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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@stroke@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{69.5pt}{51.77779pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)$ (31)

Moreover, the operator $\mathsf{J}$ is a formal inverse of the infinite Fock
Kasteleyn operator $\tilde{\mathsf{K}}$ on $\mathsf{G}_{n}$:

$\mathsf{J}\tilde{\mathsf{K}}=\operatorname{Id}_{\text{blacks}},\qquad\tilde{\mathsf{K}}\mathsf{J}=\operatorname{Id}_{\text{whites}}.$

###### Proof.

The proof follows from the construction and the computations below:

  * •

If $\mathsf{b}$ is inside the Aztec diamond, we can extend the value of
$\mathsf{K}^{-1}_{\mathsf{b},\mathsf{w}}$ for $\mathsf{w}$ in $Q_{N}$ and
$Q_{S}$.

    * –

If $\mathsf{w}$ is in $Q_{S}$, the train-tracks of type $\alpha$ separating
$\mathsf{b}$ from $\mathsf{w}$ contribute as zeros of
$g_{\mathsf{b},\mathsf{w}}$. The contour $\mathscr{C}_{2}$ does not contain
any pole of $g_{\mathsf{b},\mathsf{w}}$ and thus, the integral over
$\mathscr{C}_{2}$ of $g_{\mathsf{b},\mathsf{w}}$ in this case is zero. So we
do not need to care about the meaning of “$\mathsf{w}$ is on the right of
$\mathsf{b}$” in this case. The double integral is also zero by a similar
argument: there is no train-track of type $\alpha$ passing below $\mathsf{w}$
(in this quadrant, we see only train-tracks of type $\beta,\gamma,\delta$).
Therefore, the contour $\mathscr{C}_{2}$ does not contain any of the poles in
$v$ of the integrand (for any fixed $u$). Thus the whole integral is $0$.

    * –

A similar result holds when $\mathsf{w}$ belongs to $Q_{N}$, but for a
slightly different reason. Let us evaluate first the double integral in
Equation (17). The contour $\mathscr{C}_{2}$ for the $v$ variable in Equation
(17) contains all the poles of the integrand except the one at $v=u$. So for a
fixed $u$, the integral over $v$ is equal to $-2i\pi$ times the residue at
$v=u$, which is equal to $g_{\mathsf{b},\mathsf{w}}(u)$. We need to integrate
this residue over $u$ along $\mathscr{C}_{1}$.

      * *

If $\mathsf{b}$ is “on the left of $\mathsf{w}$” (meaning here that no
parameter from ${\bm{\gamma}}$ appears as a pole of
$g_{\mathsf{b},\mathsf{w}}$), there is no pole inside $\mathscr{C}_{1}$ so the
double integral is 0. Moreover, the indicator function in front of the single
integral is also zero.

      * *

If on the contrary, $\mathsf{w}$ is “on the right” of $\mathsf{b}$ (there are
poles of type ${\bm{\gamma}}$, but then none of type ${\bm{\delta}}$), So the
only poles of $g_{\mathsf{b},\mathsf{w}}$ are either some $\alpha_{j}$ or some
$\gamma_{j}$. The indicator function is equal to 1, so the final formula is
$-\frac{1}{2i\pi}$ times an integral surrounding all the parameters from
${\bm{\alpha}}$ and all the parameters from ${\bm{\gamma}}$ of
$g_{\mathsf{b},\mathsf{w}}$ but since all the poles are encloses, the total is
again equal to 0.

In this case again, $\mathsf{J}_{\mathsf{b},\mathsf{w}}=0$ for all
$\mathsf{w}\in Q_{N}$.

  * •

We can repeat the same arguments when $\mathsf{w}$ is in the Aztec diamond,
and $\mathsf{b}$ is either in $Q_{W}$ or $Q_{E}$, by looking now first at the
variable $u$ in the double integral. As a consequence we have the four cyan
zero blocks in this infinite inverse.

  * •

We now look at the formula when:

    * –

$\mathsf{w}$ is fixed in $Q_{N}$ (resp. $Q_{S}$) and $\mathsf{b}$ is allowed
to exit the Aztec region to be in $Q_{W}$, $Q_{E}$ and $Q_{S}$ (resp. and
$Q_{N}$);

    * –

$\mathsf{b}$ is fixed in $Q_{W}$ (resp. $Q_{E}$) and $\mathsf{w}$ is allowed
to exit the Aztec region to be in $Q_{N}$, $Q_{S}$ and $Q_{E}$ (resp. and
$Q_{W}$).

The previous arguments again show that in these situations, the value of
$\mathsf{J}_{\mathsf{b},\mathsf{w}}$ is also 0. This is represented by the
orange blocks.

  * •

We then continue computing by taking $\mathsf{b}$ in the Aztec region, and
$\mathsf{w}\in Q_{W}$: We can compute iteratively the value on every white
vertex column by column, which leads at each step a linear equation with a
single unknown. We do the same for $\mathsf{w}\in Q_{E}$. This corresponds to
the green block. We do the same for $\mathsf{w}$ in the Aztec region, and
$\mathsf{b}\in Q_{W}\cup Q_{N}$. This corresponds to the purple region.

