$L_{k}$ which consist of unions of adjacent planes $L_{k}:=C_{2k}\cup
C_{2k+1}$, $k\in\mathbb{Z}$ (see Section 4). By Proposition 4.2.1, each slab
is a copy of the 2-dimensional hexagonal lattice. There are three 3D dimer
tilings of $C_{2k}\cup C_{2k+1}$ consisting of only one type of dimer, and
these correspond to the three brickwork lozenge tilings using one type of
lozenge. (See Figure 19 for a review of the correspondence between 3D dimers
and lozenges.)

Restricted to each slab $L_{k}$, $\tau_{v}$ will be one of the three brickwork
lozenge tilings. On $L_{0}=C_{0}\cup C_{1}$, $\tau_{v}$ will be the ${a}_{1}$
brickwork lozenge tiling. On $L_{1}=C_{2}\cup C_{3}$, $\tau_{v}$ will be the
${a}_{2}$ brickwork lozenge tiling. We continue this by repeating the periodic
sequence $a_{1},...,a_{r}$ forwards and backwards in $k$ to choose the tile
type for $\tau_{v}$ on all other slabs $L_{k}=C_{2k}\cup C_{2k+1}$.

The reference tiling $\tau_{1}$ consists of all $-\eta_{1}$ tiles, which
connect $C_{2k}$ to $C_{2k-1}$. Subtracting $\tau_{1}$, the tiles in
$-\tau_{1}$ connect $C_{2k-1}$ to $C_{2k}$, meaning that they connected the
“odd” half of $L_{k-1}$ to the “even” half of $L_{k}$. Hence in the double
dimer tiling $(\tau_{v},\tau_{1})$, every tile is on an infinite path. Along
each infinite path, $(\tau_{v},\tau_{1})$ consists of the periodic sequence of
tiles parallel to $\ldots
a_{1},\eta_{1},a_{2},\eta_{1},\dots,\eta_{1},a_{r},\ldots$. In particular, all
infinite paths are parallel to ${v}+{\eta}_{1}$. This completes the
construction for $v\in\partial\mathcal{O}\cap\mathbb{Q}^{3}$, $v_{1}\geq 0$.

Figure 35: Above is an example of $v=(v_{1},v_{2},v_{3})$ and its relationship
to $w(v)=w$.

Now we extend the construction to any $v\in\mathcal{O}\cap\mathbb{Q}^{3}$,
$v\neq-\eta_{1}$. Let $p_{v}$ be the line through $-\eta_{1}=(-1,0,0)$ and
$v=(v_{1},v_{2},v_{3})$. Let $w=w(v)$ be the intersection of $p_{v}$ with
$\\{u=(u_{1},u_{2},u_{3})\in\partial\mathcal{O}:u_{1}\geq 0\\}$. See Figure
35. The relationship between $v,w$ will be sufficiently important that we
record it as a definition.

######  Definition 8.4.3.

Fix $v\in\mathcal{O}$, $v\neq-\eta_{1}$, and let $p_{v}$ be the line through
$-\eta_{1}$ and $v$. We define $w(v)$ to be the intersection of $p_{v}$ with
the part of $\partial\mathcal{O}$ with non-negative first coordinate.

Note that if $v$ is rational, $w=w(v)\in\partial\mathcal{O}$ is rational.
Since the first coordinate of $w$ is non-negative, we can construct $\tau_{w}$
as described above. In $(\tau_{w},\tau_{1})$, every tile is along an infinite
path. On the other hand, in $(\tau_{1},\tau_{1})$ none of the tiles are along
infinite paths. To construct $(\tau_{v},\tau_{1})$, we interpolate between
these two options by choosing an intermediate density of infinite paths.

If the line $p_{v}(t)$ is parameterized so that $p_{v}(1)=(-1,0,0)$ and
$p_{v}(0)=w$, let $a\in[0,1]$ be such that $p_{v}(a)=v$. Since $v$ rational,
$w$ and $a$ are also rational. Thus there exist periodic patterns of cubes in
$C_{0}$ with density $a$. To construct $(\tau_{v},\tau_{1})$, we fix a choice
of periodic pattern of cubes on $C_{0}$ with density $a$. We delete all the
infinite paths in $(\tau_{w},\tau_{1})$ which do not go through one of the
chosen cubes on $C_{0}$ and replace them with tiles parallel to $-\eta_{1}$.
The resulting tiling is $\tau_{v}$.

The tilings $\tau_{v}$ have a few important properties which we highlight.

######  Lemma 8.4.4.

Let $v\in\mathcal{O};v\neq-\eta_{1}$.

  1. 1.

$\tau_{v}$ has mean current $v$;

  2. 2.

Let $w=w(v)$ be as in Definition 8.4.3. The infinite paths in $\tau_{v}$ are
parallel to $w+\eta_{1}$.

  3. 3.

For any rational plane $P$, the restriction of $\tau_{v}$ to $P$ is doubly
periodic, with period depending on $r$ (the number of tiles $a_{1},...,a_{r}$
used to approximate $w(v)$), the choice of periodic pattern of cubes in
$C_{0}$ and $P$.

######  Remark 8.4.5.

Note that $\tau_{v}$ is not uniquely determined by $v$. It depends on the
sequence of tiles $a_{1},...,a_{r}$ used to approximate ${v}$, and on and
periodic pattern of initial sites on $C_{0}$.

We now show that pieces of $\tau_{v},\tau_{u}$ can “glued” along a plane $P$,
as long as $v,u$ have the same flow through $P$. The amount of space $k$ we
need to glue is a constant depending only on the period of the tilings
$\tau_{u}$ and $\tau_{v}$.

######  Lemma 8.4.6.

Suppose that $u,v\in\mathcal{O}\cap\mathbb{Q}^{3};u,v\neq\eta_{1}$ and
$\tau_{u},\tau_{v}$ are tilings as in Lemma 8.4.4. Suppose that $P$ is a
coordinate plane or plane with normal vector of the form $(\pm 1,\pm 1,\pm
1)$. In both cases we denote the normal vector by $n_{P}$. Let $r$ be such
that $\tau_{u}$ and $\tau_{v}$ are periodic in $P$ with fundamental domain an
$r\times r$ parallelogram. If $v\cdot n_{P}=u\cdot n_{P}$, then there is an
even integer $k>0$ (depending on $r$ and $P$) such that $\tau_{v}$ restricted
to the left half-space of $P$ can be connected to $\tau_{u}$ restricted to the
right half-space of $P+k$ for some $k=O(r)$. Further, the connecting tiling
$\tau$ is also periodic in $P$ with fundamental domain an $r\times r$
parallelogram.

######  Remark 8.4.7.

We restrict to these two types of planes $P$ since the the faces of tetrahedra
in the mesh used to define the piecewise constant approximation (Proposition
8.3.1) are contained in one of these two types of planes; see Remark 8.3.3.
The analogous result for other planes also holds, but we do not need it.

###### Proof.

By Lemma 8.4.4, $\tau_{u},\tau_{v}$ are periodic, so there exists $r>0$ finite
and determined by $\tau_{u},\tau_{v},P$ such that $\tau_{u}$ is periodic on
$P$ with fundamental domain $R_{0}\subset P$, where $R_{0}$ an $r\times r$
parallelogram contained in $P$, and similarly $\tau_{v}$ is periodic on $P+k$
with fundamental domain also an $r\times r$ parallelogram in $P+k$. Let $R$ be
the parallelopiped region parallel to $n_{P}$ between one fundamental domain
$R_{0}\subset P$ and another $R_{k}\subset P+k$.

Let $R/\sim$ be $R$ with opposite faces other than $R_{0}$ and $R_{k}$ paired
(i.e., $R/\sim$ is a 2-dimensional torus crossed with an interval). Given the
periodicity of $\tau_{u},\tau_{v}$, to that show the region between $P$ and
$P+k$ is tileable with $\tau_{u}\mid_{P}$ and $\tau_{v}\mid_{P+k}$, it
suffices to show that $R/\sim$ is tileable with $\tau_{u}\mid_{R_{0}}$ and
$\tau_{v}\mid_{R_{k}}$.

To show that this region is tileable we use the same techniques as in Section
6. In other words, first we show that $R/\sim$ with $\tau_{u}\mid_{R_{0}}$ and
$\tau_{v}\mid_{R_{k}}$ is balanced, and then use Hall’s matching theorem
(6.1.3). The setting here is more elementary than what we consider in Section
6, since here the tilings defining the boundary conditions are completely
periodic.

Since $k$ is even, any perpendicular slice of $R/\sim$ is a fundamental domain
for $\tau_{u}$ or $\tau_{v}$, the condition $v\cdot n_{P}=u\cdot n_{P}$ is
equivalent to

$\sum_{e\text{ intersecting }R_{0}}v_{\tau_{u}}(e)\cdot n_{P}=\sum_{e\text{
intersecting }R_{k}}v_{\tau_{v}}(e)\cdot n_{P}.$ (63)

Equation (63) is in turn equivalent to $R/\sim$ with boundary conditions
$\tau_{u}\mid_{R_{0}}$ and $\tau_{v}\mid_{R_{k}}$ being balanced.

Since the region is balanced, by Hall’s matching theorem (Theorem 6.1.3) it is
not tileable if and only if there exists a counterexample region $U$ which is
a strict subset. Since $U$ is a strict subset, $U$ has a nonempty interior
boundary $S\subset\partial U$. Let $T=\partial U\setminus S$. By Proposition
6.2.3,

$\displaystyle\text{imbalance}(U)=\frac{1}{6}\bigg{(}\text{white}(T)-\text{black}(T)-\text{area}(S)\bigg{)}.$

Since $\text{area}(T)\leq 2r^{2}$, $\text{white}(T)\leq 2r^{2}$. Since the
region is tileable with boundary condition from just one of the tilings, $S$
must connect $R_{0}$ and $R_{k}$, if $k>r$, by Proposition 6.5.1,there is a
universal constant $\kappa$ such that $\text{area}(S)\geq\kappa kr$. Therefore
by Proposition 6.2.3,

$\displaystyle\text{imbalance}(U)\leq\frac{2r^{2}-\kappa kr}{6}.$

Choosing $k=cr$ for some constant $c>2/\kappa$, $U$ is not a counterexample
which contradicts the assumption that the region is not tileable. This
completes the proof. ∎

Using the tilings $\tau_{v}$ as our building blocks and their gluing
properties to put them together, we now proceed to prove Theorem 8.4.1.

###### Proof of Theorem 8.4.1.

Choose a scale $\epsilon>0$ so that the piecewise-constant approximation
$\widetilde{g}$ from Proposition 8.3.1 on an $\epsilon$-scale tetrahedral mesh
$\mathcal{X}=\\{X_{1},...,X_{M}\\}$ satisfies
$d_{W}(g,\widetilde{g})<\delta/2$ and hence
$A_{\delta/2}(\widetilde{g})\subset A_{\delta}(g)$. We assume that all
$X\in\mathcal{X}$ are regular or right-angled tetrahedra so that all their
faces are contained in coordinate planes or planes with normal vector $(\pm
1,\pm 1,\pm 1)$. Recall that $\widetilde{g}\in AF(\widetilde{R})$ and that
$R\subset\widetilde{R}$, so any free boundary tiling of $\widetilde{R}$ can be
restricted to a free boundary tiling of $R$. To prove the theorem, it suffices
to construct $\tau\in T_{n}(\widetilde{R})$ with
$d_{W}(f_{\tau},\widetilde{g})<\delta/2$.

Constructing channels. We construct channels $C_{1},...,C_{K}$ which are
disjoint, partition $\widetilde{R}$ and will be nicely chosen tubular
neighborhoods of a modification of the flow lines of $\widetilde{g}+\eta_{1}$.
For any $X_{j}\in\mathcal{X}$, let $v_{j}:=\widetilde{g}\mid_{X_{j}}$. Recall
Definition 8.4.3, which relates a vector $v$ with $w(v)$, which is the
direction of the infinite paths in a periodic tiling $\tau_{v}$. For each
channel $C_{i}$, the intersection $C_{i}\cap X_{j}$ will be a tube parallel to

$w(v_{j})+\eta_{1}.$

Since $\widetilde{g}$ is valued strictly in $\text{Int}(\mathcal{O})$,
$v_{j}\neq-\eta_{1}$ for all $X_{j}\in\mathcal{X}$, and hence $w(v_{j})$ is
well-defined everywhere. As shorthand, we let $w(\widetilde{g})$ be the
piecewise-constant flow equal to $w(v_{j})$ on $X_{j}$. The definitions are
made so that if $\tau_{v_{j}}$ is a periodic tiling built by the construction
earlier in this section, the infinite paths in $(\tau_{v_{j}},\tau_{1})$ move
parallel to the direction of the channel on $X_{j}$. The values of
$\widetilde{g}$ change on the boundaries $\partial X_{j}$ of tetrahedra in the
mesh. We choose the channels $C_{i}$ to be thin enough as follows so that,
viewing $C_{i}$ as a sequence of open tubes, each end of the tube $C_{i}\cap
X_{j}$ is contained in a single plane (i.e., each end is contained in a single
face of $\partial X_{j}$).

Project the corners and edges of $X_{j}$ onto $\partial X_{j}$ along
$w(v_{j})+\eta_{1}$. Call these projections $\gamma_{j}$. The points
$\gamma_{j}\subset\partial X_{j}$ are the ones along a flow line of
$w(\widetilde{g})+\eta_{1}$ that goes through an edge of $X_{j}$. The lines
$\gamma_{j}$ divide the faces of $X_{j}$ into between $1$ and $3$ sections.

We further divide $X_{j}$ by taking into account the iterated projections of
the corners and edges of all the other tetrahedra in the mesh. In other words
for all $j$, if $X_{k}$ is a neighbor of $X_{j}$, then we project $\gamma_{j}$
onto $\partial X_{k}$ by orthogonal projection along $w(v_{k})+\eta_{1}$. We
iterate this for all tetrahedra until there is a projection of $\gamma_{j}$ on
$\partial X_{k}$ for all $\\{k,j\\}$ pairs. See the left figure in Figure 36.

