along two marked geodesics is shown in a similar fashion. We leave the details
to the reader. ∎

##  4 Proof of Theorem 1

Let $g$, $k\in\mathbb{Z}_{+}$ be fixed; as in Section 2.2, we exclude the
cases $(g,k)\in\\{(0,0),(0,1)\\}$ of the sphere, the pointed sphere, or the
disk. Recall that the case $(g,k)=(0,0)$ of the sphere is already known. The
case $(g,b,k)=(0,1,1)$ of the disk is partially known but has not been treated
in the complete setting of Theorem 1. In fact, it may actually enter the
following framework: in this case, the decomposition of Section 2.2 yields
only one well-labeled forest (and thus one unique slice) indexed by a
degenerate scheme with one external face having one unique vertex with a self-
loop edge. This creates a small ambiguity coming from the choice of the first
tree in the forest, which can be overcome by randomization when considering
random maps. The case $(g,b,k)=(0,0,1)$ of the pointed sphere would yield an
even more degenerate decomposition with a one-vertex map as a scheme and a
unique composite slice of width $0$, object that is not introduced in this
work.

Instead of considering these extensions and objects, we rather obtain these
cases by using Proposition 3 at the end of Section 1.4. More precisely,
provided Theorem 1 holds for $(g,k)=(0,2)$, we infer the case $(g,k)=(0,1)$ as
follows. Let $l_{n}^{1}\in\mathbb{Z}_{+}$ be such that
$l_{n}^{1}/{\sqrt{2n}}\to L^{1}$ as $n\to\infty$ and $Q_{n}$ be uniform in
$\smash{{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,(l_{n}^{1})}^{[0]}}$.
Then let us choose a vertex uniformly at random among the internal vertices of
$Q_{n}$ and denote by $Q_{n}^{\bullet}$ the map obtained from $Q_{n}$ by
declaring the chosen vertex as a second hole. Since $Q_{n}^{\bullet}$ is
clearly uniform in
$\smash{{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,(l_{n}^{1},0)}^{[0]}}$,
the case $(g,k)=(0,2)$ of Theorem 1 implies the convergence of the
corresponding metric measure space toward
$\smash{\mathsf{S}^{[0]}_{(L^{1},0)}}$, and finally the convergence of the
metric measure space corresponding to $Q_{n}$ toward the space
$\mathsf{S}^{[0]}_{(L^{1})}$, defined as $\mathsf{S}^{[0]}_{(L^{1},0)}$ with
its second mark and second boundary measure forgotten.

###  4.1 Gluing quadrangulations from elementary pieces

We start by interpreting the observations of Section 2.2 in the light of the
previous section for a deterministic map. Let $n\in\mathbb{Z}_{+}$ and
${\boldsymbol{l}}\in\mathbb{N}^{k}$ be fixed, let
$\mathbf{q}\in\mathbf{Q}^{[g]}_{n,{\boldsymbol{l}}}$ and $v_{*}\in
V(\mathbf{q})$. The CVS construction being one-to-one, there is a unique
labeled map $(\mathbf{m},\lambda)\in\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}$
that corresponds to $(\mathbf{q},v_{*})$. We denote by $\mathbf{s}$ the scheme
of $\mathbf{m}$ and by $(\mathrm{EP}^{e},e\in\vec{E}(\mathbf{s}))$ the
collection of elementary pieces of $(\mathbf{q},v_{*})$. We emphasize that the
decomposition strongly depends on the distinguished point $v_{*}$ and not only
on $\mathbf{q}$.

If $e\in\vec{B}(\mathbf{s})$, we let $\gamma^{e}$, $\xi^{e}$ and $\beta^{e}$
be the maximal geodesic, shuttle, and base of the slice $\mathrm{EP}^{e}$, and
we let $\mu^{e}$, $\nu^{e}$ be the associated area and base measures defined
at (6). If $e\in\vec{I}(\mathbf{s})$, we let $\gamma^{e}$, $\xi^{e}$,
$\bar{\gamma}^{e}$, $\bar{\xi}^{e}$ be the maximal geodesics and shuttles of
$\mathrm{EP}^{e}$, where the first two correspond to $I_{e}$ and the latter
two to $I_{\bar{e}}$, in the notation of Section 2.2; we also let $\mu^{e}$ be
the associated area measure defined at (7). Note that $\mathrm{EP}^{\bar{e}}$
yield the same map as $\mathrm{EP}^{e}$ with the same measure; only the marks
are ordered differently, namely $\bar{\gamma}^{e}$, $\bar{\xi}^{e}$,
$\gamma^{e}$, $\xi^{e}$.

The construction of $\mathbf{q}$ from $(\mathbf{m},\lambda)$ consists in
connecting every corner of $\mathbf{m}$ to its successor, and the paths
following consecutive successors are geodesic paths all aiming toward $v_{*}$.
On the other hand, the construction of the elementary pieces from
$(\mathbf{m},\lambda)$ consists in performing the interval CVS bijection on
every interval $I_{e}$, in the notation of Section 2.2. The only difference
between these constructions lies on the shuttles of these elementary pieces:
if $c$ is a corner in some interval $I_{e}$ whose successor in the interval
bijection belongs to the shuttle $\xi^{e}$, then in order to obtain
$\mathbf{q}$ we should rather connect $c$ to its successor $s(c)$ in the
contour order around $f_{*}$. This successor will belong to some interval
$I_{e^{\prime}}$ arriving later in contour order around $f_{*}$ – note that
this interval can be $I_{e}$ itself. Note also that this successor $s(c)$
belongs to the maximal geodesic $\gamma^{e^{\prime}}$. Moreover, interpreting
$\mathbf{q}$ and its elementary pieces as compact geodesic marked metric
spaces, it is straightforward to see that the identifications correspond to
metric gluings along geodesics.