  * •

We then discuss the entries when $\mathsf{b}$ and $\mathsf{w}$ are in the same
quadrant. Fix for example $\mathsf{w}$ in $Q_{N}$. We know from computations
above that the value of $\mathsf{J}_{\mathsf{b},\mathsf{w}}$ is 0 if
$\mathsf{b}$ is just outside of $Q_{N}$ (but connected to a white vertex
$\mathsf{w}^{\prime}$ of $Q_{N}$). We can then solve iteratively for all the
values of $\mathsf{J}_{\mathsf{b},\mathsf{w}}$ by solving
$\sum_{\mathsf{b}}\mathsf{K}_{\mathsf{w}^{\prime},\mathsf{b}}A_{\mathsf{b},\mathsf{w}}=\delta_{\mathsf{w}^{\prime},\mathsf{w}}$,
as at each step there will be always a white vertex $\mathsf{w}^{\prime}$ such
that we know the value of $\mathsf{J}_{\mathsf{b},\mathsf{w}}$ for two black
neighbors of $\mathsf{w}^{\prime}$, but the value of the third value is not
computed yet, but is completely determined by the linear equation above. In
particular, we obtain that $\mathsf{J}_{\mathsf{b},\mathsf{w}}=0$ if
$\mathsf{b}$ is not in the light cone of $\mathsf{w}$, and is equal to
$\frac{1}{\mathsf{K}_{\mathsf{w},\mathsf{b}}}$ if $(\mathsf{w},\mathsf{b})$ is
the icy edge attached to $\mathsf{w}$. We proceed in the same way for $Q_{S}$
and, after exchanging the roles of the colors, for $Q_{E}$ and $Q_{W}$. This
determines the entries of the four diagonal blocks $D_{N}$, $D_{S}$, $D_{W}$,
$D_{E}$.

  * •

Finally, we proceed as before to compute the entries when $\mathsf{b}\in
Q_{N}\cup Q_{S}$ and $\mathsf{w}\in Q_{W}\cup Q_{E}$, by propagating known
values on the boundary to the bulk by the linear equation. The actual entries
of that block will not matter for our purposes.∎

It turns out that this inverse of the Kasteleyn operator has a probabilistic
meaning. More precisely, the determinant of minors of $\mathsf{J}$ are related
to local statistics of the Boltzmann measure on the Aztec diamond, as follows:

######  Lemma 24.

Let $e_{1}=(\mathsf{w}_{1},\mathsf{b}_{1}),\ldots
e_{k}=(\mathsf{w}_{k},\mathsf{b}_{k})$ be distinct edges of the Aztec diamond
of size $n$, and
$e_{k+1}=(\mathsf{w}_{k+1},\mathsf{b}_{k+1}),\ldots,e_{l}=(\mathsf{w}_{l},\mathsf{b}_{l})$
be distinct edges in the complement of $\mathsf{A}_{n}$ in $\mathsf{G}_{n}$.
Then

$\left(\prod_{i=1}^{l}\tilde{\mathsf{K}}_{\mathsf{w}_{i},\mathsf{b}_{i}}\right)\det_{1\leq
i,j\leq
l}\mathsf{J}_{\mathsf{b}_{i},\mathsf{w}_{j}}=\mathbb{P}_{\mathsf{A}_{n}}[\text{$e_{1},\ldots,e_{k}$
are dimers}]\times\prod_{i=k+1}^{l}\mathbb{I}_{\\{\text{$e_{i}$ is icy}\\}}$
(32)

where $\mathbb{P}_{\mathsf{A}_{n}}$ is the Boltzmann probability measure on
perfect matchings of the Aztec diamond of size $n$, computed via Theorem 5.

###### Proof.

We can assume that the edges $e_{k+1},\ldots,e_{l}$ are ordered in such a way
that they are grouped in four blocks corresponding to the four quadrants:
$Q_{N}$, $Q_{S}$, $Q_{W}$, $Q_{E}$ (for edges joining a vertex of a quadrant
to a vertex of a neighboring quadrant, consider them as belonging to either
quadrant). In each block, order the edges in such a way that an edge
$e^{\prime}$ in the light-cone of a vertex of another edge $e$ of the same
quadrant should come after $e$ in the list.

Due to the structure of the matrix $\mathsf{J}$, and the order we put on the
edges, the entries of the submatrix

$(\mathsf{J}_{\mathsf{b}_{i},\mathsf{w}_{j}})_{1\leq i,j\leq l}$

in columns indexed by white vertices in the quadrants $Q_{N}$ and $Q_{S}$ are
zero, except maybe in the lower triangular part of the diagonal blocks. The
diagonal entries for these columns are
$\mathbb{I}_{\\{\text{$(\mathsf{w}_{i},\mathsf{b}_{i})$ is
icy}\\}}\frac{1}{\mathsf{K}_{\mathsf{w}_{i},\mathsf{b}_{i}}}$. We can
therefore expand the determinant

$\det_{1\leq i,j\leq l}\mathsf{J}_{\mathsf{b}_{i},\mathsf{w}_{j}}$

first along the columns corresponding to edges in the quadrants $Q_{N}$ and
$Q_{S}$. Then, for similar reasons, we can expand the along the rows
corresponding to edges in the quadrants $Q_{W}$ and $Q_{E}$. Therefore,