The result is that for each $j\in\\{1,...,M\\}$, each triangular face of
$\partial X_{j}$ is partitioned into between $1$ and $3^{M}$ pieces, and
$X_{j}$ is partitioned into tubes parallel to $w(v_{j})+\eta_{1}$ with these
pieces as their ends. Each sequence of successive tubes glued on their
intersections with $\partial X_{j}$ is a channel. The collection of channels
$C_{1},...,C_{K}$ is pairwise disjoint and covers $\widetilde{R}$. Since
$w(\widetilde{g})+\eta_{1}$ has positive first coordinate everywhere, each
channel connects a patch on $\partial\widetilde{R}$ to another.

Tiling a channel. Fix $n$ large and a choice of channel $C$. We construct a
scale $n$ tiling of $C$ which has only $-\eta_{1}$ tiles in a neighborhood of
$\partial C$ of constant-order width in $n$, and use this to say that we can
put together the tilings of the channels together to construct one tiling of
the whole region.

Let $T_{1},...,T_{m}$ be the sequence of tubes of the form $X_{j}\cap C$ in
order from one intersection of $C$ with $\partial\widetilde{R}$ to the other.
Let $v_{1},...,v_{m}$ be the corresponding values of $\widetilde{g}$ on the
tubes. Consider the $\mathbb{Z}^{3}$ tilings $\tau_{v_{1}},...,\tau_{v_{m}}$
constructed earlier in this section using the reference tiling $\tau_{1}$.
Recall that for each $v_{i}$, all tiles in $(\tau_{v_{i}},\tau_{1})$ are
either double tiles (which must be $-\eta_{1}$ tiles) or are on an infinite
path, and that all infinite paths follow the same periodic sequence. Let
$r_{j}$ be the period of $\tau_{v_{j}}$, i.e. if $\tau(x)$ denotes the tile at
$x$ in $\tau$, then for $j=1,...,m$, $r_{j}$ is such that translates
$\tau_{v_{j}}(x+r_{j}\eta)=\tau_{v_{j}}(x)$ for all unit vectors $\eta$.

The main operation we will use is that for any infinite path
$\ell\subset(\tau_{v_{j}},\tau_{1})$, we can modify $\tau_{v_{j}}$ by
“shifting” all the tiles along $\ell$, i.e. by removing all the tiles on
$\tau_{v_{j}}$ along $\ell$ and replacing them with $-\eta_{1}$ tiles. We
refer to this as deleting the path $\ell$. The idea is to delete paths that
would exit the channel before hitting $\partial\widetilde{R}$, and then to
bound the number of paths that we delete.

Let $\pi_{j}\subset\partial X_{j}$ be the starting end of $T_{j}$, so that
$T_{j}$ is a tube connecting $\pi_{j}$ to $\pi_{j+1}$. For each $j$, we start
by restricting $\tau_{v_{j}}$ to $T_{j}$. Any infinite path
$\ell\subset(\tau_{v_{i}},\tau_{1})$ which enters $T_{j}$ in $\pi_{j}$ must
exit through $\partial T_{j}\setminus\pi_{j}$, since paths in
$(\tau_{v_{j}},\tau_{1})$ always have direction with positive $\eta_{1}$
component.

Figure 36: On the left is a face of one tetrahedron. The segments are the ends
of channels, the smaller blue regions are places where we do not delete
infinite paths. The width of the white area is $O(r/n)$. The figure on the
right is a 2D schematic showing two channels $C,C^{\prime}$, with the tubes
$T_{j}$ and connector regions $B_{j}$ labeled along $C$. The width of the
white area between $C$ and $C^{\prime}$ is $O(r/n$).

First, we delete all infinite paths $\ell\subset(\tau_{v_{j}},\tau_{1})$ which
do not enter $T_{j}$ through $\pi_{j}$ and exit for the first time through
$\pi_{j+1}$, replacing the tiles of $\tau_{v_{j}}$ along these paths with
$-\eta_{1}$ tiles. Note that this includes deleting all infinite paths which
do not intersect $T_{j}$.

By Lemma 8.4.4, the infinite paths in $(\tau_{v_{j}},\tau_{1})$ have
asymptotic direction $w(v_{j})$, with oscillation bounded by the length of the
periodic sequence used to construct $\tau_{v_{j}}$, which is $O(r_{j})$. Since
the direction of the tube $T_{j}$ is also $w(v_{j})$, any infinite path in
$(\tau_{v_{j}},\tau_{1})$ that enters $T_{j}$ through $\pi_{j}$ and exits
through $T_{j}\setminus\pi_{j+1}$ is within $O(r_{j})$ distance counted as
number of edges in $\frac{1}{n}\mathbb{Z}^{3}$ of $\partial C$ along $T_{j}$.
Similarly, any path which enters $T_{j}$ through $\partial
T_{j}\setminus\pi_{j}$ would also remain within $O(r_{j})$ distance in number
of $\frac{1}{n}\mathbb{Z}^{3}$ edges of $\partial C$ along $T_{j}$. In
summary, the paths that we delete which intersect $T_{j}$ are all contained in
an neighborhood of $\partial C\cap T_{j}$ of width $O(r)$ in edge distance in
$\frac{1}{n}\mathbb{Z}^{3}$, corresponding to a neighborhood of Euclidean
width $O(r/n)$ (recall that $r$ is a constant independent of $n$).

Second (to avoid issues with corners and edges of tetrahedra, and to isolate
channels from each other), we delete all infinite paths which are within a
Euclidean neighborhood of width $1000/n$ of $\partial
T_{j}\setminus\\{\pi_{j}\cup\pi_{j+1}\\}$ (i.e., $1000$ lattice cubes in
$\frac{1}{n}\mathbb{Z}^{3}$). By the same logic as above, these are still
contained in an $O(r/n)$-width neighborhood of $\partial C$. We call this
tiling $\tau_{v_{j}}^{\prime}$.

Third, we want to glue the tiling on $T_{j}$ to the tiling on $T_{j+1}$. To do
this, we cut out a neighborhood of width $O(r)=O(r_{1},...,r_{m})$ in
$\frac{1}{n}\mathbb{Z}^{3}$ lattice cubes around $\pi_{j+1}$ (the face shared
by $T_{j}$ and $T_{j+1}$) which we call the connector region $B_{j}$. Let
$\alpha_{j},\alpha_{j+1}$ be the ends of the connector region (i.e. translates
of $\pi_{j+1}$), see the purple region in right side figure in Figure 36. Let
$P_{j}$ be the plane containing $\alpha_{j}$ and $P_{j+1}$ be the plane
containing $\alpha_{j+1}$.

Since $\tau_{v_{j}}$ and $\tau_{v_{j+1}}$ are periodic tilings of periods
$r_{j},r_{j+1}$, and since $B_{j}$ has width $O(r)$, by Lemma 8.4.6 we can
construct a tiling $\sigma_{j}$ of $\mathbb{Z}^{3}$ such that it agrees with
$\tau_{v_{j}}$ in the left half-space of $P_{j}$ and $\tau_{v_{j+1}}$ on the
right half-space of $P_{j+1}$, and fills in the region in-between in a
periodic way with period $O(r)$. We can do this for all $j=1,...,m$.

Overlaying $(\sigma_{j},\tau_{1})$, we again get a collection of infinite
paths and double tiles. First, we delete all infinite paths in $\sigma_{j}$
which were deleted to construct $\tau_{v_{j}}^{\prime}$ and
$\tau_{v_{j+1}}^{\prime}$ from $\tau_{v_{j}}$ and $\tau_{v_{j+1}}$. Second, we
delete any infinite paths which exit $C$ between $\alpha_{j}$ and
$\alpha_{j+1}$ (i.e., any paths which exits $C$ along the connector $B_{j}$).
Since $B_{j}$ has length $O(r)$ and since $\sigma_{j}$ is periodic with period
$O(r)$, again any infinite path which exits in $B_{j}$ is contained in an
$O(r)$ neighborhood of $\partial C$ along $B_{j}$.

Finally, we can glue together the tilings $\sigma_{1},...,\sigma_{m}$ by going
back and deleting any infinite path in $(\sigma_{j},\tau_{1})$ which connects
to one which would have been deleted in $(\sigma_{i},\tau_{1})$ for all other
$i\neq j$. Since the number of tubes $m$ is constant, in the end we have a
tiling $\tau$ of $C$ where we have deleted infinite paths of $(\tau,\tau_{1})$
in a neighborhood of width at most constant-order in $n$ (concretely
$1000+O(r)$, where $r$ is constant in $n$) in distance measuring in edges of
$\frac{1}{n}\mathbb{Z}^{3}$, corresponding to a neighborhood of Euclidean
width of $O(r/n)$.

Since all channels $C$ are tiled so that they have only $-\eta_{1}$ tiles in a
neighborhood of $\partial C$, we can put them together. Therefore we have
constructed a tiling $\tau\in T_{n}(\widetilde{R})$.

Bounding the final distance. To emphasize the dependence on $n$, let
$\tau^{n}$ be the tiling at scale $n$ constructed above and let $\tau_{1}^{n}$
be the $-\eta_{1}$ brickwork pattern at scale $n$. On one hand, the total flow
of $\widetilde{g}+\eta_{1}$ over any $X_{j}\in\mathcal{X}$ is

$\displaystyle\text{vol}(X_{j})(v_{j}+\eta_{1}).$

We claim that the double dimer flow
$f_{(\tau^{n},\tau_{1}^{n})}=f_{\tau^{n}}-f_{\tau_{1}^{n}}$ has the same total
flow, up to an $O(n^{-1})$ error. To explain the order of error, recall that
for a scale $n$ tiling flow, each $1/n^{3}$-volume lattice cube has flow of
order $1/n^{3}$. The error comes from the region around the boundary of the
channels where some infinite paths were deleted and replaced with $-\eta_{1}$
tiles. The number of lattice sites on the boundary of the channel is order
$n^{2}$, and the region has width constant order in $n$ in lattice cubes from
$\frac{1}{n}\mathbb{Z}^{3}$, so the amount of deleted flow in this region has
order $n^{2}/n^{3}=1/n$. Therefore there is a constant $K$ such that

$\bigg{|}\text{vol}(X_{j})(\widetilde{g}_{j}+\eta_{1})-\sum_{e\in\frac{1}{n}\mathbb{Z}^{3},\,e\cap
X_{j}\neq\emptyset}(f_{\tau^{n}}-f_{\tau_{1}^{n}})(e)\bigg{|}<Kn^{-1}.$

There is also a constant $C=C(\widetilde{R})$ such that $M=C\epsilon^{-3}$
(recall that $M=|\mathcal{X}|$ is the number of tetrahedra in the mesh). By
Lemma 5.2.7 applied to the partition $X_{1},...,X_{M}$ of $\widetilde{R}$,

$\displaystyle
d_{W}(f_{\tau^{n}}-f_{\tau_{1}^{n}},\widetilde{g}+\eta_{1})<3M(10\epsilon^{4}+Kn^{-1})<30C\epsilon+3CK\epsilon^{-3}n^{-1}.$

Taking $n$ large enough so that $1/n<\epsilon^{4}$, this becomes a bound which
is a constant times $\epsilon$. A few applications of the triangle inequality
and the “mass shift” property of Wasserstein distance, i.e. that
$\mathbb{W}_{1}^{1,1}(\mu,\nu)=\mathbb{W}_{1}^{1,1}(\mu+\rho,\nu+\rho)$ (see
Lemma 5.1.1), give that

$\displaystyle
d_{W}(f_{\tau^{n}},\widetilde{g})<d_{W}(f_{\tau^{n}}-f_{\tau_{1}^{n}},\widetilde{g}+\eta_{1})+d_{W}(f_{\tau_{1}^{n}},-\eta_{1}).$

Since $d_{W}(f_{\tau_{1}^{n}},-\eta_{1})\to 0$ as $n\to\infty$, we can make
this as small as needed as $n\to\infty$. Therefore we can choose $\epsilon$
small enough and $n$ large enough given $\delta$ so that $\tau^{n}\in
T_{n}(\widetilde{R})$ has $d_{W}(f_{\tau^{n}},\widetilde{g})<\delta/2$.
Restricting $\tau^{n}$ to $R$ completes the proof. ∎

###  8.5 Soft boundary lower bound

With the machinery developed in the previous section we can now prove the soft
boundary lower bound, namely Theorem 8.1.10. In particular we show that for
$(R,b)$ with $b$ extendable outside, then for any $g\in AF(R,b)$,

$\displaystyle\lim_{\delta\to
0}\liminf_{n\to\infty}v_{n}^{-1}\log\mu_{n}(A_{\delta}(g))\geq\text{Ent}(g).$

Recall that $\mu_{n}$ is counting measure on $TF_{n}(R,b,\theta_{n})$, the set
of free boundary tiling flows on $R$ at scale $n$ with boundary values within
$\theta_{n}$ of $b$. The main idea of the proof is to show that from the one
free boundary tiling flow $f_{\tau}\in A_{\delta}(g)\cap TF_{n}(R)$
constructed in previous section (Theorem 8.4.1), we can use the patching
theorem (Theorem 6.3.5) to show that there are actually many tiling flows in
$A_{\delta}(g)$.

###### Proof of Theorem 8.1.10.

By Proposition 8.3.1, there exists $\delta_{1}>0$ such that there is a
$\delta_{1}$-tetrahedral mesh $\mathcal{X}=\\{X_{1},...,X_{M}\\}$, region
$\widetilde{R}=\cup_{X\in\mathcal{X}}X$ containing $R$, and an asymptotic flow
$\widetilde{g}\in AF(\widetilde{R})$ taking constant values on tetrahedra in
$\mathcal{X}$ with $d_{W}(g,\widetilde{g})<\delta/2$ so that

$\displaystyle A_{\delta/2}(\widetilde{g})\subset A_{\delta}(g).$

Let $\widetilde{g}_{i}:=\widetilde{g}\mid_{X_{i}}$. Computing directly,

$\displaystyle\text{Ent}(\widetilde{g})=\frac{1}{\text{Vol}(R)}\sum_{i=1}^{M}\text{Vol}(X_{i})\text{ent}(\widetilde{g}_{i}).$

On the other hand by Proposition 7.6.3,

$\displaystyle\text{Ent}(\widetilde{g})=\text{Ent}(g)+o_{\delta}(1).$

Using the shining light construction from the proof of Theorem 8.4.1, for any
$n$ large enough there exists a tiling $\tau\in T_{n}(R)$ such that
$f_{\tau}\in A_{\delta/2}(\widetilde{g})$ has a particular form. Let
$C_{1},...,C_{K}$ denote the channels in the shining light construction. For
each tetrahedron $X$ and channel $C$ that intersect, $X\cap C$ is a tube. As
in the proof of Theorem 8.4.1, $\tau\mid_{X\cap C}$ is periodic at a scale
independent of $n$, and has mean current
$\widetilde{g}_{X}\in\text{Int}(\mathcal{O})$ on $X\cap C$ outside a
neighborhood of $\partial(X\cap C)$ of width constant order in $n$.