##### Iterative gluing procedure.

In order to reconstruct $\mathbf{q}$ from $\mathrm{EP}^{e}$,
$e\in\vec{E}(\mathbf{s})$, rather than connecting the shuttle vertices to
their actual successors all at once, we will proceed progressively by first
connecting only those whose successors belong to the maximal geodesic of the
elementary piece that arrives immediately after in contour order around
$f_{*}$.

We now formalise this idea. Let $\kappa$ denote the cardinality of
$\vec{E}(\mathbf{s})$. We arrange the half-edges $e_{1}$, …, $e_{\kappa}$
incident to the internal face of $\mathbf{s}$ according to the contour order,
starting at an arbitrarily chosen half-edge. While following the contour of
the internal face $f_{*}$ of $\mathbf{m}$, we successively visit the
elementary pieces $\mathrm{EP}^{e_{i}}$, $1\leq i\leq\kappa$, which are
themselves viewed as marked measured geodesic metric spaces. The
reconstruction of $\mathbf{q}$ will be done recursively in $\kappa$ steps,
resulting in a sequence of marked $k+1$-measured metric spaces
$\mathbf{q}_{0}$, …, $\mathbf{q}_{\kappa}$. At the $i$-th step,
$\mathbf{q}_{i+1}$ will be obtained from $\mathbf{q}_{i}$ by gluing
$\mathrm{EP}^{e_{i}}$ along (part of) its marked maximal geodesic
$\gamma^{e_{i}}$. At the same time, we will do some operations on the markings
and measures, namely reorderings, unions of marks and sums of measures, which
are all continuous by Lemma 18.

We need to keep track of the boundary marks, as well as the geodesics yet to
be glued as marks. More precisely, the marking of $\mathbf{q}_{i}$ is
$(\gamma_{i}^{0},\xi_{i}^{0},\gamma_{i}^{1},\xi_{i}^{1},\ldots,\gamma_{i}^{u_{i}},\xi_{i}^{u_{i}},\beta_{i}^{1},\ldots,\beta_{i}^{k})$,
where

  * •

$\gamma_{i}^{j}$, $\xi_{i}^{j}$, $0\leq j\leq u_{i}$, are geodesic marks,

  * •

$\beta_{i}^{1}$, …, $\beta_{i}^{k}$ are called the _boundary marks_. By
convention, certain of these marks may be empty, in which case they are simply
discarded from the marks.

The mark $\xi_{i}^{0}$ is the mark along which the subsequent gluing producing
$\mathbf{q}_{i+1}$ will occur, and $u_{i}$ represents the number of
quadrilaterals that have been involved only once in the gluing procedures up
to the $i$-th step. Each of these quadrilateral yields two marks
$\gamma_{i}^{j}$, $\xi_{i}^{j}$ for some $j\in\\{1,\dots,u_{i}\\}$,
corresponding to the unvisited half of the quadrilateral, which will have to
be glued at a further step. Finally, each $\mathbf{q}_{i}$ will come with
measures $\mu_{i}$, ${\boldsymbol{\nu}}_{i}$ where $\mu_{i}$ is called the
_area measure_ and ${\boldsymbol{\nu}}_{i}=(\nu_{i}^{1},\ldots,\nu_{i}^{k})$
is the $k$-tuple of _boundary measures_.

We initiate the construction by letting $u_{0}=0$,
$\mathbf{q}_{0}\in\mathbb{M}^{(2,k+1)}$ be the point space with the two marks
$\gamma_{0}^{0}$, $\xi_{0}^{0}$ being the unique point, and measures
$\mu_{0}=0$, ${\boldsymbol{\nu}}_{0}={\boldsymbol{0}}^{k}$. We also let all
the boundary marks be empty.

Next, provided that $\mathbf{q}_{i-1}$ has been constructed for some
$i\in\\{1,\ldots,\kappa\\}$, we define $\mathbf{q}_{i}$ by considering the
following cases, depicted on Figures 8 to 10.

  * •

If $e_{i}\in\vec{B}_{r}(\mathbf{s})$ for some $r\in\\{1,\ldots,k\\}$, meaning
in particular that $\mathrm{EP}^{e_{1}}$ is a slice, we set

$\mathbf{q}_{i}=G\big{(}\mathbf{q}_{i-1},\mathrm{EP}^{e_{i}};\xi_{i-1}^{0},\gamma^{e_{i}}\big{)}\,,$

and mark it as follows. We update the boundary marks by setting

$\beta_{i}^{r}=\beta_{i-1}^{r}\cup\beta^{e_{i}}\,,\qquad\beta_{i}^{r^{\prime}}=\beta_{i-1}^{r^{\prime}}\
\text{ for }r^{\prime}\in\\{1,\ldots,k\\}\setminus\\{r\\}\,.$

We update the geodesic marks by letting888In (17), we use the convention set
in Section 3.3 for marks in a glued space: in particular, after gluing, one of
the marks $\gamma^{e_{i}}$, $\xi_{i-1}^{0}$ is contained in the other,
depending on which is longest.