$\det_{1\leq i,j\leq l}\mathsf{J}_{\mathsf{b}_{i},\mathsf{w}_{j}}=\det_{1\leq
i,j\leq
k}\mathsf{J}_{\mathsf{b}_{i},\mathsf{w}_{j}}\times\prod_{i=k+1}^{l}\frac{\mathbb{I}_{\\{\text{$(\mathsf{w}_{i},\mathsf{b}_{i})$
is icy}\\}}}{\tilde{\mathsf{K}}_{\mathsf{w}_{i},\mathsf{b}_{i}}},$

which, once multiplied on both sides by
$\left(\prod_{i=1}^{l}\tilde{\mathsf{K}}_{\mathsf{w}_{i},\mathsf{b}_{i}}\right)$
is exactly Equation (32). ∎

The structure of the graph $\mathsf{G}_{n}$ (as of the other infinite graphs
from [Spe07]) is such that constraints at infinity propagate in all the
quadrants up to the boundary of the Aztec region: suppose we are given a dimer
configuration of $\mathsf{G}_{n}$, which is such that in an annular region
(large enough to contain the Aztec diamond part in its interior), all the
dimers are icy edges. Then necessarily, all the icy edges in the four
quadrants (and no other in these parts of the graph) are present in this dimer
configuration. Such a configuration is called _ultimately frozen_. Using the
determinantal formula from Theorem 5 with $\tilde{\mathsf{K}}$ and $J$ turns
out to define a Gibbs measure on dimer configurations on $\mathsf{G}_{n}$,
supported on ultimately frozen configurations.

######  Proposition 25.

Equation (32) from the previous lemma defines a determinantal point process on
edges of $\mathsf{G}_{n}$, which gives a Gibbs measure on the set of dimers
configurations of $\mathsf{G}_{n}$ for the specification given by Fock’s
weights on edges.

This probability measure $\mathbb{P}$ has the property to be _frozen_ outside
of the Aztec diamond: every icy edge appear with probability one, and the
other edges of the four quadrants appear with probability zero, in a random
configuration sampled according to that measure. In other words, $\mathbb{P}$
is the product measure of the Dirac mass supported on the set of icy edges in
the complement of the Aztec diamond and the Boltzmann measure on dimer
configurations of the Aztec diamond $\mathbb{P}_{\mathsf{A}_{n}}$ given by
Fock’s weights.

###### Proof.

The collection of relations $\eqref{eq:infinite_meas_1}$ for all finite
subsets of edges of $\mathsf{G}_{n}$ is a consistent set of finite-dimensional
probability distributions, by the multilinearity of the determinant.

By Kolmogorov’s extension theorem, there is a unique probability measure on
the subset of edges of $\mathsf{G}_{n}$ with those finite dimensional
marginals, which is automatically a determinantal process, supported on dimer
configurations of $\mathsf{G}_{n}$.

The fact that the configuration is frozen outside of the Aztec diamond is a
consequence of Equation (32). This implies that it automatically satisfies the
Dobrushin-Lanford-Ruelle [Dob68, LIR69] condition for large enough annuli,
containing the Aztec diamond in their interior. Therefore, it is automatically
Gibbs. ∎

This inverse $\mathsf{J}$ of the Kasteleyn matrix $\tilde{\mathsf{K}}$ is of
different nature than the inverses defined [BCdT23a]. Whereas the measures
constructed from the latter would correspond to “almost linear” height
profile, the one constructed in the proposition above has a different,
extremal slope in each of the quadrants.

In both cases, the inverses are constructed as contour integrals involving the
family of functions $g_{\mathsf{b},\mathsf{w}}(u)$ in the kernel of
$\tilde{\mathsf{K}}$. This raises the question of constructing other inverses
with that mechanism for other boundary conditions at infinity, for this family
of graphs or generalizations, and further to classify inverses of
$\tilde{\mathsf{K}}$ with a probabilistic meaning.

##  6 Limit shapes

The formula for the inverse of the Kasteleyn matrix is very well suited for
asymptotics analysis.

In this section, we discuss results about limit shapes which can be obtained
directly from the analysis of this formula. Most of them are already present
in the literature. We mostly focus on a setting where parameters
${\bm{\alpha}}$, ${\bm{\beta}}$, ${\bm{\gamma}}$, ${\bm{\delta}}$ are chosen
in a periodic way (which is more general than having periodic weights on
edges, see [BCdT23a, Section 4]).

From far away all typical random dimer configuration of a large Aztec look
almost the same. This statement can be made rigorous by looking at the _height
function_ [Thu90], and saying that this random rescaled height function
converges in probability to a continuous deterministic function, _the limit
shape_ , which maximizes a certain functional.

This has been established for the uniform measure on dimer configurations of
simply connected subgraphs of the square lattice [CKP01], generalized to more
general periodic weights [KOS06, Kuc17].

In the original setup of the uniform measure for the Aztec diamond, Jokush
Propp and Shor [JPS98] proved that the behaviour of the limit shape varies
depending on the position in the domain:

  * •

Outside of the inscribed circle (the _frozen regions_): the limit shape is
linear, and the corresponding dominos display a brickwall pattern, which has
different orientations in each corner.