We choose $\epsilon\ll\delta_{1}$, and partition the interior of $X\cap C$
(where $\tau$ has mean current in $\text{Int}(\mathcal{O})$) into small cubes
with side length $\leq\epsilon$. For each $i=\\{1,...,M\\}$, call the pieces
of the partition contained in $X_{i}$

$\\{Q_{1}^{i},...Q_{k_{i}}^{i}\\}_{i=1}^{M}.$

For any $(i,k)$ pair, $\tau\mid_{Q_{k}^{i}}$ is periodic. Recall that
$Q_{k}^{i}$ has diameter $<\epsilon$. For $\epsilon_{1}\ll\epsilon$, for $n$
large enough $\tau\mid_{\partial Q_{k}^{i}}$ is $\epsilon_{1}$-nearly-constant
with value $\widetilde{g}_{i}$ (see Definition 6.3.3), so it satisfies the
conditions for the outer boundary condition in the patching theorem (Theorem
6.3.5). Fix $c\in(0,1)$. For each $(i,k)$ pair, we choose an EGM $\mu_{i,k}$
of mean current $\widetilde{g}_{i}$ (these exist by Corollary 7.5.4). Since
$\widetilde{g}_{i}\in\text{Int}(\mathcal{O})$, a sample from $\mu_{i,k}$
satisfies the conditions of Theorem 6.3.5 with probability going to $1$ as
$n\to\infty$ (Corollary 6.6.5). Therefore by Theorem 6.3.5, for $n$ large
enough, with probability $(1-c)$, $\tau$ restricted to $\partial Q_{k}^{i}$
can be patched with a sample $\sigma$ from $\mu_{i,k}$ on an annulus of width
$cn$. By the ergodic theorem, given any $\epsilon_{2}>0$, for $n$ large enough
we can assume that

$d_{W}(f_{\sigma}\mid_{Q_{k}^{i}},\widetilde{g}_{i}\mid_{Q_{k}^{i}})<\epsilon_{2}$
(64)

with probability $1-c$. Therefore with probability $1-2c$, Equation (64) holds
and $\sigma$ can be patched with $\tau$.

Let $\pi_{i,k,n}$ denote uniform measure on tilings $\sigma$ of $Q_{k}^{i}$ at
scale $n$ with $\sigma\mid_{\partial Q_{k}^{i}}=\tau$ and satisfying Equation
(64), and let $Z_{n}(Q_{k}^{i})$ be its partition function. The additional
constraint that Equation (64) is satisfied does not change the exponential
order of the number of tilings, hence by Proposition 6.8.3, we get the
following consequences for entropy:

$\displaystyle(1+O(c))h(\mu_{i,k})\leq
n^{-3}\text{Vol}(Q_{k}^{i})^{-1}H(\pi_{i,k,n})=n^{-3}\text{Vol}(Q_{k}^{i})^{-1}\log
Z_{n}(Q_{k}^{i}).$

By Lemma 5.2.7 applied to the partition
$\\{Q_{1}^{i},....,Q_{k_{i}}^{i}\\}_{i=1}^{M}\cup\\{R\setminus\cup_{i=1}^{M}\cup_{k=1}^{k_{i}}Q_{k}^{i}\\}$,
if $\sigma\in T_{n}(R)$ is a free boundary tiling of $R$ whose restrictions to
each $Q_{k}^{i}$ are in the support of $\pi_{i,k,n}$ then using Equation (64),

$d_{W}(f_{\tau},f_{\sigma})<3\epsilon^{-3}(10\epsilon^{4}+\epsilon_{2})+3C(\partial
R)\epsilon$

where $C(\partial R)$ is a constant depending only on $R$. In particular,
choosing $\epsilon_{2}=\epsilon^{4}$ and taking $\epsilon$ sufficiently small,
$f_{\sigma}\in A_{\delta/2}(\widetilde{g})$. By Theorem 5.5.7 (uniform
continuity of the boundary value operator $T(\cdot,\partial R)$), we can
choose $\epsilon$ small enough so that the boundary values
$T(f_{\sigma},\partial R)$ are within $\theta_{n}$ of $b$ for all $\sigma$ in
the support of $\pi_{i,k,n}$. Therefore for $n$ large enough,

$\displaystyle\mu_{n}(A_{\delta/2}(\widetilde{g}))\geq\prod_{i=1}^{M}\prod_{k=1}^{k_{i}}Z_{n}(Q_{k}^{i})\geq\prod_{i=1}^{M}\prod_{k=1}^{k_{i}}\exp\bigg{(}n^{3}\text{Vol}(Q_{k}^{i})h(\mu_{i,k})(1+O(c))\bigg{)}.$

Recall that $v_{n}^{-1}=n^{-3}\text{Vol}(R)^{-1}$. Since $\mu_{i,k}$ is an EGM
of mean current $\widetilde{g}_{i}\in\text{Int}(\mathcal{O})$,
$h(\mu_{i,k})=\text{ent}(\widetilde{g}_{i})$ by Theorem 7.5.2. Rearranging and
taking into account the $O(\epsilon)$ proportion of each tetrahedron
$X\in\mathcal{X}$ that is not included in the patched regions,

$\displaystyle v_{n}^{-1}\log\mu_{n}(A_{\delta/2}(\widetilde{g}))$
$\displaystyle\geq\sum_{i=1}^{M}\sum_{k=1}^{k_{i}}\frac{\text{Vol}(Q_{k}^{i})}{\text{Vol}(R)}\text{ent}(\widetilde{g}_{i})(1+O(c))$
$\displaystyle\geq\sum_{i=1}^{M}\frac{\text{Vol}(X_{i})}{\text{Vol}(R)}\text{ent}(\widetilde{g}_{i})(1+O(c))(1-O(\epsilon))$
$\displaystyle=\text{Ent}(\widetilde{g})(1+O(c))(1-O(\epsilon)).$

Recall that $\epsilon$ is the size of the patched regions, and $c<\epsilon$ is
the patching error. All of $\epsilon,c,\epsilon_{2}$ are much smaller than
$\delta>0$, and go to $0$ as $\delta\to 0$. In particular for any fixed
$\delta>0$, and $\epsilon>0$ fixed small enough given $\delta>0$, there is a
$n(\delta)$ such that for all $n>n(\delta)$,

$\displaystyle\liminf_{n\to\infty}v_{n}^{-1}\log\mu_{n}(A_{\delta}(g))\geq\liminf_{n\to\infty}v_{n}^{-1}\log\mu_{n}(A_{\delta/2}(\widetilde{g}))\geq\text{Ent}(\widetilde{g})+O(\epsilon)=\text{Ent}(g)+o_{\delta}(1).$

Taking $\delta\to 0$ completes the proof. ∎

###  8.6 Generalized patching and hard boundary lower bound

To prove the hard boundary lower bound (Theorem 8.2.8), we need one more tool.
Recall that $\overline{\rho}_{n}$ is the uniform probability measure on
tilings of a fixed region $R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$.

The shining light construction (Theorem 8.4.1) shows that for any $\delta>0$
and $g\in AF(R,b)$, for $n$ large enough there exists a free-boundary tiling
$\tau\in T_{n}(R)$ such that $d_{W}(f_{\tau},g)<\delta$. For hard boundary
conditions, we need to know that every $g\in AF(R,b)$ can be approximated by a
tiling of the fixed region $R_{n}$. To do this, we prove a generalized
patching theorem (Theorem 8.6.2).

Let $B_{n}=[-n,n]^{3}$. Recall that the patching theorem (Theorem 6.3.5) says
that if two tilings $\tau_{1},\tau_{2}$ are nearly constant with value
$s\in\text{Int}(\mathcal{O})$ (Definition 6.3.3), then for any $c>0$ there is
$n$ large enough that we can patch together $\tau_{2}\mid_{B_{(1-c)n}}$ and
$\tau_{1}\mid_{\mathbb{Z}^{3}\setminus B_{n}}$ by tiling the width-$cn$
annulus between them.

In Section 6 where the regular patching theorem (Theorem 6.3.5) was proved,
all our tools were combinatorial, and we thought of tilings $\tau$ of
$\mathbb{Z}^{3}$ without rescaling to $\frac{1}{n}\mathbb{Z}^{3}$. Here we
look at the tileability of more general “annular regions,” where the tilings
are rescaled to live in $\frac{1}{n}\mathbb{Z}^{3}$.

Let $R\subset\mathbb{R}^{3}$ be a compact set which is the closure of a
connected domain and has piecewise-smooth boundary $\partial R$ (i.e., the
sort of region to which our LDP applies). For any small $c>0$, we define

$\displaystyle R^{c}=\\{x\in R:d(x,\partial R)\geq c\\}.$

The set $R\setminus R^{c}$ is an annular region. On the discrete side, given a
free-boundary tiling $\tau\in T_{n}(R)$, it restricts to $\tau^{\prime}\in
T_{n}(R^{c})$ which is a free boundary tiling of $R^{c}$. We let
$R_{n}^{c}\subset\frac{1}{n}\mathbb{Z}^{3}$ be the region covered by
$\tau^{\prime}$. Given another free-boundary tiling $\sigma\in T_{n}(R)$, let
$R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ be the region covered by $\sigma$. The
region $A=R_{n}\setminus R_{n}^{c}\subset\frac{1}{n}\mathbb{Z}^{3}$ is the
type of annular region we study here.

Let $(R,b)$ be flexible and suppose that
$R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ is a sequence of regions satisfying
the conditions of the hard boundary LDP (Theorem 8.2.4), i.e. regions with
tileable boundary values $b_{n}$ on $\partial R$ converging to $b$ in
$\mathbb{W}_{1}^{1,1}$. To prove any $g\in AF(R,b)$ can be approximated by
$f_{\tau}$ with $\tau$ a tiling of $R_{n}$, we show that we can patch together
suitable tilings on annuli of the form $R_{n}\setminus R_{n}^{c}$, with hard
boundary condition on the outside.

######  Definition 8.6.1.

We say that a flow $g\in AF(R,b)$ is flexible if $g$ satisfies the condition
that for any compact set $D\subset\text{Int}(R)$,
$\overline{g(D)}\subset\text{Int}(\mathcal{O})$.

The pair $(R,b)$ is flexible (see Definition 8.2.1 and Lemma 8.2.2) if and
only if there exists $g\in AF(R,b)$ which is flexible.

######  Theorem 8.6.2 (Generalized patching theorem).

Fix $c>0$. Let $(R,b)$ be flexible, with $b$ a boundary asymptotic flow which
is extendable outside. Let $R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ be a
sequence of regions with tileable boundary values $b_{n}$ on $\partial R$
converging to $b$ in $\mathbb{W}_{1}^{1,1}$. Let $\sigma_{n}$ be a sequence of
tilings of $R_{n}$.

Let $\tau_{n}\in T_{n}(R)$ be a sequence of tilings such that
$d_{W}(g,f_{\tau_{n}})\to 0$ as $n\to\infty$ for $g\in AF(R,b)$ flexible. Let
$\tau_{n}^{\prime}$ be $\tau_{n}$ restricted to a free boundary tiling of
$R^{c}$, and let $R_{n}^{c}\subset\frac{1}{n}\mathbb{Z}^{3}$ be the cubes
covered by $\tau_{n}^{\prime}$.

For $n$ large enough, $R_{n}\setminus R_{n}^{c}$ is tileable.

######  Remark 8.6.3.

The flexible condition here is analogous to the $s\in\text{Int}(\mathcal{O})$
condition in the original patching theorem (Theorem 6.3.5). The generalized
patching theorem is the reason the hard boundary LDP requires that $(R,b)$ is
flexible.

Before we prove this, we explain how it can be used to prove the hard boundary
lower bound (Theorem 8.2.8). First, from the generalized patching theorem, it
is straightforward to prove the fixed boundary analog of Theorem 8.4.1.

######  Corollary 8.6.4.

Suppose that $(R,b)$ is flexible and $b$ is a boundary asymptotic flow which
is extendable outside. Fix a sequence of regions $R_{n}$ with tileable
boundary values $b_{n}$ on $\partial R$ converging to $b$ in
$\mathbb{W}_{1}^{1,1}$. For any $\delta>0$ and any $g\in AF(R,b)$, there is
$n$ large enough such that there exists a tiling $\tau$ of $R_{n}$ with
$d_{W}(f_{\tau},g)<\delta$.

###### Proof.

Since $(R,b)$ is flexible, there exists $g_{0}\in AF(R,b)$ which is flexible.
For any $\epsilon>0$, the new flow $g_{\epsilon}=\epsilon g_{0}+(1-\epsilon)g$
satisfies $d_{W}(g,g_{\epsilon})<C\epsilon$ for some constant $C>0$. Taking
$\epsilon$ small enough, we can guarantee that
$d_{W}(g,g_{\epsilon})<\delta/2$.

Since $g_{\epsilon}$ is flexible and $R_{n}$ is tileable, by Theorem 8.6.2 for
$n$ large enough there exists a tiling $\tau$ of $R_{n}$ such that
$d_{W}(g_{\epsilon},f_{\tau})<\delta/2$. By the triangle inequality
$d_{W}(g,f_{\tau})<\delta$, which completes the proof. ∎

Adding Corollary 8.6.4 as the first step, the proof of Theorem 8.2.8 (hard
boundary lower bound) is the same as the proof of Theorem 8.1.10 (soft
boundary lower bound).

###### Proof of Theorem 8.2.8.

By Corollary 8.6.4, given any $\delta>0$, for $n$ large enough and any $g\in
AF(R,b)$ we can find a tiling $\tau$ of $R_{n}$ such that
$d_{W}(f_{\tau},g)<\delta$. Further, this tiling is of the form given in the
shining light construction (proof of Theorem 8.4.1), other than in an annulus
of width $c\in(0,1)$ where $c$ can be taken arbitrarily small. The remainder
of the proof is identical to the proof of Theorem 8.1.10, where we use the
regular patching theorem to patch in samples from ergodic Gibbs measures of
appropriate mean currents. ∎

Figure 37: 2D schematic picture for the proof of the generalized patching
theorem.