$\gamma_{i}^{0}=\gamma_{i-1}^{0}\cup\big{(}\gamma^{e_{i}}\setminus\xi_{i-1}^{0}\big{)}\,,\qquad\xi_{i}^{0}=\xi^{e_{i}}\cup\big{(}\xi_{i-1}^{0}\setminus\gamma^{e_{i}}\big{)}\,,$
(17)

and, setting $u_{i}=u_{i-1}$, we let $\gamma_{i}^{j}=\gamma_{i-1}^{j}$ and
$\xi_{i}^{j}=\xi_{i-1}^{j}$ for $1\leq j\leq u_{i}$. Finally, we update the
measures by

$\mu_{i}=\mu_{i-1}+\mu^{e_{i}}\,,\qquad\nu_{i}^{r}=\nu_{i-1}^{r}+\nu^{e_{i}}\,,\qquad\nu_{i}^{r^{\prime}}=\nu_{i-1}^{r^{\prime}}\
\text{ for }r^{\prime}\in\\{1,\ldots,k\\}\setminus\\{r\\}\,.$

This case is illustrated in Figure 8.

  * •

If $e_{i}\in\vec{I}(\mathbf{s})$, meaning in particular that
$\mathrm{EP}^{e_{1}}$ is a quadrilateral, we keep the boundary marks unchanged
by setting $\beta_{i}^{r}=\beta_{i-1}^{r}$ for $1\leq r\leq k$, we set
${\boldsymbol{\nu}}_{i}={\boldsymbol{\nu}}_{i-1}$, and consider the following
two possible situations.

    * –

If $e_{i}\notin\\{\bar{e}_{j},\,1\leq j<i\\}$, that is, if the unoriented edge
corresponding to $e_{i}$ is visited for the first time, we let again

$\mathbf{q}_{i}=G\big{(}\mathbf{q}_{i-1},\mathrm{EP}^{e_{i}};\xi_{i-1}^{0},\gamma^{e_{i}}\big{)}\,,$

and update its geodesic marks as follows. We update the first two geodesic
marks by (17). We set $u_{i}=u_{i-1}+1$ and let
$\gamma_{i}^{j}=\gamma_{i-1}^{j}$ and $\xi_{i}^{j}=\xi_{i-1}^{j}$ for $1\leq
j\leq u_{i}-1$. Finally, we set $\gamma_{i}^{u_{i}}=\bar{\gamma}^{e_{i}}$,
$\xi_{i}^{u_{i}}=\bar{\xi}^{e_{i}}$, and $\mu_{i}=\mu_{i-1}+\mu^{e_{i}}$. This
case is illustrated in Figure 9.

    * –

If $e_{i}\in\\{\bar{e}_{j},\,1\leq j<i\\}$, say $e_{i}=\bar{e}_{\ell}$, that
is, if the unoriented edge corresponding to $e_{i}$ is visited for the second
time, then $\gamma^{e_{i}}=\bar{\gamma}^{e_{\ell}}$ is a mark of
$\mathbf{q}_{i-1}$: it is the mark $\gamma^{e_{i}}=\gamma_{\ell}^{u_{\ell}}$
of $\mathbf{q}_{\ell}$ and stays a mark of the subsequent spaces
$\mathbf{q}_{\ell+1}$, …, $\mathbf{q}_{i-1}$. Similarly,
$\xi^{e_{i}}=\bar{\xi}^{e_{\ell}}$ is a mark of $\mathbf{q}_{i-1}$. We let

$\mathbf{q}_{i}=G\big{(}\mathbf{q}_{i-1};\xi_{i-1}^{0},\gamma^{e_{i}}\big{)}\,,$

we update the first two geodesic marks by (17), and, setting
$u_{i}=u_{i-1}-1$, we let $(\gamma_{i}^{j},\xi_{i}^{j},1\leq j\leq u_{i})$ be
the the sequence $(\gamma_{i-1}^{j},\xi_{i-1}^{j},1\leq j\leq u_{i-1})$ from
which the terms $\gamma^{e_{i}}$ and $\xi^{e_{i}}$ have been removed. Finally,
we set $\mu_{i}=\mu_{i-1}$. This case is illustrated in Figure 10.

