alpha^{+})\nu(\beta^{+})}}\lim_{j\rightarrow\infty}\left|\overline{D}_{\tilde{G}_{j,s}}^{-1}(w_{1},b_{2})\overline{D}_{\tilde{G}_{j,s}}^{-1}(w_{2},b_{1})\right|$

By Lemma 7.4,
$|\lim_{j\rightarrow\infty}\lim_{s\rightarrow\infty}\overline{D}^{-1}_{\tilde{G}_{j,s}}(b_{2},w_{1})|$
is uniformly bounded for all $b_{2},w_{1}$. Consider
$\left|\overline{D}_{\tilde{G}_{j,s}}^{-1}(w_{2},b_{1})\right|$. The following
cases might occur:

  1. (1)

$b_{1}$ is a vertex of $G$. By Lemma 4.22(1) and the decay rate of the
Dirichlet Green’s function $G_{G_{j}}$ given by Lemma 4.26, we have

$\displaystyle|\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{1},w)|\leq\left|\frac{G_{G_{j}}(b_{1},v_{1})}{\Delta(b_{1},v_{1})}\right|+\left|\frac{G_{G_{j}}(b_{1},v_{2})}{\Delta(b_{1},v_{2})}\right|\leq
C^{\prime\prime}\left[[d_{G}(b_{1},v_{1})]^{-\frac{d-2}{2}}+[d_{G}(b_{1},v_{2})]^{-\frac{d-2}{2}}\right]$

where $(v_{1},v_{2})$ is the edge of $G$ corresponding to $w$.

  2. (2)

$b_{1}$ is a vertex of $G^{+}$. Assume $b_{1}=b_{0,i}$. In this case we have

$\displaystyle\lim_{j\rightarrow\infty}\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{1},w)=\sum_{c=i+1}^{m_{0}}\left[\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0,c-1},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0,c},w)\right]+\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0},w)$

Recall that $\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0},w)=0$. For $1\leq k\leq
a_{0}$, let $x_{k},y_{k}$ be the dual edge of $(b_{3,k-1},b_{3,k})$. For
$m_{0}+1\leq k\leq m_{0}+i$, let $x_{k},y_{k}$ be the dual edge of
$(b_{1,k-m_{0}},b_{1,k-m_{0}-1})$.

Assume for $1\leq k\leq m_{0}$, $x_{k},b_{0,k-1},y_{k},b_{0,k}$ are adjacent
to a white vertex of $\overline{G}$ in clockwise order. Then we obtain

(7.9)
$\displaystyle\nu(b_{0,k}b_{0,k-1})[\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0,k},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(b_{0,k-1},w)]=-\mathbf{i}\nu(x_{k}y_{k})[\overline{D}_{\tilde{G}_{j,s}}^{-1}(y_{k},w)-\overline{D}_{\tilde{G}_{j,s}}^{-1}(x_{k},w)]$

Given that $l_{f,g_{0}}\cup l_{f,g_{1}}$ is a geodesic joining $g_{0}$ and
$g_{1}$, and $d>8$, we infer that ${\mathbb{E}}[h_{M_{1},M_{2}}(f)]^{2}$ is
uniformly bounded for all $j,s$; then Part (1) follows.

Now we prove Part (2) by contradiction. Assume that with strictly positive
probability, a face $f$ is enclosed by infinitely many self-avoiding cycles in
$M_{1}\triangle M_{2}$. Then there exists $c_{0}>0$, such that for any $N$,

$\displaystyle\lim_{j\rightarrow\infty}\mathbb{P}(f\ \mathrm{is\ enclosed\ by\
at\ least}\ N\ \mathrm{self-avoiding\ cycles\ in}\ M_{1}\triangle M_{2}\
\mathrm{of}\ \mathcal{G}_{j})>c_{0}.$

Note that crossing each self-avoiding cycle enclosing $f_{0}$ in
$M_{1}\triangle M_{2}$, with probability $\frac{1}{2}$, the $h_{M_{1},M_{2}}$
increases by 1, and with probability $\frac{1}{2}$, $h_{M_{1},M_{2}}$
decreases by 1. Then $f$ is enclosed by at least $N$ self-avoiding cycles in
$M_{1}\triangle M_{2}$, and the boundary heights are 0,
$h_{M_{1},M_{2}}(f_{n})$ is the sum of at least $N$ i.i.d. random variables
$\xi_{1},\ldots,\xi_{N},\ldots$, each of which takes value $1$ with
probability $\frac{1}{2}$ and takes value $0$ with probability $\frac{1}{2}$.