It remains to prove the generalized patching theorem (Theorem 8.6.2). The
proof is structurally analogous to the proof of the regular patching theorem
for cubes $B_{n}=[-n,n]^{3}$ (Theorem 6.3.5), and relies on a sequence of
lemmas. We give an outline of the main ideas to explain where each of the
lemmas is used, accompanied by the schematic picture in Figure 37. We then
state and prove each of the lemmas, followed by a proof of Theorem 8.6.2.

Note that there are some superficial changes between the results here and
their analogs in Section 6, since here our regions
$R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ instead of
$B_{n}\subset\mathbb{Z}^{3}$. Basically this corresponds to a change in units.
We introduce a few new pieces of notation to make it easier to work with the
tilings $\tau$ of $\frac{1}{n}Z^{3}$ instead of $\mathbb{Z}^{3}$.

  * •

If $Q$ is a discrete surface built out of lattice squares in
$\frac{1}{n}\mathbb{Z}^{3}$, we define $\text{area}_{n}(Q)$ to be the number
of lattice squares on $Q$. This is $n^{2}$ times the Euclidean area of $Q$. If
$Q$ is a surface not built of lattice squares, we can still use
$\text{area}_{n}(Q)$ to mean the Euclidean area of $Q$ times $n^{2}$.

  * •

If $\ell$ is a discrete curve built out of edges of squares in
$\frac{1}{n}\mathbb{Z}^{3}$, we defined $\text{length}_{n}(\ell)$ to be number
of lattice edges in $L$. This is $n$ times the Euclidean length of $\ell$.

  * •

If $\tau$ is a tiling of $\frac{1}{n}\mathbb{Z}^{3}$, then the tiling flow
$f_{\tau}$ and pretiling flow $v_{\tau}$ are typically rescaled so that for
$e\in\frac{1}{n}\mathbb{Z}^{3}$, $v_{\tau}(e)=\pm 1/n^{3}$ or $0$, and
$f_{\tau}(e)=\pm 5/6n^{3}$ or $\pm 1/6n^{3}$. We define $\widetilde{v}_{\tau}$
to be unrescaled flow $\widetilde{v}_{\tau}(e)=\pm 1$ or $0$ for
$e\in\frac{1}{n}\mathbb{Z}^{3}$, and similarly $\widetilde{f}_{\tau}(e)=\pm
5/6$ or $\pm 1/6$.

The proof of Theorem 8.6.2 uses a mixture of combinatorial results like in
Section 6 and more analytic results about Wasserstein distance, which are for
rescaled tiling flows. These pieces of notation make it easier to go between
these points of view, and to explain the purely combinatorial arguments in a
way more analogous to Section 6.

The main combinatorial tool is again Hall’s matching theorem (Theorem 6.1.3),
which says that if $A=R_{n}\setminus R_{n}^{c}$ is not tileable, then there
exists a counterexample set $U\subset A$ which proves it. The interior
boundary $S$ of $U$ is without loss of generality a minimal monochromatic
discrete surface with number of squares from $\frac{1}{n}\mathbb{Z}^{3}$ on
the surface bounded above and below by a constant times $n^{2}$ (Lemma 8.6.5).
The first step is to indent slightly and let $A_{\text{mid}}\subset A$ be a
slightly smaller annulus where $U$ is well-behaved (Lemma 8.6.6). We then
define $U_{\text{mid}}=U\cap A_{\text{mid}}$.

We find a test tiling $\tau_{\text{SL}}$ using a shining light construction
(Definition 8.6.8) and show that it satisfies a flow bound (Lemma 8.6.9). This
is where we use the condition that $g$ is flexible. Using Lemma 8.6.9, we show
that there is a constant $K\in(0,1)$ such that
$\text{imbalance}(U_{\text{mid}}^{\prime})\leq-Kn^{2}+O(n)$, where
$U_{\text{mid}}^{\prime}$ is $U_{\text{mid}}$ plus a few cubes from the rest
of $U$ (determined by $\tau_{\text{SL}}$). This “slack” corresponds to flow
from $\tau_{\text{SL}}$ which exits through the boundary of $U_{\text{mid}}$
in the interior, see the pink arrows in Figure 37.

It remains to bound the imbalance in $U_{\text{shell}}:=U\setminus
U_{\text{mid}}^{\prime}$. This we break into two pieces:

  * •

Regions of $U_{\text{shell}}$ contained in nice “cylinders” connecting
$\partial A$ to $\partial A_{\text{mid}}$. These are the regions where there
are black and blue arrows in Figure 37, the sides of the cylinder are the
orange regions which we call the “ribbon surface.”

  * •

The rest of $U_{\text{shell}}$, which we call the “leftover region.”

Up to error related to the orange area in Figure 37 (the ribbon surface), we
show that the imbalance of $U_{\text{shell}}$ is the same as the imbalance in
the cylinder regions. Finally we relate the imbalance in the cylinders to the
flux of tiling flows $f_{\sigma_{n}}$, $f_{\tau_{n}}$, and
$f_{\tau_{\text{SL}}}$, which we can bound using Wasserstein convergence
considerations using Lemma 5.2.8, up to an error proportional to the green
area in Figure 37.

We now proceed to the lemmas. We first note that Lemma 6.5.2 for cubes has an
analog for general regions. The only difference is that the constants
$c_{1},c_{2}$ change since they can depend on the regions.

######  Lemma 8.6.5.

Let $R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ be a region of diameter in lattice
squares at least $n$ such that $R_{n}$ are regions approximating a fixed
region $R\subset\mathbb{R}^{3}$ with $\partial R$ a piecewise smooth surface.
Define $A=R_{n}\setminus R_{n}^{c}$ for some $c>0$. Suppose that $S\subset A$
is a monochromatic minimal discrete surface in $\frac{1}{n}\mathbb{Z}^{3}$
with connects the inner and outer boundaries of $A$. Then there exist constant
$c_{1},c_{2}$ independent of $S$ and $n$ such that

$c_{1}n^{2}\geq\text{area}_{n}(S)\geq c_{2}n^{2}$

where $c_{2}\sim c^{2}$.

###### Proof.

This proof is completely analogous to the proof of Lemma 6.5.2.

For the upper bound, we use that $S$ is minimal to get a bound which is a
constant times the surface area of $R_{n}$. Since
$R_{n}\subset\frac{1}{n}\mathbb{Z}^{3}$ are regions approximating $R$, and
since $\partial R$ is piecewise smooth, $\text{area}(R_{n})$ is bounded by a
constant times $n^{2}$, where this constant is determined by $\partial R$.

For the lower bound, we apply Proposition 6.5.1 to a point $p\in S$ which is
distance at least $c/3$ from $\partial A$ to get that

$\displaystyle\text{area}(S)\geq\kappa((c/3)\,n)^{2}=c_{2}n^{2},$

where $\kappa$ is a universal constant coming from the isopertimetric
inequality. From this expression we see that $c_{2}$ is order $c^{2}$. ∎

######  Lemma 8.6.6 (Generalized indenting lemma).

Let $A=R_{n}\setminus R_{n}^{c}$, and fix $\beta>0$ small. Let $S$ be a
minimal monochromatic surface connecting inner and outer boundaries of $A$.
There exists $\epsilon<c$ independent of $S$ such that the following hold for
$n$ large enough:

  1. 1.

Let $A_{a,b}$ denote the annulus between layers $\partial R_{n}^{a}$ and
$\partial R_{n}^{b}$. Then $\text{area}_{n}(S\cap
A_{\epsilon,2\epsilon})<\beta n^{2}$.

  2. 2.

There exists $a\in(\epsilon,2\epsilon)$ such that
$\text{length}_{n}(\partial(\partial R_{n}^{a}\cap S))\leq(\beta/\epsilon)n$.

  3. 3.

A “ribbon surface” $\gamma$ for $S\cap\partial R_{n}^{a}$ is a surface
connecting $\partial R_{n}^{a}$ to $\partial R_{n}$ with boundary
$\partial(S\cap\partial R_{n}^{a})$ on $\partial R_{n}^{a}$. The “ribbon area”
is the minimal $\text{area}_{n}$ of a ribbon surface, and is bounded by
$2\beta n^{2}$.

The analogous bounds hold for some $a^{\prime}\in(c-2\epsilon,c-\epsilon)$.

######  Remark 8.6.7.

For two surfaces $A,B$, the set $A\cap B$ is either a surface (2-dimensional),
a curve (1-dimensional), or a combination of the two. In any of these cases,
we take $\partial(A\cap B)$ to mean that we take union of the curve part of
$A\cap B$ and the boundary of the surface part of $A\cap B$.

###### Proof.

By Lemma 8.6.5, $c_{2}n^{2}\leq\text{area}_{n}(S)\leq c_{1}n^{2}$. For a given
$b>0$, divide a band of the form $A_{0,b}$ in half. Each time we divide, by
Lemma 8.6.5 both the halves of $S$ have area bounded below by a fixed constant
times $b^{2}$. On the other hand, one of the halves of $S$ can have at most
$1/2$ the original area. Iterating this, we can find $\epsilon>0$ small enough
so that the outer band after we split has area at most $\beta n^{2}$.

By the pigeonhole principle, there exists a layer $an$ between $\epsilon n$
and $2\epsilon n$ where

$\displaystyle\text{length}_{n}(\partial(\partial R_{n}^{a}\cap S))\leq(\beta
n^{2})/(\epsilon n)=(\beta/\epsilon)n.$

Given this, we can find a ribbon surface $\gamma$ (not necessarily built from
lattice squares) with $\text{area}_{n}(\gamma)\leq(\beta/\epsilon)n(2\epsilon
n)=2\beta n^{2}$, and hence the ribbon area is bounded by $2\beta n^{2}$. ∎

Suppose that $g\in AF(R,b)$ is flexible. For any $\delta>0$, we can find a
piecewise-constant flow $\widetilde{g}$ with $d_{W}(g,\widetilde{g})<\delta$
(Proposition 8.3.1). More precisely, $\widetilde{g}$ is piecewise-constant on
a mesh of small tetrahedra $\mathcal{X}$. The region
$\widetilde{R}=\cup_{X\in\mathcal{X}}X$ contains $R$, and $\widetilde{g}\in
AF(\widetilde{R})$. In the proof of Theorem 8.4.1, to construct a tiling
approximation $\tau\in T_{n}(R)$ of $g$ for $n$ large, we construct a tiling
approximation $\widetilde{\tau}\in T_{n}(\widetilde{R})$ of $\widetilde{g}$
for $n$ large, and restrict it to $R$.

The flexible condition passes from $g$ to $\widetilde{g}$ as follows. For any
compact set $D\subset\text{Int}(R)$, $g$ flexible means that
$\overline{g(D)}\subset\text{Int}(\mathcal{O})$. In particular there is a
constant $k_{D}\in[0,1)$ such that if $x\in D$ then

$|g(x)|_{1}\leq k_{D}<1,$

where $|\cdot|_{1}$ denotes the $L^{1}$ norm. When $g$ is flexible, we can
choose $\widetilde{g}$ so that for any compact set $D\subset\text{Int}(R)$,
for all tetrahedra $X\in\mathcal{X}$ such that $X\subset D$,
$|\widetilde{g}_{X}|_{1}\leq k_{D}$. When this holds, we say that
$\widetilde{g}$ is a piecewise constant approximation of $g$ inheriting the
flexible condition.

For any $\epsilon>0$ and $D\subset\text{Int}(R)$ compact, for $\delta>0$ small
enough we can find $\widetilde{g}$ which is piecewise-constant on a scale
$\delta$ tetrahedral mesh $\mathcal{X}$, has
$d_{W}(g,\widetilde{g})<\epsilon$, and inherits the flexible condition on $D$,
i.e. $|\widetilde{g}_{X}|\leq k_{D}$ for all $X\in\mathcal{X}$ such that
$X\subset D$.

In the shining light construction in the proof of Theorem 8.4.1, we cut the
tetrahedra into tubes (with thin space between them), and construct the tiling
approximation $\widetilde{\tau}$ of $\widetilde{g}$ by filing the tubes in $X$
with periodic tilings of mean current approximating $\widetilde{g}_{X}$ on
$X$, and fill the thin area in between with all $-\eta_{1}$ tiles. There is
some maximum period $r$ that we use to construct the periodic tilings on the
tubes, and $r$ is independent of $n$. For all $n$, at all the sites where
$\widetilde{\tau}$ does not have mean current $\widetilde{g}$,
$\widetilde{\tau}$ looks locally like the $-\eta_{1}$ brickwork pattern. The
width of the region containing the places where $\widetilde{\tau}$ looks like
the $-\eta_{1}$ brickwork pattern is $O(r)$ and therefore independent of $n$.

######  Definition 8.6.8 (Shining light measures).

Let $g\in AF(R,b)$ and let $\widetilde{g}$ be an approximation as discussed
above. For each $n$, let $\widetilde{\tau}_{n}\in T_{n}(\widetilde{R})$ be a
tiling produced by the shining light construction with $\widetilde{g}$, where
the periodic tilings in the tubes have maximum period $r$. Fix a large
constant $C=O(r)$. We define a sequence of shining light measures
$\lambda_{n}$ for $g$ using $\widetilde{g}$ so that for each $n$,
$\lambda_{n}$ is uniform measure on tilings of the form
$(\widetilde{\tau}_{n}+x)\mid_{R}\in T_{n}(R)$ for $x\in\mathbb{R}^{3}$ with
$|x|\leq C$.

Using this, we prove a lemma analogous to Lemma 6.6.7 from Section 6. Instead
of a result for ergodic measures of mean current
$s\in\text{Int}(\mathcal{O})$, this lemma is for a sequence of shining light
measures $\lambda_{n}$ for a flexible flow $g$. This lemma is why the flexible
condition is needed for generalized patching and hence for the hard boundary
LDP.

######  Lemma 8.6.9 (Shining light flow bound).

Let $D\subset\text{Int}(R)$, and let $g\in AF(R,b)$ be flexible. Let
$\widetilde{g}$ be piecewise constant approximation of $g$ on a tetrahedral
mesh $\mathcal{X}$ such that $\widetilde{g}$ inherits the flexible condition,
and such that the union of tetrahedra $\mathcal{D}\subset\mathcal{X}$ covering
$D$ is still contained in $\text{Int}(R)$.