Figure 8: The gluing procedure in the case where $\mathrm{EP}^{e_{i}}$ is a
slice. In this picture and the following ones, the black wiggly curve depicts
all the marks different from $\gamma_{i-1}^{0}$, $\xi_{i-1}^{0}$. The reader
should bear in mind that, in general, $\mathbf{q}_{i-1}$ has no reason to
present a planar topology as in these pictures. The boundary $\xi_{i-1}^{0}$
of $\mathbf{q}_{i-1}$ is glued to the maximal geodesic $\gamma^{e_{i}}$, and
the base of $\mathrm{EP}^{e_{i}}$ is added to the $r$-th boundary mark of
$\mathbf{q}_{i-1}$ whenever $\bar{e}_{i}$ is incident to $\mathlcal{h}_{r}$.
The first two geodesic marks are updated according to (17), which leads to the
two alternative situations described in this figure, depending on which of
$\xi_{i-1}^{0}$ and $\gamma^{e_{i}}$ is the longest: the unglued part of these
geodesics becomes part of $\xi_{i}^{0}$ or of $\gamma_{i}^{0}$. Figure 9: The
gluing procedure in the case where $\mathrm{EP}^{e_{i}}$ is a quadrilateral
that was not involved previously in the construction. The boundary
$\xi_{i-1}^{0}$ of $\mathbf{q}_{i-1}$ is glued to the maximal geodesic
$\gamma^{e_{i}}$. The first two geodesic marks are again updated according to
(17), leading to two possible situations depending on which of $\xi_{i-1}^{0}$
and $\gamma^{e_{i}}$ is the longest. Only one of these situations is
represented on this figure. In this case, the two geodesic boundary marks
$\bar{\gamma}^{e_{i}}$ and $\bar{\xi}^{e_{i}}$ are added to the marking; they
will be involved in a later construction step. Figure 10: The gluing procedure
in the case where $\mathrm{EP}^{e_{i}}$ is a quadrilateral, one side of which
was already involved in a previous construction step. The boundary
$\xi_{i-1}^{0}$ of $\mathbf{q}_{i-1}$ is glued to the maximal geodesic
$\gamma^{e_{i}}$, which had been introduced as a geodesic mark in this
previous construction step. The first two geodesic marks are again updated
according to (17), and the geodesic marks $\gamma^{e_{i}}$, $\xi^{e_{i}}$ are
removed from the remaining marks.

It is important to notice that, in $\mathbf{q}_{i}$, all the marks
$\gamma_{i}^{j}$, $\xi_{i}^{j}$, $0\leq j\leq u_{i}$, are geodesic marks.
Indeed, each of these paths always take the form of a chain of consecutive
successors, which therefore must be a geodesic; more precisely, these are the
maximal geodesics and shuttles of the interval CVS bijection on the intervals
$I_{e_{1}}\cup\ldots\cup I_{e_{i}}$ and $I_{\bar{e}_{j}}$ for each $j\leq i$
such that
$e_{j}\in\vec{I}(\mathbf{s})\setminus\\{\bar{e}_{1},\dots,\bar{e}_{i}\\}$.

At the end of this inductive procedure, we have connected all shuttle corners
of some interval $I_{e_{i}}$, to their actual successors in $\mathbf{m}$
whenever these lie on some $I_{e_{i^{\prime}}}$ with $1\leq
i<i^{\prime}\leq\kappa$. It remains to connect the shuttle corners in some
$I_{e_{i}}$ whose actual successor in $\mathbf{m}$ lies in some
$I_{e_{i^{\prime}}}$ with $1\leq i^{\prime}\leq i\leq\kappa$. But one can
observe that $u_{\kappa}=0$, so that $\mathbf{q}_{\kappa}$ carries exactly two
geodesic marks $\gamma_{\kappa}^{0}$, $\xi_{\kappa}^{0}$. The shuttle corners
yet to be connected are exactly those of $\xi_{\kappa}^{0}$, and should be
matched to the successive corners of $\gamma_{\kappa}^{0}$. Therefore, as
marked metric spaces, we have
$\mathbf{q}=G(\mathbf{q}_{\kappa};\gamma_{\kappa}^{0},\xi_{\kappa}^{0})$, with
marks $\beta_{\kappa}^{r}$, $1\leq r\leq k$, which are precisely the connected
components of the boundary $\partial\mathbf{q}$, ordered as they should.

It is also possible to view all these gluing operations at once, as shown on
Figure 11.

Figure 11: Reconstructing $\mathbf{q}$ by gluing its elementary pieces along
geodesics. On this example, we have $\kappa=14$. Although we used the same
scheme as in Figure 3 without $\mathlcal{h}_{3}$ (for lower complexity),
beware that the labels here do not match those from the left of Figure 3. On
the top, the half-edges of $\vec{E}(\mathbf{s})$ are arranged according to the
contour order. With each one of them corresponds a “triangle,” which is either
a slice (here, with $e_{1}$, $e_{5}$, $e_{8}$, $e_{11}$) or “half” a
quadrilateral, depending whether the half-edge belongs to
$\vec{B}(\mathbf{s})$ or $\vec{I}(\mathbf{s})$. The five matchings of the
corresponding “halves” of quadrilaterals, that is, the matchings of half-edges
in $\vec{I}(\mathbf{s})$, are represented with light colors. The vertices of
these triangles are represented at a height corresponding to their label,
where $\lambda_{*}^{i}=\min_{I_{e_{i}}}\lambda-1$. The geodesic marks of the
pieces may be involved in multiple gluings. For instance, the shuttle of the
leftmost triangle is involved in three gluings; it is split in three parts,
corresponding to the triangles that can be “seen” to its right (those
corresponding to $e_{2}$, $e_{5}$, and $e_{7}$).