Let $\mathcal{S}_{N,j}$ be the event that $f$ is enclosed by at least $N$
self-avoiding cycles in $M_{1}\triangle M_{2}$ of $\mathcal{G}_{j}$, then we
have

$\displaystyle\mathbb{E}[h_{M_{1},M_{2}}(f)]^{2}\geq\mathbb{E}[(\xi_{1}+\ldots+\xi_{N})^{2}|\mathcal{S}_{N,j}]\mathbb{P}(\mathcal{S}_{N,j})=\sum_{j=1}^{N}\mathbb{E}(\xi_{j})^{2}\mathbb{P}(\mathcal{S}_{N,j})=c_{0}N$

Since $N$ is arbitrary we obtain

$\displaystyle\lim_{j\rightarrow\infty}\mathbb{E}[h_{M_{1},M_{2}}(f)]^{2}=\infty;$

which contradicts Part (1). The contradiction implies Part (2). ∎

######  Example 7.6.

We now give an example in which (7.5) holds with a universal constant $c$ for
all the $G_{j}$’s. Consider for example, a nonamenable, vertex-transitive
regular tiling of the hyperbolic plane with vertex degree $D$ and face degree
$D^{+}$. By [26],

$\displaystyle\inf\left\\{\frac{|\partial_{E}K|}{|K|}:K\subset V\
\mathrm{finite\ and\
nonempty}\right\\}=(D-2)\sqrt{1-\frac{4}{(D-2)(D^{+}-2)}};=i_{E}(G)$

Let $v_{0}$ be a fixed vertex of $G$. Let $G_{j,s}$ be the finite subgraph of
$G$ constructed as follows

  * •

Let $F_{j,s}$ be the union of all the faces containing at least one vertex in
$G_{j,s}$. Let $\partial F_{j,s}$ be all the edges in $F_{j,s}$ incident to at
least one face in $F_{j,s}$ and one face not in $F_{j,s}$ as well as their
endvertices. Assume $\partial F_{j,s}$ is a simple closed curve.

the boundary $\partial^{V}G_{j,s}$ of $G_{j,s}$, defined to be the union of
vertices in $G_{j,s}$ incident to at least a vertex not in $G_{j,s}$, as well
as edges in $G_{*}$ (the matching graph) joining two such vertices sharing a
face. There are 4 vertices $z_{1},z_{2},z_{3},z_{4}$ in $\partial
F_{j,s}\cap\partial^{V}G_{j,s}$ in clockwise order

    * –

$\partial F_{j,s}$ is divided by $u_{1},u_{2},u_{3},u_{4}$ into 4 segments
$L_{u_{1}u_{2}}$, $L_{u_{2}u_{3}}$, $L_{u_{3}u_{4}}$, and $L_{u_{4},u_{1}}$;

    * –

$d_{G}(u_{1},v_{0})=d_{G}(u_{2},v_{0})=j$;

    * –

$L_{u_{1}u_{2}}$ is a geodesic joining $u_{1}$ and $u_{2}$;

    * –

all the vertices in $L_{u_{3}u_{4}}\cap\partial^{V}G_{j,s}$ have graph
distance exactly $s$ from $v_{0}$;

    * –

$L_{u_{3}u_{3}}$ is a geodesic in $G$ joining $u_{1}$ and $u_{2}$;

    * –

$L_{u_{4}u_{1}}$ is a geodesic in $G$ joining $u_{4}$ and $u_{1}$;

    * –

Let $Q_{j}$ be the union of all the faces containing at least one vertex with
distance at most $j$ to $v_{0}$. Let $\partial Q_{j}$ be the union of all the
edges incident to at least one face in $Q_{j}$ and one face not in $Q_{j}$.
Let $R_{z_{1}z_{2}}$ be the portion of $\partial Q_{j}$ from $z_{1}$ to
$z_{2}$ in clockwise order.

Let $N_{j}$ be the number of vertices in $R_{z_{1}z_{2}}$, then

$\displaystyle
1<\limsup_{j\rightarrow\infty}N_{j}^{\frac{1}{j}}<1+i_{E}(G);\qquad\forall j$

where $\alpha>0$ is a constant independent of $j$

As in Assumption 5.5, let $\hat{G}_{j,s}^{+}$ be the subgraph of $G^{+}$ that
contains $G_{j,s}$ as an interior dual graph. Let $v_{1,j,s}=z_{1}$ and
$v_{2,j,s}=z_{2}$, $C_{1,j,s}=L_{z_{1}z_{2}}$. Let $G_{j}$ in Assumption 3.15
be obtained from $G_{j,s}$ by letting $s\rightarrow\infty$. Then by Theorem
5.6, the limit measure obtained by such an approximation with
$j,s\rightarrow\infty$ is different from the one obtained with Temperley
boundary conditions as in Theorem 4.22. However, one can check that for any
$d>8$, (7.5) holds with a universal constant $c$ independent of $j$. By
Theorem 7.5, the variance of double dimer height function is finite for any
face.

Acknowledgements. ZL acknowledges support from National Science Foundation DMS
1608896 and Simons Foundation grant 638143. ZL thanks Gourab Ray for
discussions.

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