  * •

inside the inscribed circle (the _liquid region_): the slope of the height
function varies continuously, and all type of edges have a positive
probability of appearance.

The inscribed circle separates _frozen regions_ from a _temperate_ (or
_liquid_ , using the terminology of [KOS06]). It is referred to as the
_arctic circle_ , and the main result of [JPS98] is known as the _arctic
circle theorem_.

This has been extended to periodic weights for which the liquid region is an
ellipse [Joh02], where the fundamental domain still contains a single pair of
white and black vertices. Then it has been obtained for the 2-periodic case
[CJ16] and biased variant [BD23], where the underlying spectral curve has
genus 1, and more recently by Berggren and Borodin [BB23] for a generic
arbitrary genus spectral curve, but with additional assumptions which seem
purely technical.

We claim that Formula (17) allows us to recover and go beyond the results
cited above: having the inverse Kasteleyn operator as an explicit contour
integral is particularly well-suited for asymptotic analysis: it is then
possible to extract from standard saddle point analysis the limiting behaviour
of local probabilities, and from them reconstruct the rescaled expected height
function which would give the limit shape.

We first give in Section 6.1 a short derivation of the arctic circle (ellipse)
theorem, by the saddle point method, as presented in [BBC+17] for the uniform
measure, inspired by its use in [OR03] for plane partitions. Then, in Section
6.2 and 6.3, we explore some limit shape results for Fock’s weights given by
genus 0 and 1 M-curves. Finally we discuss in Section 6.4 how the geometric
arguments of [BB23] can be adapted to this more general context to give a less
restrictive result, see also the forthecoming paper [BBS].

###  6.1 A short derivation of the arctic circle theorem

A short derivation of the arctic circle theorem for uniform weights is
presented in [BBC+17]. We give briefly here a variant for Kenyon’s _critical_
(genus 0) weights , where all the parameters $\alpha_{j}$ (resp. $\beta_{j}$,
$\gamma_{j}$, $\delta_{j}$) are equal to a single value $\alpha$ (resp.
$\beta$, $\gamma$, $\delta$) satisfying the following cyclic order

$\alpha<\gamma<\beta<\delta.$

By [CKP01], we know that the rescaled height function converges in
probability, for the uniform topology, to a deterministic continuous function,
with a gradient contained in some polygon. Our goal is to determine here the
interface between the region where the gradient of the limiting height
function is extremal and where it is not.

The expected gradient of the height function is directly related to the
probability that an edge of given type/orientation at that position is a
dimer. Let us look at a particular edge $\mathsf{e}=(\mathsf{w},\mathsf{b})$
crossed by train-tracks with parameters $\delta$ and $\alpha$, where
$\mathsf{w}$ has coordinates $(\mathsf{w}_{x},\mathsf{w}_{y})=(2i,2j+1)$ and
$\mathsf{b}$ has coordinates $(\mathsf{b}_{x},\mathsf{b}_{y})=(2i-1,2j)$. By
Equation (26), the probability to see this edge in a random dimer
configuration is given by:

$\mathsf{K}_{\mathsf{w},\mathsf{b}}\mathsf{K}^{-1}_{\mathsf{b},\mathsf{w}}=(\alpha-\delta)\frac{1}{(2i\pi)^{2}}\int_{\mathscr{C}_{2}}\int_{\mathscr{C}_{1}}\frac{g_{\mathsf{b},0}(u)g_{0,\mathsf{w}}(v)}{v-u}\left(\frac{(u-\beta)(v-\delta)}{(u-\delta)(v-\beta)}\right)^{n}du\,dv$

where

$\displaystyle g_{0,\mathsf{b}}(u)$
$\displaystyle=\frac{(u-\delta)^{i-1}(u-\alpha)^{j}}{(u-\gamma)^{i}(u-\beta)^{j}},$
$\displaystyle g_{0,\mathsf{w}}(v)$
$\displaystyle=\frac{(v-\gamma)^{i}(v-\beta)^{j}}{(v-\delta)^{i}(v-\alpha)^{j+1}}.$

We are interested in the behaviour of the integrand when $n$ is large, and
$(i/n,j/n)\to(x,y)$, which can be rewritten in this asymptotic regime as:

$\exp(n(F(u;x,y)-F(v;x,y)+o(1)))$ (33)

where

$F(u;x,y)=y\log(u-\alpha)+(1-y)\log(u-\beta)-x\log(u-\gamma)-(1-x)\log(u-\delta).$

The critical points of $F$ are given by a quadratic equation in $u$: there are
therefore two of them (counted with multiplicity). In order to apply the
saddle point, one needs to move continuously the contour to make them pass
through the critical points in the direction for which the critical point will
be a maximum (resp. a minimum) for the real part of $F$ for the variable $u$
(resp. for $v$). In that configuration, the double integral tends to 0
exponentially fast as $N$ goes to infinity. See for example [OR03] where this
technique has been introduced in the context of tilings. When the two critical
points are not on the real line, one needs to make one of the contour cross
the other, at least partially, to make the two contours pass through the two
critical points (and cross orthogonally). By doing so, we get an additional
contribution, given by the integral along the path between the two critical
point of the residue of the integrand when $u=v$, which becomes the main
contribution (the remaining double integral goes to zero by the saddle point
method). See Figure 14.