##### Measures.

We claim that the previous equality
$\mathbf{q}=G\big{(}\mathbf{q}_{\kappa};\gamma_{\kappa}^{0},\xi_{\kappa}^{0}\big{)}$
only holds as $k$-marked $k+1$-measured metric spaces up to a difference in
the supports consisting of a bounded number of vertices. When considering
rescaled measures through the operator $\Omega_{n}$, this small difference
will be of no importance in our limiting arguments.

First of all, the boundary measures of the external faces match. This is
because the boundary of a given external face of $\mathbf{q}$ is made of the
bases, say $\beta^{e_{i_{1}}}$, …, $\beta^{e_{i_{j}}}$, of several slices that
satisfy
$\beta^{e_{i_{\ell}}}\cap\beta^{e_{i_{\ell+1}}}=\\{\bar{\rho}^{e_{i_{\ell}}}\\}$
for $1\leq\ell\leq j$ with $i_{j+1}=i_{1}$ and where we denoted by
$\bar{\rho}^{e}$ the final point of the base of $\mathrm{EP}^{e}$. Since the
base measure of the slice $\mathrm{EP}^{e}$ is the counting measure on
$\beta^{e}\setminus\\{\bar{\rho}^{e}\\}$, the resulting measure in the gluing
is the counting measure on the boundary of the considered external face of
$\mathbf{q}$, as desired.

For the boundary measures of the external vertices, the measure in
$\mathbf{q}$ is the counting measure on the singleton consisting of the
external vertex, while the corresponding boundary measure in the gluing is the
zero measure.

For the area measure, by convention, we decided that in the elementary pieces,
the measures were taken on vertices _outside of the shuttles_. In doing so,
after each gluing operation where a piece of a shuttle is glued to a piece of
a maximal geodesic, the corresponding vertices are counted only once, as they
should, _except possibly for the vertices $\rho$, $\bar{\rho}$ of the
quadrilaterals_, since they lie both on a maximal geodesic and on a shuttle,
and are therefore not part of the counting measures by convention. Therefore,
the final gluing is naturally equipped with an area measure that is the
counting measure on all but at most $2(2g+k-1)$ vertices, which is an upper
bound on the number of vertices of the scheme by Lemma 7.

##### Conclusion.

We finally observe that the number $\kappa$ of gluing operations necessary to
obtain $\mathbf{q}$ is uniformly bounded in $n$. Indeed, the number of edges
of $\mathbf{s}$ is, by Lemma 7, smaller than $3(2g+k-1)$.

As a result, Theorem 1 will directly follow from subsequent applications of
Proposition 22 once we will have shown that, after a proper scaling, the
elementary pieces of a uniform quadrangulation jointly converge in
distribution in the GHP topology.

###  4.2 Scaling limit of the collection of elementary pieces

We now fix ${\boldsymbol{L}}=(L^{1},\dots,L^{k})\in[0,\infty)^{k}$. We let $b$
and $p$ be the numbers of indices $i$ such that $L^{i}>0$ and $L^{i}=0$
respectively. In order to ease notation, we assume that $L^{1}$, …, $L^{b}>0$
while $L^{b+1}$, …, $L^{k}=0$.

##### Limiting measure for size parameters.

We denote by
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{S}}}}^{\star}$ the
set of rooted genus $g$ schemes with $k$ holes, $\mathlcal{h}_{1}$, …,
$\mathlcal{h}_{b}$ being faces, $\mathlcal{h}_{b+1}$, …, $\mathlcal{h}_{k}$
being vertices of degree $1$, and whose internal vertices are all of degree
exactly $3$. These are called _dominant_ schemes. Let
$\mathbf{s}\in\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{S}}}}^{\star}$
be fixed and denote its root vertex by $v_{0}$. We let
$\mathcal{T}_{\mathbf{s}}$ be the set of tuples

$\Big{(}(\mathsf{a}^{e})_{e\in\vec{E}(\mathbf{s})},(\mathsf{h}^{e})_{e\in\vec{I}(\mathbf{s})},(\mathsf{l}^{e})_{e\in\vec{B}(\mathbf{s})},(\lambda^{v})_{v\in
V(\mathbf{s})}\Big{)}\in(\mathbb{R}_{+})^{\vec{E}(\mathbf{s})}\times(\mathbb{R}_{+})^{\vec{I}(\mathbf{s})}\times\mathbb{R}^{\vec{B}(\mathbf{s})}\times\mathbb{R}^{V(\mathbf{s})}$

such that

  * •

$\sum_{e\in\vec{E}(\mathbf{s})}\mathsf{a}^{e}=1$,

  * •

$\mathsf{h}^{\bar{e}}=\mathsf{h}^{e}$, for all $e\in\vec{I}(\mathbf{s})$,

  * •

$\sum_{e\in\vec{B}_{i}(\mathbf{s})}\mathsf{l}^{e}=L^{i}$, for $1\leq i\leq b$,

  * •

$\lambda^{v_{0}}=0$.