Let $S$ be a monochromatic black surface in $\frac{1}{n}\mathbb{Z}^{3}$ with
boundary $\partial S$ and $\lambda_{n}$ be a sequence of shining light
measures as in Definition 8.6.8 for $\widetilde{g}$. Let $\Theta_{D}$ be the
collection of odd cubes adjacent to $S\cap D$. Let $N=\text{area}_{n}(S\cap
D)$ be the number of squares from $\frac{1}{n}\mathbb{Z}^{3}$ on $S\cap D$.
Then there is a constant $K_{D}\in(0,1)$ independent of $S$ such that for all
$n$ large enough,

$\displaystyle\mathbb{E}_{\lambda_{n}}[|\text{flux}(\widetilde{v}_{\tau},S\cap
D)|]\geq K_{D}|\Theta_{D}|+O(n^{-1}N)+O(\text{length}_{n}(\partial(S\cap D)).$

The constant $K_{D}$ depends only on $D$ and $g$. In particular, it is
independent of $\widetilde{g}$, as long as $\widetilde{g}$ inherits the
flexible condition and is constructed on a small enough mesh $\mathcal{X}$.

######  Remark 8.6.10.

Recall that for $\tau$ a tiling in $\frac{1}{n}\mathbb{Z}^{3}$, the flow
$\widetilde{v}_{\tau}$ is the non-rescaled pretiling flow.

###### Proof.

Restricted to any tetrahedron $X\in\mathcal{X}$, $\lambda_{n}$ samples a
tiling which is periodic with mean current $\widetilde{g}_{X}$ up an
$O(n^{-1})$ error. In particular, there are probabilities
$p_{1}(X),....,p_{6}(X)$ such that the probability of seeing a tile of type
$i$ in $\tau$ restricted to $X$ sampled from $\lambda_{n}$ is
$p_{i}(X)+O(n^{-1})$ for $i=1,...,6$. Let $N_{1},...,N_{6}$ denote the
corresponding six types of squares $f$ on $S$. Let $N_{i}(X)$ denote the
number of each type of square in $S$ restricted to $X$.

Recall that $\mathcal{D}\subset\mathcal{X}$ is the collection of mesh
tetrahedra $X\in\mathcal{X}$ such that $X\cap D\neq\emptyset$. By assumption
$\cup_{X\in\mathcal{D}}X\subset\text{Int}(R)$. Since $\widetilde{g}$ inherits
the flexible condition, there is a constant $k_{D}\in[0,1)$ such that
$|\widetilde{g}_{X}|_{1}\leq k_{D}$ for all $X\in\mathcal{D}$. This is related
to the probabilities since

$\widetilde{g}_{X}=(p_{1}(X)-p_{2}(X),p_{3}(X)-p_{4}(X),p_{5}(X)-p_{6}(X))$
(65)

Since $S$ is monochromatic, the flux of $\widetilde{v}_{\tau}$ across $S$ is
equal to minus the number of tiles in $\tau$ which cross $S$. Thus the
expected value of the flux (up to sign) can be split as the sum of indicator
functions $\mathbbm{1}_{f}$, where $\mathbbm{1}_{f}(\tau)$ is $1$ if $f$ is
crossed by a tile in $\tau$ and $0$ otherwise. Then

$\displaystyle\mathbb{E}_{\lambda_{n}}[|\text{flux}(\widetilde{v}_{\tau},S\cap
D)|]\geq\sum_{X\in\mathcal{D}}\sum_{f\in S\cap X\cap
D}\mathbb{E}_{\lambda_{n}}[\mathbbm{1}_{f}(\tau)]=\sum_{X\in\mathcal{D}}\sum_{i=1}^{6}N_{i}(X)(p_{i}(X)+O(n^{-1})).$

The total number of squares on the surface is
$N=\sum_{X\in\mathcal{D}}\sum_{i=1}^{6}N_{i}(X)=\text{area}_{n}(S\cap D)$. By
Lemma 6.6.6, $N_{1}+N_{2},N_{3}+N_{4},N_{5}+N_{6}$ are all equal to $N/3$ up
to an error of $O(\text{length}_{n}(\partial(S\cap D))$. Let

$\displaystyle
p_{D}=\min_{X\in\mathcal{D}}\max\\{\min\\{p_{1}(X),p_{2}(X)\\},\min\\{p_{3}(X),p_{4}(X)\\},\min\\{p_{5}(X),p_{6}(X)\\}\\}.$

Since $|\widetilde{g}_{X}|_{1}\leq k_{D}<1$, for all $X\in\mathcal{D}$ at
least four of $p_{1}(X),...,p_{6}(X)$ must be nonzero, including one from each
pair. Combined with Equation (65), one can easily check from this that
$p_{D}\geq(1-k_{D})/6>0$. On the other hand by the same arguments as in Lemma
6.6.7,

$\displaystyle\mathbb{E}_{\lambda_{n}}[|\text{flux}(\widetilde{v}_{\tau},S\cap
D)|]\geq p_{D}N/3+O(n^{-1}N)+O(\text{length}_{n}(\partial(S\cap D))).$

Since $|\Theta_{D}|\leq\text{area}_{n}(S\cap D)=N$, this completes the proof.
The constant $K_{D}=p_{D}/3=(1-k_{D})/18$. As $k_{D}$ is determined by just
$g$ and $D$, we note that this is independent of the choice of
$\widetilde{g}$, as long as $\widetilde{g}$ inherits the flexible condition
and is constructed on a small enough mesh $\mathcal{X}$. ∎

Equipped with these lemmas, we can now give the proof of the generalized
patching theorem.

###### Proof of Theorem 8.6.2.

If $A=R_{n}\setminus R_{n}^{c}$ is not tileable, then by Theorem 6.1.3 there
exists a counterexample region $U\subset A$ which has only odd cubes along its
interior boundary $S$, but has

$\displaystyle\text{imbalance}(U)=\text{even}(U)-\text{odd}(U)>0.$

By Corollary 6.2.6, we can assume that the interior boundary $S\subset\partial
U$ is a minimal monochromatic discrete surface in $\frac{1}{n}\mathbb{Z}^{3}$.

By Lemma 8.6.6, for any $\beta>0$ we can find $\epsilon$ and inner and outer
layers $a_{+}\in(\epsilon,2\epsilon)$ and $a_{-}\in(c-2\epsilon,c-\epsilon)$
such that for $a=a_{+}$ or $a=a_{-}$,

$\text{length}_{n}(\partial(S\cap\partial R_{n}^{a}))\leq(\beta/\epsilon)n$
(66)

and further such that there is a ribbon surface $\gamma$ for
$\partial(S\cap\partial R_{n}^{a})$ such that

$\text{area}_{n}(\gamma)\leq 2\beta n^{2}.$ (67)

We define the middle annulus $A_{\text{mid}}=R_{n}^{a_{+}}\setminus
R_{n}^{a_{-}}$. Let $U_{\text{mid}}=U\cap A_{\text{mid}}$.

We now fix a compact set $D\subset\text{Int}(R)$. We can assume that $D$ is
contained in $A_{\text{mid}}$. We can find a piecewise-constant approximation
$\widetilde{g}$ of $g$ on a tetrahedral mesh $\mathcal{X}$ which inherits the
flexible condition. We let $\mathcal{D}\subset\mathcal{X}$ be the collection
of tetrahedra $X$ such that $X\cap D\neq\emptyset$. We can assume that the
mesh scale of $\mathcal{X}$ is small enough so that the union of all
$X\in\mathcal{D}$ is still contained in $\text{Int}(R)$.

By analogous pigeonhole principle arguments in the indenting lemmas, we can
assume that $D$ has width in squares from $\frac{1}{n}\mathbb{Z}^{3}$ of at
least $cn/2$ and has $\text{length}_{n}(\partial(S\cap D))=O(n)$.

Let $N=\text{area}_{n}(S\cap D)$. By Lemma 8.6.9, we can find a sequence of
shining light measures $\lambda_{n}$ satisfying for $n$ large enough,

$\displaystyle\mathbb{E}_{\lambda_{n}}[|\text{flux}(\widetilde{v}_{\tau},S\cap
D)|]\geq K_{D}|\Theta_{D}|+O(Nn^{-1})+O(\text{length}_{n}(\partial(S\cap D)).$

By Lemma 8.6.5, $|\Theta_{D}|\geq N=\text{area}_{n}(S\cap
D)\geq(c_{2}/4)n^{2}$. Therefore

$\displaystyle\mathbb{E}_{\lambda_{n}}[|\text{flux}(\widetilde{v}_{\tau},S\cap
D)|]\geq\frac{K_{D}c_{2}}{4}n^{2}+O(n).$

In particular, we can sample a tiling $\tau_{\text{SL}}$ from $\lambda_{n}$
such that

$|\text{flux}(\widetilde{v}_{\tau_{\text{SL}}},S\cap
D)|\geq\frac{K_{D}c_{2}}{4}n^{2}+O(n).$ (68)

Let $U_{\tau_{\text{SL}}}$ be the cubes covered by $\tau_{\text{SL}}$
restricted to $U_{\text{mid}}$. This is a tileable region, so

$\displaystyle\text{imbalance}(U_{\tau_{\text{SL}}})=0.$

Let $U_{\text{mid}}^{\prime}\subset U$ be $U_{\tau_{\text{SL}}}$ minus even
cubes in $A\setminus U$ with a face on $S$ which are connected to $U$ by
$\tau_{\text{SL}}$. By Equation (68),

$\text{imbalance}(U_{\text{mid}}^{\prime})\leq-\frac{K_{D}c_{2}}{4}n^{2}+O(n).$
(69)

Let $U_{\text{shell}}=U\setminus U_{\text{mid}}^{\prime}$. It remains to bound
the imbalance in $U_{\text{shell}}$.

Consider the set $\alpha=U\cap\partial A_{\text{mid}}$. For each connected
component $\alpha_{i}$ of $\alpha$, we form a closed surface using a cylinder
ribbon surface component $\gamma_{i}$ of $\gamma$ and corresponding patch
$\alpha_{i}^{\prime}\subset\partial A$. Let $\alpha^{\prime}$ be the union of
the $\alpha_{i}^{\prime}$ components.

Let $V_{i}$ be the region enclosed by $\alpha_{i},\gamma_{i}$, and
$\alpha_{i}^{\prime}$, and let $V=\cup_{i}V_{i}$.

The regions $U_{\text{shell}}\setminus V$ are the leftover regions. Let $W$ be
a connected component of the leftover region. By construction, $\partial W$
either intersects at most one of $\partial R_{n}$ or $\partial R^{c}_{n}$.
Thus the boundary condition on $\partial W$ comes from only one tiling, either
$\tau_{n}$ or $\sigma_{n}$.

Suppose it comes from $\sigma_{n}$, i.e. that $\partial W\cap\partial
R_{n}\neq\emptyset$ (the version where it comes from $\tau_{n}$ is identical,
we just make a choice for concreteness). Since $\sigma_{n}$ can be extended to
a tiling of all of $R_{n}$, we can extend $\sigma_{n}$ to a tiling covering
$W$. Let $W_{\sigma}$ be the region covered by the tiles from $\sigma_{n}$
which intersect $W$. Clearly $\text{imbalance}(W_{\sigma})=0$, and
$W_{\sigma}\cap\partial R_{n}=W\cap\partial R_{n}$. If a tile in $W_{\sigma}$
crosses $\partial W$, then either

  * •

It crosses $\partial W\cap S$, in which case $W_{\sigma}$ contains an even
cube which is not contained in $W\subset U$.

  * •

It crosses $\partial W\cap\gamma$ (recall that $\gamma$ is the ribbon
surface). In this case $W_{\sigma}$ could have an odd cube which is not in
$W$. However the number of these added cubes over all components $W$ of
$\text{U}_{\text{shell}}\setminus V$ is bounded by
$\text{area}_{n}(\gamma)\leq 4\beta n^{2}$.

Therefore

$\displaystyle\text{imbalance}(U_{\text{shell}}\setminus V)\leq 4\beta n^{2}.$

We now show that

$\displaystyle\text{imbalance}(U_{\text{shell}}\cap
V)\leq\text{imbalance}(V)+4\beta n^{2}.$

The $4\beta n^{2}$ term again comes from the ribbon area. We use ideas
analogous to those above. Let $Y$ be a component of
$V\setminus(U_{\text{shell}}\cap V)$. First note that $Y$ intersects at most
one of $\partial R_{n}$ and $\partial R_{n}^{c}$, so its boundary condition
comes from only one tiling.

Assume $\sigma_{n}$ is the tiling which defines the boundary condition on $Y$,
and extend it to a tiling which covers $Y$. Let $Y_{\sigma}$ be the region
covered by $\sigma_{n}$ tiles which intersect $Y$. Clearly
$\text{imbalance}(Y_{\sigma})=0$. If a tile in $Y_{\sigma}$ crosses $\partial
Y$, then it is in one of two cases:

  * •

It crosses $\partial Y\cap S$. Since $Y\subset A\setminus U$, in this case
$Y_{\sigma}$ contains an odd cube which is not in $Y$. This makes the
imbalance of $Y$ larger.

  * •

It crosses $\partial Y\cap\gamma$ (recall $\gamma$ is the ribbon surface). In
this case $Y_{\sigma}$ could have an even cube which is not in $Y$. However
the number of these added cubes over all components $Y$ of
$V\setminus(\text{U}_{\text{shell}}\cap V)$ is bounded by
$\text{area}_{n}(\gamma)\leq 4\beta n^{2}$.