Figure 14: Left: representation of the two contours of integration when
$\Sigma$ is the Riemann sphere for the inverse Kasteleyn matrix of Section
6.1. The curve $A_{0}$ is the unit circle in this case. Right: deformation of
the contours in the case when the two critical points of $F$ are not on the
unit circle (they are thus on different sides), to be in position to apply the
saddle point method. Deforming the contours like this creates unwanted
contribution to the integral of the residue at $u=v$ along the hatched segment
joining the two critical points of $F$, which has to be subtracted.

The corresponding probability of the corresponding edge converges to a number
strictly between 0 and 1: we are in the so-called _liquid region_ , or _rough
phase_. On the contrary, when the two critical points are both real, the
integral becomes trivial, and the corresponding probability is either 0 or 1:
we are then in a _frozen region_.

The _arctic curve_ describing the transition from the liquid region the a
frozen phase is thus given by the set of coordinates $(x,y)$ for which the two
critical points merge on the real axis.

In our case, these two critical points merge when $x$ and $y$ satisfy the
following equation:

$r^{2}(x-\frac{1}{2})^{2}+r^{2}(y-\frac{1}{2})^{2}+2r(2-r)(x-\frac{1}{2})(y-\frac{1}{2})=r-1$

where

$r=\frac{(\beta-\alpha)(\delta-\gamma)}{(\beta-\gamma)(\delta-\alpha)}=(\alpha,\gamma;\beta,\delta)\in(1,+\infty)$

is the cross ratio of the four points $\alpha,\gamma,\beta,\delta$.

We know by [KO06] that two sets of isoradial critical weights correspond are
gauge equivalent if the corresponding train-track parameters are related by a
Möbius transformation preserving the unit circle. The cross-ratio $r$ being
Möbius invariant, it is expected that the limit shape if a function of this
parameter.

When $r=2$ (for which a representative is when $\alpha$, $\gamma$, $\beta$,
$\delta$ cut the unit circle in four equal arcs), the corresponding Boltzmann
measure is uniform, and the arctic curve is a circle. A cross-ration $r\neq 2$
can be obtained by putting different weight on NE/SW and NW/SE dominos.
Choosing the parameters $\alpha,\gamma,\beta,\delta$ to be $\pm e^{\pm
i\theta/2}$, the ratio between the weights is given by $\tan\frac{\theta}{2}$
and $r=(\cos\frac{\theta}{2})^{-2}$. In this case, the arctic curve is an
ellipse, as proved by Johansson [Joh02, Theorem 2.4].

Figure 15: Arctic regions for (1-)periodic critical weights, for various
values of the cross-ratio $r$ between $\alpha$, $\beta$, $\gamma$ and
$\delta$. Here $r=(\cos\frac{\theta}{2})^{-2}$ and $\theta=k\frac{\pi}{20}$
for $k=1,\ldots,10$. The green circle corresponding to the classical arctic
circle for the uniform measure, for $k=10$.

###  6.2 The (periodic) critical Aztec diamond has no gaseous phase

We assume now that the parameters ${\bm{\alpha}},{\bm{\beta}}$ (resp.
${\bm{\gamma}},{\bm{\delta}}$) are periodic with period $l$ (resp. $k$), for
some $k,l\geq 1$.

The integrand in the formula describing the probability of a single edge has
the same form, as Equation (33), where now the function $F(u;x,y)$ is given by
the formula:

$F(u;x,y)=\frac{1}{l}\sum_{j=1}^{l}\left[y\log(u-\alpha_{j})+(1-y)\log(u-\beta_{j})\right]\\\
-\frac{1}{k}\sum_{j=1}^{k}\left[x\log(u-\gamma_{j})+(1-x)\log(u-\delta_{j})\right].$

The equation for critical points of $F(\cdot;x,y)$ can be rewritten as:

$\frac{1}{l}\sum_{j=1}^{l}\left[\frac{y}{u-\alpha_{j}}+\frac{1-y}{u-\beta_{j}}\right]=\frac{1}{k}\sum_{j=1}^{k}\left[\frac{x}{u-\gamma_{j}}+\frac{1-x}{u-\delta_{j}}\right],$

which by putting everything above the same denominator, becomes a polynomial
equation in $u$ of degree $2k+2l-2$. It has thus $2k+2l-2$ complex roots,
counted with multiplicity.

On the other hand, writing
$u=e^{2i\bar{u}},\alpha_{j}=e^{2i\bar{\alpha}_{j}},\ldots$, and using the fact
that

$\frac{2iu}{u-\alpha}=\frac{e^{i(\bar{u}-\bar{\alpha})}}{\sin(\bar{u}-\bar{\alpha})}=i+\cot(\bar{u}-\bar{\alpha}),$

the same critical equation has the following form:

$\frac{1}{l}\sum_{j=1}^{l}\left[y\cot(\bar{u}-\bar{\alpha}_{j})+(1-y)\cot(\bar{u}-\bar{\beta}_{j})\right]=-\frac{1}{k}\sum_{j=1}^{k}\left[x\cot(\bar{u}-\bar{\gamma}_{j})+(1-x)\cot(\bar{u}-\bar{\delta}_{j})\right]$

when $\bar{u}$ is real, both sides are real. The left-hand side takes all real
values between $-\infty$ and $+\infty$ when $\bar{u}$ is in
$(\bar{\alpha}_{j},\bar{\alpha}_{j+1})$ or
$(\bar{\beta}_{j},\bar{\beta}_{j+1})$, for some $1\leq j\leq l-1$, whereas the
right-hand side stays bounded on those intervals. As a consequence, there is
at least a solution of the critical equation on each of these intervals.
Reasoning in the same way on the intervals
$(\bar{\gamma}_{j},\bar{\gamma}_{j+1})$,
$(\bar{\delta}_{j},\bar{\delta}_{j+1})$, $1\leq j\leq k-1$, exchanging the
roles of the left- and right-hand sides, this gives in total at least
$2k+2l-4$ real roots. Therefore, there are at most two complex, non-real
roots, for the equation in $\bar{u}$, which have to be complex conjugated.
This means that for the original equation in $u=e^{2i\bar{u}}$, the two extra
solutions are not on the unit circle and the reciprocal of the complex
conjugate of one is equal to the other.

Repeating the same saddle-point analysis as above, one can see that the
probability of the considered edge, will have a limit, which will be non
trivial in $(0,1)$ if and only the two extra critical points are not on the
unit circle. When this is the case, let us call $u_{0}$ the one inside the
unit disc. Then one sees that we can find a unique point $(x,y)$ which has
$u_{0}$ (and the inverse of its complex conjugate) as critical points for $F$,
by solving the system of two real linear equations obtained by separating the
real and imaginary part of

$\frac{\partial}{\partial u}F(u_{0};x,y)=0.$

whereas, the edge will be frozen if all the critical points are real. The
argument that this linear system has a unique solution, which is furthermore
in the square $[0,1]$ is an slight adaptation of the proof of [BB23, Theorem
4.9].

The transition between the two regimes occur when the two extra complex
critical points merge at some $u_{0}=e^{2i\bar{u_{0}}}$ on the unit circle,
where they become zeros of the first, and second derivative of $F(\cdot;x,y)$.
The _arctic curve_ separating the two regimes is obtained finding $x$ and $y$
such that

$\begin{cases}\frac{\partial}{\partial u}F(u_{0};x,y)=0\\\
\frac{\partial^{2}}{\partial u^{2}}F(u_{0};x,y)=0\end{cases}$ (34)

for any point $u_{0}$ on the unit circle $S^{1}$. Since $F$ is an affine
function of $x$ and $y$, this amounts to solving again a linear system in $x$
and $y$ with coefficients which are explicit functions of $u_{0}$, yielding an
explicit parametrization of the arctic curve $u_{0}\in
S^{1}\mapsto(x(u_{0}),y(u_{0}))$.

The result above can be summarized in the following proposition:

######  Proposition 26.

For periodic “genus 0” weights, with arbitrary fundamental domain,

  * •

The map from the unit disc to the liquid region is a homeomorphism.

  * •

the arctic curve has a single component and has an explicit parametrization by
trigonometric (or rational) functions. It is thus a real algebraic curve of
genus 0. In particular there is no gaseous phase.

  * •

The tangency points with the sides of the of the square $[0,1]$ at the bottom
(resp. top, left, right) to $u_{0}$ taking one of the values of
${\bm{\alpha}}$ (resp. ${\bm{\beta}}$, ${\bm{\gamma}}$, ${\bm{\delta}}$).

The genus 0 weights are not generic, and is not covered by the results of
Berggren and Borodin [BB23]. The statement about the tangency points has been
noticed by Dan Betea for Schur measures corresponding to the Aztec diamond,
when parameters are periodic (which is essentially Stanley’s weights in the
periodic setting) [Bet15].

The discussion above is mainly about the arctic curve separating the liquid
and the frozen region. Actually, the saddle point analysis allows to find the
frequency of each type of domino near any point of the liquid region, from
which we can reconstruct the expected slope, and then the limit shape.

Figure 16: The arctic curve for critical periodic Aztec diamond (with period 3
vertically, and 2 horizontally).

###  6.3 The elliptic case

We briefly study here the case of elliptic weights, with periodic train-track
parameters. The surface $\Sigma$ is now a torus
$\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ where $\tau\in i\mathbb{R}^{*}_{+}$.

First, as in Section 6.1, let us consider the case where the parameters
$\alpha_{j}$ (resp. $\beta_{j}$, $\gamma_{j}$, $\delta_{j}$) are equal to the
same value $\alpha$ (resp. $\beta$, $\gamma$, $\delta$) belonging to
$\mathbb{R}/\mathbb{Z}$.