There is a natural Lebesgue measure $\mathcal{L}_{\mathbf{s}}$ on
$\mathcal{T}_{\mathbf{s}}$ defined as follows. First, if $J$ is a finite set,
and $L>0$ a positive real number, we let $\Delta_{J}^{L}$ be the Lebesgue
measure on the simplex $\\{(\mathsf{x}^{j},j\in
J)\in(\mathbb{R}_{+})^{J}:\sum_{j\in J}\mathsf{x}^{j}=L\\}$. The latter
measure can be defined as the image of the measure $\bigotimes_{j\in
J^{\prime}}\mathrm{d}\mathsf{x}^{j}\mathbf{1}_{\\{\sum_{j\in
J^{\prime}}\mathsf{x}^{j}<L\\}}$, where $J^{\prime}$ is obtained from $J$ by
removing one arbitrary element $j^{\prime}$, by the mapping
$(\mathsf{x}^{j},j\in J^{\prime})\mapsto(\mathsf{x}^{j},j\in J)$ where
$\mathsf{x}^{j^{\prime}}=1-\sum_{j\in J^{\prime}}\mathsf{x}^{j}$.

Next, let $I(\mathbf{s})$ be an orientation of $\vec{I}(\mathbf{s})$, that is,
a set containing exactly one element from $\\{e,\bar{e}\\}$ for every
$e\in\vec{I}(\mathbf{s})$. We let $\mathcal{L}^{+}_{I(\mathbf{s})}$ be the
measure $\bigotimes_{e\in
I(\mathbf{s})}\mathrm{d}\mathsf{h}^{e}\mathbf{1}_{\\{\mathsf{h}^{e}\geq
0\\}}$. Similarly, we let $\mathcal{L}_{V(\mathbf{s})}$ be the measure
$\bigotimes_{v\in V^{\prime}(\mathbf{s})}\mathrm{d}\lambda^{v}$, where
$V^{\prime}(\mathbf{s})=V(\mathbf{s})\setminus\\{v_{0}\\}$.

Finally, the measure $\mathcal{L}_{\mathbf{s}}$ is the the image measure of

$\Delta_{\vec{E}(\mathbf{s})}^{1}\otimes\mathcal{L}^{+}_{I(\mathbf{s})}\otimes\bigotimes_{i=1}^{b}\Delta_{\vec{B}_{i}(\mathbf{s})}^{L_{i}}\otimes\mathcal{L}_{V(\mathbf{s})}$

by the mapping that associates with

$\Big{(}(\mathsf{a}^{e})_{e\in\vec{E}(\mathbf{s})},(\mathsf{h}^{e})_{e\in
I(\mathbf{s})},\big{(}(\mathsf{l}^{e})_{e\in\vec{B}_{i}(\mathbf{s})},1\leq
i\leq b\big{)},(\lambda^{v})_{v\in V^{\prime}(\mathbf{s})}\Big{)}$

the unique compatible element of $\mathcal{T}_{\mathbf{s}}$, that is, such
that $\mathsf{h}^{e}=\mathsf{h}^{\bar{e}}$ for every
$e\in\vec{I}(\mathbf{s})$, and such that $\lambda^{v_{0}}=0$.

We let $\mathrm{Param}_{\boldsymbol{L}}$ be the probability measure on
$\bigcup_{\mathbf{s}\in\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{S}}}}^{\star}}\\{\mathbf{s}\\}\times\mathcal{T}_{\mathbf{s}}$
whose density with respect to the measure
$\sum_{\mathbf{s}\in\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{S}}}}^{\star}}\delta_{\mathbf{s}}\otimes\mathcal{L}_{\mathbf{s}}$
is

$\frac{1}{\mathcal{Z}_{\boldsymbol{L}}}\prod_{e\in\vec{I}(\mathbf{s})}q_{\mathsf{h}^{e}}(\mathsf{a}^{e})\prod_{e\in\vec{B}(\mathbf{s})}q_{\mathsf{l}^{e}}(\mathsf{a}^{e})\prod_{e\in
I(\mathbf{s})}p_{\mathsf{h}^{e}}(\smash{\delta\\!\lambda^{e}})\prod_{e\in\vec{B}(\mathbf{s})}p_{3\mathsf{l}^{e}}(\delta\\!\lambda^{e})\,,$
(18)

where $p_{t}$, $q_{x}$ are defined after (30), where
$\delta\\!\lambda^{e}=\lambda^{e^{+}}-\lambda^{e^{-}}$ for
$e\in\vec{E}(\mathbf{s})$, and where $\mathcal{Z}_{\boldsymbol{L}}$ is a
normalizing constant, equal to the integral of the remaining display. Beware
that the third product is over $I(\mathbf{s})$, not $\vec{I}(\mathbf{s})$.