Therefore in summary,

$\displaystyle\text{imbalance}(U_{\text{shell}})\leq 8\beta
n^{2}+\text{imbalance}(V).$

We now relate imbalance to flux. As the proof of Proposition 6.2.3 (where we
relate black and white surface area to imbalance), given a set $V$, we apply
the divergence theorem to the reference flow $\widetilde{r}(e)=1/6$ for all
$e$ in $\frac{1}{n}\mathbb{Z}^{3}$ oriented even to odd to get that

$\displaystyle\text{imbalance}(V)=\text{flux}(\widetilde{r},\partial
V)=\frac{1}{6}\bigg{(}\text{white}(\partial V)-\text{black}(\partial
V)\bigg{)}.$

We apply this to the set $V=\cup_{i}V_{i}$. By Equation (67), the $\gamma_{i}$
contribute at most $4\beta n^{2}$. Therefore

$\text{imbalance}(U_{\text{shell}})\leq\sum_{i}\text{flux}(\widetilde{r},\partial
V_{i})\leq 12\beta
n^{2}+|\text{flux}(\widetilde{r},\alpha)-\text{flux}(\widetilde{r},\alpha^{\prime})|.$
(70)

For the second inequality, we orient $\alpha,\alpha^{\prime}$ to always both
have inward-pointing normal vector (i.e. inward on $\partial A$ and inward on
$\partial A_{\text{mid}}$), meaning one has the opposite normal vector as when
we compute flux for $\partial V_{i}$. This is why we get a minus sign.

The non-rescaled flow $\widetilde{f}_{\tau}$ is the divergence-free version of
the pretiling flow $\widetilde{v}_{\tau}$; related by the equation
$\widetilde{f}_{\tau}(e)=\widetilde{v}_{\tau}(e)-\widetilde{r}(e)$ for all
edges $e$ oriented even to odd. The rescaled version has
$f_{\tau}=\frac{1}{n^{3}}\widetilde{f}_{\tau}$.

Since the boundary condition on $\alpha$ is given by $\tau_{\text{SL}}$ and
the boundary conditions on $\alpha^{\prime}$ are given by $\tau_{n}$ on the
inner boundary and $\sigma_{n}$ on the outer boundary, none of the tiles from
the corresponding tilings cross $\alpha,\alpha^{\prime}$ and hence

$\displaystyle\text{flux}(\widetilde{v}_{\tau_{\text{SL}}},\alpha)=\text{flux}(\widetilde{v}_{\ast},\alpha^{\prime})=0,$

where $\widetilde{v}_{\ast}$ is equal to $\widetilde{v}_{\tau_{n}}$ on the
inner boundary of $\partial A$ and is equal to $\widetilde{v}_{\sigma_{n}}$ on
the outer boundary of $\partial A$. Therefore

$\text{imbalance}(U_{\text{shell}})\leq 12\beta
n^{2}+|\text{flux}(\widetilde{f}_{\tau_{\text{SL}}},\alpha)-\text{flux}(\widetilde{f}_{\ast},\alpha^{\prime})|,$
(71)

where $\widetilde{f}_{\ast}=\widetilde{f}_{\tau_{n}}$ on the inner boundary of
$\partial A$ and $\widetilde{f}_{\ast}=\widetilde{f}_{\sigma_{n}}$ on the
outer boundary of $\partial A$.

It remains to bound these flux differences, and this is where we use
information about the boundary conditions. First note that for any surface $X$
and any tiling $\tau$ of $\frac{1}{n}\mathbb{Z}^{3}$, the flux of the non-
rescaled $\widetilde{f}_{\tau}$ and the rescaled $f_{\tau}$ are related by:

$\text{flux}(\widetilde{f}_{\tau},X)=n^{2}\text{flux}(f_{\tau},X).$ (72)

We have that the rescaled versions of the tiling flows $f_{\tau_{n}}$ and
$f_{\tau_{\text{SL}}}$ (rescaled, so without the tildes) converge as
$n\to\infty$ to the $g\in AF(R,b)$ given in the theorem statement, that is,

$\displaystyle\lim_{n\to\infty}d_{W}(f_{\tau_{n}},g)=\lim_{n\to\infty}d_{W}(f_{\tau_{\text{SL}}},g)=0.$

Recall that $T(\cdot,X)$ denotes the trace operator which takes a flow to its
restriction to a surface $X$. By Theorem 5.5.7, for $X$ fixed and any
$f_{1},f_{2}\in AF(R)$, given any $\delta>0$ there exists $\delta_{1}$ such
that if $d_{W}(f_{1},f_{2})<\delta_{1}$ then
$\mathbb{W}_{1}^{1,1}(T(f_{1},X),T(f_{2},X))<\delta$. Recall also that
$T(g,\partial R)=b$, and that we are given that $T(f_{\sigma_{n}},\partial
R)=b_{n}$ converges to $b$ in $\mathbb{W}_{1}^{1,1}$. Given these facts, we
can choose $n$ large enough to guarantee the following:

  * •

For the outer boundary $\partial R$,

$\mathbb{W}_{1}^{1,1}(T(f_{\sigma_{n}},\partial R),b)<\delta$ (73)
$\mathbb{W}_{1}^{1,1}(T(f_{\tau_{\text{SL}}},\partial R),b)<\delta.$ (74)

  * •

For the inner boundary $\partial R^{c}$,

$\mathbb{W}_{1}^{1,1}(T(f_{\tau_{n}},\partial R^{c}),T(g,\partial
R^{c}))<\delta$ (75) $\mathbb{W}_{1}^{1,1}(T(f_{\tau_{\text{SL}}},\partial
R^{c}),T(g,\partial R^{c}))<\delta.$ (76)

  * •

Finally, let $\partial R_{\text{mid}}=\partial(R^{a_{+}}\setminus R^{a_{-}})$
be the piecewise smooth surface approximated by $\partial
A_{\text{mid}}=\partial(R_{n}^{a_{+}}\setminus R_{n}^{a_{-}})$. Then

$\mathbb{W}_{1}^{1,1}(T(f_{\tau_{\text{SL}}},\partial
R_{\text{mid}}),T(g,\partial R_{\text{mid}}))<\delta.$ (77)

Recall also that boundary value flows correspond to measures, and note that
$\text{flux}(f,X)=T(f,X)(X)$ is the total mass of the measure $T(f,X)$ on $X$.
In particular, for a surface $X$ and tiling $\tau$ of
$\frac{1}{n}\mathbb{Z}^{3}$ and $B\subset X$,

$\displaystyle\text{flux}(f_{\tau},B)=T(f_{\tau},X)(B).$

Lemma 5.2.8 applied to measures $\mu,\nu$ supported on a surface $X$ says that
if $\mathbb{W}_{1}^{1,1}(\mu,\nu)<\delta$, then for any $B\subset X$,

$\displaystyle\mathbb{W}_{1}^{1,1}(\mu\mid_{B},\nu\mid_{B})\leq\delta+\delta^{1/2}(C(B)+1),$

where $\delta^{1/2}C(B)$ is bounded by $2$ times the difference of the area of
B and the $\delta^{1/2}$ neighborhood of $B$ within $X$; equivalently, by the
area of the annulus of width $\delta^{1/2}$ with inner boundary $\partial B$
(see Remark 5.2.9).

To use this, we relate $\alpha,\alpha^{\prime}$ which are contained in the
discrete surfaces $\partial A_{\text{mid}}$ and $\partial A$ built out of
$\frac{1}{n}\mathbb{Z}^{3}$ lattice squares, to $B,B^{\prime}$ on the
piecewise smooth surfaces $\partial R_{\text{mid}}$ and $\partial
R\cup\partial R^{c}$ respectively.

By Equation (66), $\text{length}_{n}(\partial\alpha)\leq(2\beta/\epsilon)n$.
Correspondingly the Euclidean length is bounded as
$\text{length}(\partial\alpha)\leq(2\beta/\epsilon)$. Given this, we can cover
$\partial\alpha$ with a collection of cubes $\mathcal{C}$ with Euclidean side
length $3\epsilon$, with $|\mathcal{C}|\leq 2\beta/(3\epsilon^{2})$. Since the
Euclidean width between $\partial A$ and $\partial A_{\text{mid}}$ is less
than $2\epsilon$, $\mathcal{C}$ also covers
$\partial\alpha^{\prime}\subset\partial A$.

The Hausdorff distances between $\partial A$ and $\partial R\cup\partial
R^{c}$ and between $\partial A_{\text{mid}}$ and $\partial R_{\text{mid}}$ are
both bounded by $2/n$. There are corresponding sets $B\subset\partial
R_{\text{mid}}$ and $B^{\prime}\subset\partial R\cup\partial R^{c}$ which
differ from $\alpha,\alpha^{\prime}$ respectively by Hausdorff distance at
most $2/n$. Thus for $n$ large enough, $\mathcal{C}$ also covers
$B,B^{\prime}$. Since $\partial R\cup\partial R^{c}$ and $\partial
R_{\text{mid}}$ are piecewise smooth, there is a constant $C^{\prime}$ such
that the area of either surface restricted to one of the cubes in
$\mathcal{C}$ is at most $C^{\prime}\epsilon^{2}$. Since $|\mathcal{C}|\leq
2\beta/(3\epsilon^{2})$, there is some constant $C$ such that if
$\delta^{1/2}\leq\epsilon$, then

$\displaystyle\delta^{1/2}C(B)$ $\displaystyle\leq C\beta$ (78)
$\displaystyle\delta^{1/2}C(B^{\prime})$ $\displaystyle\leq C\beta.$ (79)

We have that $\text{length}(\partial\alpha)\leq 2\beta/\epsilon$, so the
number of $\frac{1}{n}\mathbb{Z}^{3}$ lattice points along $\partial\alpha$ is
bounded by a constant times $n$ (the constant here depends on
$\beta/\epsilon$). Since the Hausdorff distance between $\alpha$ and $B$ is
bounded by $2/n$, for any tiling $\tau$ of $\frac{1}{n}\mathbb{Z}^{3}$, the
flux of $f_{\tau}$ through a surface is proportional to the number of
$\frac{1}{n}\mathbb{Z}^{3}$ lattice points on the surface times
$\frac{1}{n^{2}}$. Thus for any tiling $\tau$, since $f_{\tau}$ is divergence-
free,

$\displaystyle|\text{flux}(f_{\tau},\alpha)-\text{flux}(f_{\tau},B)|\leq
O(n^{-1}).$ (80)

By Equation (79), since $\text{length}(\partial B^{\prime})\leq
C(B^{\prime})$, also have that $\text{length}(\partial B^{\prime})\leq
C\beta/\epsilon$. Since the Hausdorff distance between $\alpha^{\prime}$ and
$B^{\prime}$ is bounded by $2/n$, the number of $\frac{1}{n}\mathbb{Z}^{3}$
lattice points on a surface between them is also bounded by a constant times
$n$ (the constant here depends on $\beta/\epsilon$). Thus we analogously get
that for any tiling $\tau$ of $\frac{1}{n}\mathbb{Z}^{3}$,

$\displaystyle|\text{flux}(f_{\tau},\alpha^{\prime})-\text{flux}(f_{\tau},B^{\prime})|\leq
O(n^{-1}).$ (81)

Therefore by Lemma 5.2.8, for $\delta$ such that $\delta^{1/2}<\epsilon$,
Equations (73), (75) to relate $f_{*}$ to $g$ on $B^{\prime}$, plus Equation
(79) where we determine the constant $C(B^{\prime})$, and finally Equation
(81) to relate $f_{*}$ on $\alpha^{\prime}$ to $f_{*}$ on $B^{\prime}$, we get
that

$\displaystyle|\text{flux}(f_{*},\alpha^{\prime})-\text{flux}(g,B^{\prime})|\leq\delta+\delta^{1/2}+C\beta+O(n^{-1}),$
(82)

where $f_{*}$ is $f_{\sigma_{n}}$ on the outer boundary of $\partial A$ and
$f_{\tau_{n}}$ on the inner boundary. Similarly, using Equations (74), (76),
and (77) to relate $f_{\tau_{\text{SL}}}$ to $g$, plus Equations (80), (78)
for the constant $C(B)$ and to relate $f_{\tau_{\text{SL}}}$ on $\alpha,B$,
the test tiling $\tau_{\text{SL}}$ satisfies analogous bounds on both $\alpha$
and $\alpha^{\prime}$:

$\displaystyle|\text{flux}(f_{\tau_{\text{SL}}},\alpha^{\prime})-\text{flux}(g,B^{\prime})|$
$\displaystyle\leq\delta+\delta^{1/2}+C\beta+O(n^{-1})$ (83)
$\displaystyle|\text{flux}(f_{\tau_{\text{SL}}},\alpha)-\text{flux}(g,B)|$
$\displaystyle\leq\delta+\delta^{1/2}+C\beta+O(n^{-1}).$ (84)

Since $g$ is divergence-free and takes values with norm bounded between $-1$
and $1$, and $B,B^{\prime}$ differ from $\alpha,\alpha^{\prime}$ by Hausdorff
distance bounded by $2/n$, Equation (67) implies that

$|\text{flux}(g,B)-\text{flux}(g,B^{\prime})|\leq 4\beta+O(n^{-1}).$ (85)

Combining Equation (72) with the above,

$\displaystyle|\text{flux}(\widetilde{f}_{\tau_{\text{SL}}},\alpha)-\text{flux}(\widetilde{f}_{\ast},\alpha^{\prime})|$
(86)
$\displaystyle\leq\bigg{[}|\text{flux}(f_{\tau_{\text{SL}}},\alpha)-\text{flux}(g,B)|+|\text{flux}(g,B)-\text{flux}(g,B^{\prime})|+|\text{flux}(g,B^{\prime})-\text{flux}(f_{*},\alpha)|\bigg{]}n^{2}$
(87) $\displaystyle\leq(2\delta+2\delta^{1/2}+2C\beta+O(n^{-1})+4\beta)n^{2}.$
(88)

Combining this with Equation (69) and Equation (71), we get that

$\displaystyle\text{imbalance}(U)$
$\displaystyle=\text{imbalance}(U_{\text{mid}}^{\prime})+\text{imbalance}(U_{\text{shell}})$
$\displaystyle\leq-\frac{K_{D}c_{2}}{4}n^{2}+(16+2C)\beta n^{2}+2\delta
n^{2}+2\delta^{1/2}n^{2}+O(n).$

The factor $K_{D}c_{2}/4$ and the constant $C$ are fixed independent of
$\delta,\beta$. Implicit here is also the parameter $\epsilon$ that we indent
by.