Rewriting the probability of a single edge using (25), with the same notation
as for the critical case, we get:

$\mathsf{K}_{\mathsf{w},\mathsf{b}}\mathsf{K}^{-1}_{\mathsf{b},\mathsf{w}}=\int_{\mathscr{C}_{1}}\int_{\mathscr{C}_{2}}\exp(n(F(u;x,y)-F(v;x,y))+o(1))\frac{du\,dv}{\theta_{1}(\pi(v-u))}$

where now

$F(u;x,y)=y\log\theta_{1}(\pi(u-\alpha))+(1-y)\log\theta_{1}(\pi(u-\beta))\\\
-x\log\theta_{1}(\pi(u-\gamma))-(1-x)\log\theta_{1}(\pi(u-\delta)).$

The differential of $F$ is a meromorphic 1-form on the torus $\Sigma$. It has
a divisor of degree $2g-2=0$. Since $F$ is periodic in the horizontal
direction of the torus, the integral of $dF$ along the cycle
$\frac{\tau}{2}+\mathbb{R}/\mathbb{Z}$ is equal to 0. Moreover, since $dF$ is
real on this cycle, the intermediate value theorem implies that $dF$ has at
least two zeros on this cycle. Since it has four simple poles, located at
$\alpha$, $\beta$, $\gamma$ and $\delta$, it must have two additional zeros.

These extra zeros can be either both on the same real component of the torus
(in a frozen or gaseous phase), or both non-real and symmetric, and complex
conjugated (in the liquid phase).

The boundary of the liquid phase is thus obtained by looking at the transition
between the two regimes, when the two extra critical points merge into a
double critical point. As in the critical case, asking for $u$ to be a double
critical point of $F$ yields a linear system in $x$ and $y$ with coefficients
that are given in terms of elliptic functions of $u$. When $u$ runs along the
two real components of the torus, we get two closed curves: the outer curve
separates the liquid phase from the frozen ones in the corners; the inner one
separates the liquid phase from the gas bubble near the center.

Figure 17: Three arctic curves for a torus with $\tau=i\sqrt{3}$. Left: the
limit shape for the unbiased $2\times 2$-periodic Aztec diamond
($\alpha,\beta,\gamma,\delta$ cut the circle into four equal arcs). Center:
the limit shape for the biased weights with $\rho=\frac{1}{6}$
($\alpha=0,\gamma=\frac{1}{6},\beta=\frac{1}{2},\delta=\frac{5}{6}$) and the
same value of $\tau$. Right:
$\alpha=0,\beta=\frac{3}{5},\gamma=\frac{3}{10},\delta=\frac{17}{20}$

Note that having periodic (even constant here) train-track parameters does not
imply that the edge weights are periodic [BCdT23a, Section 4]. In order for
the edge weights to be periodic, for a fundamental domain containing two white
and two black vertices, the parameters $\alpha,\beta,\gamma,\delta$ should
satisfy:

$\alpha-\beta+\gamma-\delta=0\text{ mod
}1,\qquad\alpha-\beta-\gamma+\delta=0\text{ mod }1.$ (35)

which means that $\beta-\alpha=\pm\frac{1}{2}$, and
$\delta-\gamma=\pm\frac{1}{2}$. This case contains in particular the $2\times
2$ periodic Aztec diamond, studied by

  * •

Chhita and Johansson [CJ16] by taking
$\alpha=0,\beta=\frac{1}{2},\gamma=\frac{1}{4},\delta=\frac{3}{4}$ and letting
$\tau$ vary on the imaginary axis,

  * •

Borodin and Duits [BD23], for a specific value of $t$, which in fact does not
play a role in the computation of the limit shape.

These are the cases depicted on the left and middle of Figure 17. An example
where the periodicity condition is _not_ satisfied is shown on the right of
that same figure. Note that in this case, the picture does not have axial
symmetries anymore.

One can extend this study when the parameters ${\bm{\alpha}}$, ${\bm{\beta}}$,
${\bm{\gamma}}$, and ${\bm{\delta}}$ are $k\times l$ period, like in Section
6.2. With the additional constraint that the corresponding edge weights are
periodic, this covers in particular the case $k=2,l=2$ in the study of
Berggren and Borodin [BB23] (without the technical assumption they have on the
weights to guarantee that the vertical tentacles of the amoeba of the
corresponding spectral curve are separated), but also the extension to the
case where the periodicity condition for edge weights is not met, like the one
depicted on Figure 18. One can relate here, like in the critical case, the
number of cusps along bottom, left, top, right side of the square with the
number of distinct values for ${\bm{\alpha}}$, ${\bm{\gamma}}$,
${\bm{\beta}}$, ${\bm{\delta}}$ respectively, whereas there are four of them
along the liquid/gas interface.

Figure 18: Left: a 2x2 example with ${\bm{\alpha}}=\\{0.744904,0.448131\\}$,
${\bm{\beta}}=\\{2.35367,1.58096\\}$, ${\bm{\gamma}}=\\{1.21721,0.983251\\}$,
${\bm{\delta}}=\\{3.05117,2.38214\\}$. Right: a 3x2 example where
${\bm{\alpha}}=\\{0.599648,0.0928766,0.288755\\}$,
${\bm{\beta}}=\\{2.29566,1.7928,2.20249\\}$, ${\bm{\gamma}}=\\{1.18328\\}$,
${\bm{\delta}}=\\{3.06226,2.74699\\}$. The fact that there is a single contact
point on the left boundary (whereas there are two on the right side) is a
consequence of the fact that ${\bm{\gamma}}$ is 1-periodic (constant) whereas
${\bm{\delta}}$ is 2-periodic.