##### Scaling limits for size parameters.

Next, let $({\boldsymbol{l}}_{n})=(l_{n}^{1},\ldots,l^{k}_{n})$ and $Q_{n}$ be
as in the statement of Theorem 1. We let $v^{*}_{n}$ be uniformly distributed
over the set of internal vertices of $Q_{n}$, whose cardinality given by (3)
only depends on the parameters. Consequently, $(Q_{n},v^{*}_{n})$ is uniformly
distributed over the set of quadrangulations from
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}^{[g]}_{n,{\boldsymbol{l}}_{n}0}$,
seeing $v_{*}$ as a $k+1$-th hole. The rooted labeled map
$(M_{n},\lambda_{n})$ corresponding via the CVS correspondence is thus
uniformly distributed over
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{M}}}}^{[g]}_{n,{\boldsymbol{l}}_{n}}$.
We denote by $S_{n}$ the scheme of the nonrooted map corresponding to $M_{n}$,
and we root $S_{n}$ uniformly at random among its half-edges, incident to
internal or external faces, but such that the corresponding edge does not
belong to the boundary of $\mathlcal{h}_{b+1}$, …, $\mathlcal{h}_{k}$. Note
that we could have rooted $S_{n}$ from the root of $M_{n}$ by asking that the
root of $M_{n}$ belongs to the forest indexed by the root of $S_{n}$ but this
would have introduced an undesirable bias. Here instead, from the unrooted map
corresponding to $M_{n}$, the map $M_{n}$ is rooted at a uniform corner
incident to its internal face. Furthermore, the boundaries of the holes
$\mathlcal{h}_{b+1}$, …, $\mathlcal{h}_{k}$ are excluded from the possible
rootings of $S_{n}$ since they should be thought of as having null length in
the limit.

For $i\in\\{b+1,\ldots,k\\}$, the hole $\mathlcal{h}_{i}$ of $M_{n}$ is called
a _vanishing face_ if it is a face, that is, if $l_{n}^{i}>0$. The
corresponding hole $\mathlcal{h}_{i}$ of the scheme $S_{n}$ is called a
_tadpole_ if it is made of a single self-loop edge incident to a single vertex
of degree $3$. We let $S_{n}^{\mathclap{\,\cdot}\circ}$ be the map $S_{n}$ in
which every tadpole corresponding to a vanishing face $\mathlcal{h}_{i}$ has
been shrunk into a single vertex, still denoted by $\mathlcal{h}_{i}$, in the
sense that the corresponding self-loop has been removed. Note that the root of
$S_{n}$ is never removed in this operation, so that
$S_{n}^{\mathclap{\,\cdot}\circ}$ is always rooted.

Forgetting the root of $Q_{n}$, we let
$(\mathrm{EP}^{e}_{n},e\in\vec{E}(S_{n}))$ be the collection of elementary
pieces of $(Q_{n},v^{*}_{n})$. For the half-edges $e\in\vec{B}(S_{n})$, we let
$A_{n}^{e}$, $L_{n}^{e}$ be the area and width of the slice
$\mathrm{EP}^{e}_{n}$. For $e\in\vec{I}(S_{n})$, we let $A_{n}^{e}$,
$H_{n}^{e}$ be the first half-area and width of the quadrilateral
$\mathrm{EP}^{e}_{n}$; note that $A_{n}^{\bar{e}}$ is the second half-area of
$\mathrm{EP}^{e}_{n}$. Recall that the vertices of $S_{n}$ are in one-to-one
correspondence with the nodes of $M_{n}$: for every $v\in V(S_{n})$, we denote
by $\Lambda_{n}^{v}$ the label of the corresponding node, where we choose for
the labeling function $\lambda_{n}$ the representative giving label $0$ to the
root vertex of $S_{n}$.

######  Proposition 24.

As $n\to\infty$, with probability tending to one, every vanishing face of
$M_{n}$ induces a tadpole in $S_{n}$, and, on this likely event, for every
$e\in\bigsqcup_{i=b+1}^{k}\vec{B}_{i}(S_{n})$, it holds that
$A^{e}_{n}+L_{n}^{e}=\Theta((L_{n}^{e})^{2})$ in probability.

Moreover, the following convergence in distribution holds:

$\left(S_{n}^{\mathclap{\,\cdot}\circ},\left(\frac{A_{n}^{e}}{n}\right)_{e\in\vec{E}(S^{\mathclap{\,\cdot}\circ}_{n})},\left(\frac{H_{n}^{e}}{\sqrt{2n}}\right)_{e\in\vec{I}(S^{\mathclap{\,\cdot}\circ}_{n})},\left(\frac{L^{e}_{n}}{\sqrt{2n}}\right)_{e\in\vec{B}(S^{\mathclap{\,\cdot}\circ}_{n})},\left(\Big{(}\frac{9}{8n}\Big{)}^{1/4}\Lambda_{n}^{v}\right)_{v\in
V(S^{\mathclap{\,\cdot}\circ}_{n})}\right)\\\ \mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\big{(}S,\left({A^{e}}\right)_{e\in\vec{E}(S)},\left({H^{e}}\right)_{e\in\vec{I}(S)},\left(L^{e}\right)_{e\in\vec{B}(S)},\left(\Lambda^{v}\right)_{v\in
V(S)}\big{)}\,,$ (19)

where the limiting random variable has the law
$\mathrm{Param}_{\boldsymbol{L}}$ described in the previous paragraph.

This proposition is a generalization of [Bet16, Proposition 15]. Given its
technical nature, we postpone its proof to Appendix B.