Taking $\epsilon$ small enough, we can make $\beta$ as small as needed. The
parameter $\delta>0$ is related to the distance between tiling flows and their
limits and is required to satisfy $\delta^{1/2}<\epsilon$, but this can be
guaranteed for $n$ large enough. Therefore for $n$ large enough,
$\text{imbalance}(U)$ will be non-positive, and hence $U$ is not a
counterexample. This completes the proof. ∎

###  8.7 Upper bounds

To complete the proof of the large deviation principles, we need prove the
upper bounds, namely Theorem 8.1.11 for the soft boundary LDP and Theorem
8.2.9 for the hard boundary LDP. We show that the soft boundary upper bound
implies the hard boundary one, and then prove the soft boundary one.

######  Lemma 8.7.1.

Theorem 8.1.11 implies Theorem 8.2.9.

###### Proof.

Recall that $\mu_{n}$ is counting measure on $TF_{n}(R,b,\theta_{n})$ for some
sequence of thresholds $(\theta_{n})_{n\geq 1}$ with $\theta_{n}\to 0$ as
$n\to\infty$ sufficiently slowly.

On the other hand $\overline{\mu}_{n}$ is counting measure on tilings of fixed
regions $R_{n}$ with scale $n$ tileable boundary value $b_{n}$ such that
$b_{n}\to b$ as $n\to\infty$.

We choose the sequence of thresholds $\theta_{n}$ so that there exists $N$
such that if $n\geq N$ then $\mathbb{W}_{1}^{1,1}(b_{n},b)<\theta_{n}$. In
this case, for any $g\in AF(R,b)$, for $n$ large enough,

$\displaystyle\overline{\mu}_{n}(A_{\delta}(g))\leq\mu_{n}(A_{\delta}(g)).$

Therefore if Theorem 8.1.11 holds, then for all $g\in AF(R,b)$,

$\displaystyle\lim_{\delta\to
0}\limsup_{n\to\infty}v_{n}^{-1}\log\overline{\mu}_{n}(A_{\delta}(g))\leq\lim_{\delta\to
0}\limsup_{n\to\infty}v_{n}^{-1}\log\mu_{n}(A_{\delta}(g))\leq\text{Ent}(g).$

This completes the proof of Theorem 8.2.9 from Theorem 8.1.11. ∎

It remains to prove Theorem 8.1.11, namely that for any $g\in AF(R,b)$,

$\displaystyle\lim_{\delta\to
0}\limsup_{n\to\infty}v_{n}^{-1}\log\mu_{n}(A_{\delta}(g))\leq\text{Ent}(g).$

The main idea is “coarse graining”, i.e. that on a very small box, a uniform
random tiling of $R$ looks approximately like a random tiling sampled from a
$\mathbb{Z}^{3}_{\text{even}}$-invariant Gibbs measure of mean current $s$,
where $s$ is the expected mean current on the box.

###### Proof of Theorem 8.1.11.

In this proof, we assume without loss of generality that $R$ is contained in
the unit cube $B=[0,1]^{2}$ (this is just to avoid complicating the proof with
an extra scaling parameter).

Let $\pi_{n,\delta}$ be the uniform probability measure on the set of tilings
$\tau$ with tiling flow $f_{\tau}\in A_{\delta}(g)\cap TF_{n}(R)$ and
satisfying $\mathbb{W}_{1}^{1,1}(T(f_{\tau}),b)<\theta_{n}$. The purpose of
this is so that the partition function of $\pi_{n,\delta}$ is
$Z_{n,\delta}=\mu_{n}(A_{\delta}(g))$.

Tile $\mathbb{R}^{3}$ by translated copies of $B$, each with a translated copy
of $R$ inside it. Let $\Lambda_{n}=\frac{1}{n}\mathbb{Z}^{3}\cap B$. We define
a $\mathbb{Z}^{3}_{\text{even}}$-invariant measure $\nu_{n}$ on tilings of
$\mathbb{Z}^{3}$ as follows (this measure can sample tilings with some double
tiles and some untiled sites). We take an independent sample from
$\pi_{n,\delta}$ on each copy of $R$, and then average over translations by
$x\in\text{even}(\Lambda_{n})$. The measure $\nu_{n}$ samples tilings that are
perfect matchings on the interior of each copy of $R$. All sites in each copy
of $B\setminus R$ which are not covered by a tile connecting it to a copy of
$R$ are empty. Two copies of $R$ might intersect on their boundaries (e.g. in
the case $R=B$), in which case $\nu_{n}$ can sample double tiles. However the
fraction of possible sites where $\nu_{n}$ samples double tiles is bounded by
the fraction of sites in $\partial B$, namely

$\displaystyle\frac{6n^{2}}{n^{3}}=\frac{6}{n}.$

We define a subsequential limit

$\displaystyle\nu:=\lim_{j\to\infty}\nu_{n_{j}}.$

Note that $\nu$ is a $\mathbb{Z}^{3}_{\text{even}}$-invariant measure on
tilings of $\mathbb{Z}^{3}$, allowed to have untiled sites. It can written as
a weighted average of a measure on dimer tilings and the empty ensemble.

Let $\nu_{n,0}$ be defined analogously to $\nu_{n}$, but without averaging
over translations. Let $\nu_{n,x}$ be the version where all tilings are
translated by a fixed $x\in\text{even}(\Lambda_{n})$. The Shannon entropy of
$\pi_{n,\delta}$ is

$\displaystyle n^{-3}\text{Vol}(R)^{-1}\log
Z_{n,\delta}=v_{n}^{-1}\log\mu_{n}(A_{\delta}(g)).$

By construction,

$|\Lambda_{n}|^{-1}H_{\Lambda_{n}}(\nu_{n,0})=n^{-3}\log\mu_{n}(A_{\delta}(g)).$

For any other $x\in\text{even}(\Lambda_{n})$, a sample on $B$ contains pieces
from up to $8$ samples of $\pi_{n,\delta}$. Since $\nu_{n}$ is a uniform
measure, and since there are more tilings when we are allowed more double
tiles,

$\displaystyle H_{\Lambda_{n}}(\nu_{n,x})\geq
H_{\Lambda_{n}}(\nu_{n,0})\qquad\forall\,x\in\text{even}(\Lambda_{n}).$

The specific entropy of $\nu_{n}$ can be computed using any sequence of boxes
$\Delta_{M}$ with $|\Delta_{M}|\to\infty$ as $M\to\infty$. In particular, we
can choose $\Delta_{M}=\Lambda_{Mn}$ so that

$\displaystyle
h(\nu_{n})=\lim_{M\to\infty}|\Lambda_{Mn}|^{-1}H_{\Lambda_{Mn}}(\nu_{n}).$

On each of the $M^{3}$ copies of $\Lambda_{n}$ in $\Lambda_{Mn}$, $\nu_{n}$
samples an independent draw from $\pi_{n,\delta}$. Thus

$\displaystyle
h(\nu_{n})\geq\lim_{M\to\infty}M^{3}|\Lambda_{Mn}|^{-1}H_{\Lambda_{n}}(\nu_{n,0})=|\Lambda_{n}|^{-1}H_{\Lambda_{n}}(\nu_{n,0})=n^{-3}\log\mu_{n}(A_{\delta}(g)).$

On the other hand, since $h$ is upper-semicontinuous,

$\displaystyle\limsup_{j\to\infty}h(\nu_{n_{j}})\leq h(\nu).$

Therefore

$\displaystyle\limsup_{j\to\infty}v_{n_{j}}^{-1}\mu_{n_{j}}(A_{\delta}(g))\leq\frac{1}{\text{Vol}(R)}h(\nu),$

and we have reduced the problem to bounding $h(\nu)$. Define $\varphi_{n}$ to
be the $\nu_{n}$-expected flow, namely

$\displaystyle\varphi_{n}:=Z_{n,\delta}^{-1}\sum_{\tau}f_{\tau},$

where the sum is over tilings $\tau$ in the support of $\pi_{n,\delta}$. We
define a subsequential limit

$\varphi:=\lim_{j\to\infty}\varphi_{n_{j}}.$

Up to taking additional subsequences we can assume that the subsequences for
$\varphi_{n}$ and $\nu_{n}$ are the same. Note that $\varphi\in AF(R)$. Since
$A_{\delta}(g)$ is convex, Theorem 5.3.1 implies that
$\varphi\in\overline{A_{\delta}(g)}$. Therefore

$\frac{1}{\text{Vol}(R)}\int_{R}\text{ent}(\varphi(x))\,\mathrm{d}x=\text{Ent}(\varphi)\leq\sup_{h\in\overline{A_{\delta}(g)}}\text{Ent}(h)=\text{Ent}(g)+o_{\delta}(1).$

The last equality uses that Ent is upper semi-continuous in the Wasserstein
topology (Proposition 7.6.3). This reduces the problem to showing that
$\frac{1}{\text{Vol}(R)}h(\nu)$ is bounded by $\text{Ent}(\varphi)$. To this
end, we partition $B$ into a collection $\mathcal{C}$ of $k^{3}$ smaller cubes
of size $1/k^{3}$. We define a new flow $\alpha_{k}$ supported in $R$ by, for
all $C\in\mathcal{C}$ such that $C\cap R\neq\emptyset$,

$\displaystyle\alpha_{k}(x)=\frac{1}{|C\cap R|}\int_{C\cap
R}\varphi(y)\,\mathrm{d}y\qquad\forall x\in C\cap R.$

For $x\not\in R$, $\alpha_{k}(x)=0$. Since ent is concave (Lemma 7.2.1), by
Jensen’s inequality,

$\displaystyle\int_{C\cap
R}\text{ent}(\alpha_{k}(x))\mathrm{d}x\geq\int_{C\cap
R}\text{ent}(\varphi(x))\mathrm{d}x.$

On the other hand, $\alpha_{k}$ converges to $\varphi$ a.s. and
$|\alpha_{k}|\leq 1$, so $\alpha_{k}$ converges to $\varphi$ in $L^{1}$, hence
by Corollary 7.6.2,

$\displaystyle\lim_{k\to\infty}\text{Ent}(\alpha_{k})=\text{Ent}(\varphi).$

Therefore it is sufficient to show that $\text{Ent}(\alpha_{k})$ is an upper
bound for all $k$. We now define $\nu_{n,C}$ to be $\nu_{n}$ but averaged only
over the translations $x\in\text{even}(\Lambda_{n}\cap C)$ (equivalently,
conditioned on the origin being in $C$). For each $C\in\mathcal{C}$, let
$\nu_{C}$ be a subsequential limit of $\nu_{n,C}$. The measures $\nu_{C}$ are
$\mathbb{Z}^{3}_{\text{even}}$-invariant, and we can choose the subsequences
so that

$\displaystyle\nu=k^{-3}\sum_{C\in\mathcal{C}}\nu_{C}.$

Therefore

$\displaystyle h(\nu)=k^{-3}\sum_{C\in\mathcal{C}}h(\nu_{C}).$

Note that if $C\cap R=\emptyset$, then $\nu_{C}$ is the empty ensemble and
hence in that case $h(\nu_{C})=0$. When $C\cap R\neq\emptyset$, then $\nu_{C}$
splits as the sum of an empty ensemble (corresponding to selecting the origin
in $C\setminus C\cap R$) and a measure on dimer tilings (corresponding to
selecting the origin in $C\cap R$). In a slight abuse of notation we refer to
the mean current of $\nu_{C}$ as the mean current of its component which is a
measure on dimer tilings. To bound $h(\nu_{C})$ when $C\cap R\neq\emptyset$ we
compute this mean current. Recall from Section 2.2 that $s_{0}(\tau)$ is the
vector of the tile at the origin in $\tau$, and the mean current of a
$\mathbb{Z}^{3}_{\text{even}}$-invariant measure $\mu$ can be computed as

$\displaystyle
s(\mu)=\int_{\Omega}s_{0}(\tau)\,\mathrm{d}\mu(\tau)=\mathbb{E}_{\mu}[s_{0}(\tau)].$

We can also compute the mean current by looking at the expected tile direction
over a set of points instead of just looking at the origin. Let
$E(\Lambda_{n})$ denote the edges in $\Lambda_{n}$ oriented from even to odd.
We can similarly define

$s_{n,C}(\tau):=|\text{even}(\Lambda_{n}\cap C\cap R)|^{-1}\sum_{e\in
E(\Lambda_{n}\cap C\cap R)}f_{\tau}(e)e.$

Here note that we intersect with $R$ because if the origin is chosen in
$C\setminus C\cap R$ we get the empty ensemble, and hence the mean current is
not defined. This is the average direction of $f_{\tau}$ over $\Lambda_{n}\cap
C\cap R$, and by $\mathbb{Z}^{3}_{\text{even}}$-invariance $s(\mu)$ can also
be computed

$\displaystyle s(\mu)=|\text{even}(\Lambda_{n}\cap C\cap
R)|^{-1}\sum_{x\in\text{even}(\Lambda_{n}\cap C\cap
R)}\int_{\Omega}s_{0}(\tau+x)\,\mathrm{d}\mu(\tau)=\int_{\Omega}s_{n,C}(\tau)\,\mathrm{d}\mu(\tau).$

Using this, we compute that

$\displaystyle\mathbb{E}_{\nu_{n,C}}[s_{n,C}(\tau)]=Z_{n,\delta}^{-1}\sum_{\tau}\sum_{e\in
E(\frac{1}{n}\mathbb{Z}^{3}\cap C\cap R)}|C\cap
R|^{-1}f_{\tau}(e)e=\text{avg}_{C\cap R}(\varphi_{n}).$

Since $\nu_{C}$ is a subsequential limit of $\nu_{n,C}$ (up to choice of
another subsequence),

$\displaystyle
s_{C}:=s(\nu_{C})=\int_{\Omega}s_{0}(\tau)\,\mathrm{d}\nu_{C}(\tau)=\lim_{j\to\infty}\int_{\Omega}s_{0}(\tau)\,\mathrm{d}\nu_{n_{j},C}(\tau)$
$\displaystyle=\lim_{j\to\infty}\int_{\Omega}s_{n_{j},C}(\tau)\,\mathrm{d}\nu_{n_{j},C}(\tau)$
$\displaystyle=\lim_{j\to\infty}\text{avg}_{C\cap R}(\varphi_{n_{j}}).$

On the other hand, since $\varphi_{n_{j}}\to\varphi$ in $L^{1}$,

$\displaystyle
s_{C}=\limsup_{j\to\infty}\text{avg}_{C}(\varphi_{n_{j}})=\text{avg}_{C}(\varphi).$

Finally we relate $h(\nu_{C})$ to $\text{ent}(s_{C})$. Recall that
$\text{ent}(s):=\max_{\rho\in\mathcal{P}^{s}}h(\rho)$, where $\mathcal{P}^{s}$
is the space of $\mathbb{Z}^{3}_{\text{even}}$-invariant probability measures
on dimer tilings of mean current $s$. The measure $\nu_{C}$ is a sum of an
empty ensemble (corresponding to the origin being chosen in $C\setminus R$)
which has zero entropy and a $\mathbb{Z}^{3}_{\text{even}}$-invariant on dimer
tilings (corresponding to the origin being chosen in $C\cap R$) which has mean
current $s_{C}$. Thus

$\displaystyle h(\nu_{C})\leq\text{ent}(s_{C}).$

Therefore for all $k>1$,

$\displaystyle\limsup_{j\to\infty}v_{n_{j}}^{-1}\mu_{n_{j}}(A_{\delta}(g))$
$\displaystyle\leq\frac{1}{\text{Vol}(R)}h(\nu)=\frac{1}{\text{Vol}(R)}\frac{1}{k^{3}}\sum_{C\in\mathcal{C}}h(\nu_{C})$
$\displaystyle\leq\frac{1}{\text{Vol}(R)}\frac{1}{k^{3}}\sum_{C\in\mathcal{C}}\text{Vol}(R\cap
C)\text{ent}(s_{C})=\text{Ent}(\alpha_{k}).$

Taking $k\to\infty$, this shows that

$\displaystyle\limsup_{n_{j}\to\infty}v_{n_{j}}^{-1}\mu_{n_{j}}(A_{\delta}(g))\leq\text{Ent}(\varphi)=\text{Ent}(g)+o_{\delta}(1).$

Since this holds for any convergent subsequence $n_{j}$, taking $\delta\to 0$
completes the proof.