###  6.4 Higher genus

The study above can be done on surfaces with higher genus, with train-track
parameters assigned in a periodic way, even though it can be technically
challenging to work with prime forms to control the saddle point method.
Imposing periodicity for weights from periodicity of train-track parameters
implies that the fundamental domain is large enough (one needs to pick $g$
distinct integer points in the interior points of the corresponding Newton
polygon), but we can relax this condition (and get just quasi-periodic
weights), as in the genus 1 case above.

In [BB23], the authors give a similar looking formula for the inverse of the
Kasteleyn matrix of the Aztec diamond with periodic weights. The main
differences are the fact that the integrals are written with respect to the
coordinates $z$ (and $w$) of the spectral curve instead of an intrisic
coordinate of the Riemann surface $\Sigma$, and the fact that the integrand is
a block matrix111In particular compare the term $E(u,v)$ in Equation (17) and
the term $z_{1}-z_{2}$ in their formula. . Otherwise, unsurprisingly, the
function $F$ appearing in the asymptotics of the integral is the same as here.
We refer to the paper [BB23] for a detailed discussion of the saddle point
method in the periodic case under additional technical assumptions. Working
with formula (17) can be seen as a way to circumvent the technical assumptions
they have to extend their results to non-generic cases, where the spectral
curve is singular or the amoeba has a degenerate behaviour. In that case, the
geometric correspondence is to be made between the limit shape of the Aztec
diamond and the “upper part” $\Sigma^{+}$ of $\Sigma$, which is the connected
component of $\Sigma\setminus\bigcup_{i=0}^{g}A_{i}$ whose oriented boundary
is $A_{0}-\sum_{i=1}^{g}A_{i}$ (instead of the amoeba of the spectral curve).
Most of their geometric arguments can be carried or adapted in this slightly
changed context to obtain the following:

######  Proposition 27 ([BB23]).

  * •

The map $\psi$ from $(x,y)$ in the liquid region to the critical point of $F$
in the interior of $\Sigma^{+}$ is a homeomorphism.

  * •

There is a bijective correspondence between gaseous bubbles and the real ovals
$A_{1},\cdots A_{g}$.

  * •

There is a bijective correspondence between frozen regions and connected
components of $A_{0}\setminus\\{\alpha_{j},\beta_{j},\gamma_{j},\delta_{j}\ ;\
1\leq j\leq n\\}$.

The saddle point method allows one to not only compute the probability that a
given edge occurs, but also to derive the asymptotics for the entry of inverse
Kasteleyn matrix for vertices $\mathsf{b}$ and $\mathsf{w}$ at finite
distance, near the same macroscopic point $(x,y)$ in the liquid region: it
converges to $A^{u_{0}}_{\mathsf{b},\mathsf{w}}$, where $u_{0}=\psi(x,y)$ is
the critical point of $F(\cdot;x,y)$ in $\Sigma^{+}$, and $A^{u_{0}}$ is the
inverse from the family of inverses introduced in [BCdT23a] for the Fock
Kasteleyn operator on any infinite minimal graph containing as a subgraph a
neighborhood of the Aztec diamond containing $\mathsf{w}$ and $\mathsf{b}$
(for example the infinite square lattice). The inverse $A^{u_{0}}$ on the
infinite square defines a Gibbs measure $\mathbb{P}^{u_{0}}$ on dimer
configurations by the usual determinantal formula:

$\mathbb{P}^{u_{0}}((\mathsf{w}_{1},\mathsf{b}_{1}),\dots,(\mathsf{w}_{k},\mathsf{b}_{k}))=\Bigl{(}\prod_{j=1}^{k}\mathsf{K}_{\mathsf{w}_{j},\mathsf{b}_{j}}\Bigr{)}\det_{1\leq
i,j\leq k}\left(A^{u_{0}}_{\mathsf{b}_{i},\mathsf{w}_{j}}\right)$

Therefore, one can obtain in this generalized setup the analogue of [BB23,
Theorem 1.10]:

######  Proposition 28.

The local statistics of dimers around a point $(x,y)$ of the liquid region
converge to the infinite Gibbs measure $\mathbb{P}^{u_{0}}$ on the square
lattice, with $u_{0}=\psi(x,y)$.

We also note that the imaginary part of the equation used to determine the
critical points of $F$, is the same as the master equation for the _tangent
plane method_ by Kenyon and Prause [KP22b, KP22a, KP24]: indeed, for a fixed
$u$, the action $F$ is an affine function of $x$ and $y$, and the coefficients
of $x$ and $y$ (respectively $\log z(u)$ and $\log w(u)$) which are analytic
functions of $u$, and whose imaginary parts give the slope of the limit shape
at the point $(x,y)$, see [BCdT23a, Section 4.5]. The remaining constant part
is encoding the boundary conditions and should be directly related to the
_intercept function_. We have thus two interpretations of the same equation:
one coming from finding critical points of the action functional $F$, the
other other one by looking at _isothermal coordinates_ in which the slope and
the intercept of the asymptotic height function describing the limit shape are
harmonic. Further investigation is needed to make a full connection between
the two approaches. A starting point is the expression of the expression of
the surface tension in terms of geometric quantities on $\Sigma$, obtained in
[BCdT23a, Section 4.6], see also [BBS].

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