∎

##  9 Open problems

We mentioned in the introduction that there is literature exploring the local
move connectivity problem, considering moves such as the “flip” and “trit”
illustrated below.

Figure 38: Flip and trit.

Both the flip and the trit amount to finding a cycle in $\mathbb{Z}^{3}$ (of
length $4$ or $6$ respectively) that alternates between membership and non-
membership in $\tau$, and then swapping the members and non-members.
Generally, a cycle swap is a swap of an alternating cycle of length $k$, and a
$k$-swap is a cycle swap for which the cycle has length $k$. It is clear that
any two perfect matchings of the same region can be connected by a sequence of
such swaps of this form (simply by applying swaps to all of the cycles
contained in the union of the two perfect matchings). But it is in general not
so clear whether one can get from any matching to any other using only
$k$-swaps for small $k$.

######  Problem 9.0.1.

Is there a finite $K$ such that for any positive $j$, $m$ and $n$ (at least
one of which is even) it possible to get from any dimer configuration of an
$j\times m\times n$ box to any other via sequence of $k$-swaps with $k\leq K$?
Is this possible using only flips and trits?

The examples we have presented in Section 3 already show the answer to both
questions is no if one replaces boxes with general simply connected regions,
such as those that can be tiled with alternating slabs of brickwork, each
oriented a different direction. If we think in terms of the non-intersecting
path interpretation from Section 1.5, we can see that the existence of taut
patterns like the ones shown there are an obstruction to local move
connectedness.

Progress was made on Problem 9.0.1 just after the first draft of this paper
was released in [HLT23]. In particular their results show that any dimer
tiling of a $j\times m\times n$ box (for $jmn$ even, $j,m,n\geq 2$) admits at
least one flip or trit [HLT23, Theorem 1]. See Section 3 for further
description of their results.

######  Problem 9.0.2.

What can be said about the convergence rate of the mixing algorithm described
in Section 3.3? Is there a more efficient way to sample random perfect
matchings of 3D regions?

######  Problem 9.0.3.

Is there a unique ergodic Gibbs measure corresponding to each mean current in
the interior of $\mathcal{O}$?

######  Problem 9.0.4.

If $\nu_{1}$ and $\nu_{2}$ are ergodic Gibbs measures of the same mean
current, and $(\tau_{1},\tau_{2})$ is sampled uniformly from
$(\nu_{1}\otimes\nu_{2})$, are there necessarily infinitely many infinite
paths in the union of $\tau_{1}$ and $\tau_{2}$?

######  Problem 9.0.5.

What can be said about the typical fluctuations of the flow associated to a
uniformly random perfect matching of a simple region such as a cube or torus?
Do they converge to a natural Gaussian process?

In 2D, Kenyon showed that domino tiling height functions converge in law to
the Gaussian free field [Ken00]. This suggests that the discrete gradients of
the height functions should converge (at least in some sense) to the gradient
of the Gaussian free field. The dual of the discrete gradient (i.e., the
discrete flow) should converge in some sense to the dual of the gradient of
the Gaussian field—which can be shown to be equivalent to the field obtained
by projecting vector-valued white noise orthogonally onto the space of
divergence-free fields. It seems reasonable to conjecture that the same holds
in any dimension.

######  Problem 9.0.6.

Is it the case for $d\geq 3$ that the discrete divergence-free flows obtained
from uniformly random perfect matchings (on a torus or box, say, or in the
$\mathbb{Z}^{3}$ Gibbs measure setting) converge in the fine mesh limit to the
Gaussian random generalized flow obtained by projecting vector-valued white
noise onto the space of divergence-free flows?

######  Problem 9.0.7.

Does there exist a three-dimensional region $R\subset\mathbb{R}^{3}$ and a
boundary condition $b$ for which the Ent maximizer for $(R,b)$ is not unique?
We have shown that such a system would have to be “rigid” in the sense defined
in the introduction (i.e., there is an interior point $x$ such that for any
neighborhood $U$ of $x$ the set $\overline{g(U)}$ must intersect one of the
edges of $\mathcal{O}$). But we have not ruled out the existence of multiple
Ent maximizers.

In fact there do exist two dimensional surfaces $R$ where the corresponding
Ent maximizer is not unique. Consider the “slanted cylinder” below, where the
left and right edges are glued following the numbers in the diagram. Here are
two possible tilings of the slanted cylinder.

Figure 39: Two tilings of the slanted cylinder. The left and right edges are
glued.

Any tiling of the slanted cylinder consists of a choice of north (N) or east
(E) tile for each diagonal, so if the cylinder has height $m$ then it has
$2^{m}$ distinct tilings. Since there is only one choice to make on each of
the diagonal “stripes” (deciding whether to color it blue or orange) the
entropy per site tends to zero as the width of the cylinder tends to infinity,
and the functions obtained as fine-mesh limits of these constructions are all
maximizers of Ent. A slanted cylinder can also be realized as an induced
subgraph of $\mathbb{Z}^{3}$ as shown below.

Figure 40: Front and back sides of the surface of a cube, with five “stripes”
wrapping around it, whose vertices correspond to the squares in Figure 39 and
form a slanted cylinder embedded in $\mathbb{Z}^{3}$. Hall’s matching theorem
implies that every perfect matching of the set of vertices hit by these
stripes is obtained by choosing one of the two possible perfect matchings
within each stripe.

If we try to take a fine mesh limit of this example, we get a region with zero
volume in $\mathbb{R}^{3}$. The question is whether this kind of phenomenon
can arise for regions with non-zero volume that are ordinary subsets of
$\mathbb{R}^{3}$ (as opposed to, say, 3D analogs of the slanted cylinder).
“Thickening” the ribbon-like example above (by taking the union of multiple
ribbon layers, taken on different concentric cubes) does not seem to work, as
a tiling obtained that way need not be locally frozen (trit moves may be
possible at the corners).

######  Problem 9.0.8.

Is there a region $R\subset\mathbb{R}^{3}$ and boundary asymptotic flow $b$
where the Ent maximizing flow takes values on a face of $\partial\mathcal{O}$
within a strict subset of the interior of $R$? Or within a strict subset of
all of $R$? (Does this happen on the boundary of the Aztec octahedron?) For an
example of $(R,b)$ where the limit shape takes values in a face of
$\partial\mathcal{O}$ on all of $R$ (i.e., not only a strict subset of the
region), see Example 8.2.6.

######  Problem 9.0.9.

Given a region $R\subset\mathbb{R}^{3}$ and a flow $b$ on $\partial R$, is
there an elegant way to describe the conditions under which $AF(R,b)$ is
nonempty? In other words, under what conditions does $b$ admit an extension to
$R$ which is an asymptotic flow (measurable, divergence-free, and valued in
$\mathcal{O}$)? Recall that if $R\subset\mathbb{Z}^{3}$ is a discrete region
and $b$ is a discrete vector field on $\partial R\subset\mathbb{Z}^{3}$, then
Hall’s matching theorem or the min cut, max flow principle say that $b$ is
extendable if and only if there is no counterexample region $U\subset R$ such
that $S=\partial U\cap R$ is a type of discrete minimal surface, and any
extension of $b$ would be required to have too much flow across $S$. Is there
a continuum version of Hall’s matching theorem and the min cut, max flow
principle that characterizes when $b$ on $\partial R$ can be extended to an
asymptotic flow—i.e., a statement that $b$ is extendable as long as there is
no “minimal surface” cut $S$ such that any extension of $b$ would be required
to have too much flow across $S$? See e.g. [Str10] for discussion of related
problems.

A particularly simple case of interest is that where $R$ is a polyhedron and
the boundary value $b$ is constant on the faces of the polyhedron.

######  Problem 9.0.10.

Let us try to generalize Aztec prism example from the introduction. Suppose
$R\subset\mathbb{R}^{3}$ is a prism of the form $S\times[0,1]$ (where $S$ is a
two-dimensional region) and $b$ is equal to $0$ on the top and bottom faces of
the prism. Alternatively, one may identify the top and bottom of the prism, to
obtain $S$ cross a circle. We expect that one can show from basic symmetry
that the Ent minimizing flow $g$ has zero flow in the vertical direction, that
its restriction to a slice $S\times\\{x\\}$ does not depend on $x$.
Understanding the behavior within this slice is then a two-dimensional flow
problem. Is this behavior the same as what one would see for the corresponding
two-dimensional dimer model on the slice?

######  Problem 9.0.11.

What can be said about the interfaces between frozen regions on the boundaries
of limit shapes (such as those apparent in the figures in the introduction)?
How large do the fluctuations tend to be?

######  Problem 9.0.12.

The 2D Aztec diamond has four frozen regions (one for each vertex) and the 3D
Aztec octahedron appears to have twelve frozen regions (one for each edge).
One might guess that in the $k$-dimensional analog we would see
$4\binom{k}{2}$ frozen regions, one for each co-dimension-two boundary
simplex. Can anything along these lines be proved, either in 3D or in higher
dimensions?

######  Problem 9.0.13.

In two dimensions, the large deviation theory [CKP01] can be generalized to
many other types of random height function models [She05], even though for
most of these models we cannot compute ent explicitly. For example, instead of
having height differences constrained to $\\{3/4,-1/4\\}$ as in the 2D dimer
model, they could be constrained to some other set, like $\\{-1,1\\}$ or
$\\{-1,0,1\\}$. That raises a natural question for us. To what other discrete
divergence-free flow models in 3D (or in higher dimensions) can the results of
this paper be extended? For example, what if instead of restricting the even-
to-odd flows to lie in $\\{5/6,-1/6\\}$ we restrict them to $\\{-1,1\\}$ or to
some other set? Would the max-flow-min-cut theory available in these settings
allow us to complete the steps that relied on Hall’s matching theorem in this
paper? Could the “chain swapping” arguments used in this paper be adapted to
establish the strict concavity of ent in these settings?

As we mentioned earlier, given a lattice flow $v$ on $\mathbb{Z}^{3}$ one can
define a discrete “curl” that assigns to each oriented plaquette—which
corresponds to an oriented edge of the dual lattice—the flow of $v$ around
that plaquette. One can then a define vector potential function $A_{\tau}$ on
the dual lattice of $\mathbb{Z}^{3}$ whose curl corresponds to the flow
$f_{\tau}$ on $\mathbb{Z}^{3}$, though $A_{\tau}$ is a priori only determined
up to the addition of a vector field with curl zero. Restricting the flow
$f_{\tau}$ to take values in $\\{5/6,-1/6\\}$ is then equivalent restricting
the curl of $A_{\tau}$ to lie in $\\{5/6,-1/6\\}$.

Readers familiar with lattice gauge theory (see [Cha19] for a survey) can tell
a similar story about a constrained lattice connection with gauge group $U(1)$
(the complex unit circle) as follows. Fix some small constant
$\alpha\in(0,\pi)$ and constrain the holonomy around every plaquette (oriented
clockwise as one looks from the even to the odd incident cube) to lie in
$\\{e^{5\alpha i/6},e^{-\alpha i/6}\\}$. Then define a domino to be a pair of
cubes separated by a plaquette with holonomy $e^{5\alpha i/6}$. Since the
product of oriented holonomies around a single cube is zero, each interior
cube belongs to exactly one domino, and (up to boundary conditions) one
expects a uniformly random constrained connection to correspond to a uniformly
random 3D domino tiling.

######  Problem 9.0.14.

Can our large deviation theory be extended to any other types of holonomy-
constrained random connections, Abelian or otherwise? Are there other aspects
of gauge theory for which this perspective is useful?

Although we have not explained this in detail, we believe that all the
arguments of this paper will still apply to the setting where the edges are
periodically “weighted” in the manner described in [KOS06]. For example, one
might consider a weighting that strongly favors edges whose vertices have the
form $(x,y,z)$ and $(x,y,z+1)$ where $z$ is even. If the weight is strong
enough, one can use a standard Peierls argument to show if we are given two
independent samples from the minimal-specific-free-energy ergodic Gibbs
measure, then there are a.s. no infinite paths in their union.

######  Problem 9.0.15.

If we allow periodic weights, as in [KOS06], what can we say about the phase
diagram? Are there some choices of weights for which the double dimer model
a.s. contains no infinite paths and others for which it a.s. contains
infinitely many infinite path? Are there any other possibilities? Can one say,
even on a rough qualitative level, how similar the function ent described here
(and its periodically-edge-weighted analogs) will be to the surface tension
functions described in [KOS06] (which are interesting algebraic geometry
constructions with finitely many singular cusps)? In this generalized setting,
can one say anything about the magnitude of the typical fluctuations of a
random flow, or how such fluctuations might depend on the edge weights?

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