##### Scaling limits of the elementary pieces.

As $(M_{n},\lambda_{n})$ is uniformly distributed over the set
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{M}}}}^{[g]}_{n,{\boldsymbol{l}}_{n}}$,
conditionally given (19), the random variables $\mathrm{EP}^{e}_{n}$,
$e\in\vec{E}(S_{n})$, are only dependent through the relations linking
$\mathrm{EP}^{\bar{e}}_{n}$ with $\mathrm{EP}^{e}_{n}$ for
$e\in\vec{I}(S_{n})$. Moreover,

  * •

if $e\in\vec{B}(S_{n})$, then $\mathrm{EP}^{e}_{n}$ is uniformly distributed
among slices with area $A_{n}^{e}$, width $L_{n}^{e}$ and tilt
$\Lambda_{n}^{e^{+}}-\Lambda_{n}^{e^{-}}$;

  * •

if $e\in\vec{I}(S_{n})$, then $\mathrm{EP}^{e}_{n}$ is uniformly distributed
among quadrilaterals with half-areas $A_{n}^{e}$ and $A_{n}^{\bar{e}}$, width
$H_{n}^{e}$ and tilt $\Lambda_{n}^{e^{+}}-\Lambda_{n}^{e^{-}}$.

Applying the Skorokhod representation theorem, we may and will assume that the
convergence of (19) holds almost surely. Since, by Lemma 7, there are finitely
many possible schemes, this furthermore implies that
$S_{n}^{\mathclap{\,\cdot}\circ}=S$ for $n$ sufficiently large. Together with
Theorems 12 and 14, the above observations entail that the collection of
resclaed random metric spaces
$(\Omega_{n}(\mathrm{EP}^{e}_{n}),e\in\vec{E}(S^{\mathclap{\,\cdot}\circ}_{n}))$,
converge in distribution in the GHP topology toward a family
$\big{(}\mathsf{EP}^{e},e\in\vec{E}(S)\big{)}$ of continuum elementary pieces
with the following law conditionally given the right-hand side of (19):

  * •

if $e\in\vec{B}(S)$, then $\mathsf{EP}^{e}$ is a continuum slice with area
$A^{e}$, width $H^{e}$ and tilt $\Lambda^{e^{+}}-\Lambda^{e^{-}}$;

  * •

if $e\in\vec{I}(S)$, then $\mathsf{EP}^{e}$ is a continuum quadrilateral with
half-areas $A^{e}$ and $A^{\bar{e}}$, width $H^{e}$ and tilt
$\Lambda^{e^{+}}-\Lambda^{e^{-}}$.

We write $\gamma(\mathsf{EP}^{e})$, $\xi(\mathsf{EP}^{e})$,
$\mu(\mathsf{EP}^{e})$, and either $\beta(\mathsf{EP}^{e})$,
$\nu(\mathsf{EP}^{e})$ or $\bar{\gamma}(\mathsf{EP}^{e})$,
$\bar{\xi}(\mathsf{EP}^{e})$ the marks and measures of $\mathsf{EP}^{e}$, with
an obvious choice of notation. As continuum elementary pieces have only been
defined as limits in the GHP topology of discrete elementary pieces so far,
these marks and measures are the limits of the corresponding marks and
measures of the discrete pieces.

We treat the elementary pieces corresponding to the vanishing faces thanks to
Corollary 13. We define, for every hole $\mathlcal{h}_{i}$ of $S_{n}$ with
$b+1\leq i\leq k$, the elementary piece $\mathrm{EP}^{\mathlcal{h}_{i}}_{n}$
as $\mathrm{EP}^{e}_{n}$ if $\vec{B}_{i}(S_{n})=\\{e\\}$ or the vertex map
otherwise, marked five times at its unique vertex and endowed twice with the
zero measure (thinking of it as an “empty slice”). By Proposition 24 and
Corollary 13, with probability tending to $1$, each one of the rescaled random
metric spaces $\Omega_{n}(\mathrm{EP}^{\mathlcal{h}_{i}}_{n})$, $b+1\leq i\leq
k$, converges in distribution in the GHP topology toward the point space (note
that the tilt of $\mathrm{EP}^{\mathlcal{h}_{i}}_{n}$ is always equal to $0$).

###  4.3 Gluing pieces together

We can now complete the proof of Theorem 1 by applying Proposition 22 at every
step of the inductive construction of Section 4.1.

We work on the event of asymptotic full probability of Proposition 24 and
assume that $n$ is large enough so that $S^{\mathclap{\,\cdot}\circ}_{n}=S$.
We denote by $\kappa=|\vec{E}(S)|+p$ and let $e_{1}$, …, $e_{\kappa}$ be the
sequence made of the half-edges of $\vec{E}(S)$, as well as the external
vertices of $S$, listed in contour order. Since $S$ is dominant, these
external vertices are $\mathlcal{h}_{b+1}$, …, $\mathlcal{h}_{k}$. Applying
the construction of Section 4.1 to the random quadrangulation $Q_{n}$, up to
adding the gluing of the point space for each external vertex (not changing
the markings and measures), we obtain a sequence $Q_{n,1}$, …, $Q_{n,\kappa}$
of marked measured metric spaces.

The limiting marked measured metric space
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ is obtained from
$(S,(\mathsf{EP}^{e},e\in\vec{E}(S))$ by recursively defining a sequence of
marked measured metric spaces $\mathsf{S}_{0}$, …, $\mathsf{S}_{\kappa}$, in
the following way. For $0\leq i\leq\kappa$, the space $\mathsf{S}_{i}$ will
carry geodesic marks $\gamma_{i}^{j}$, $\xi_{i}^{j}$, $0\leq j\leq u_{i}$,
boundary marks $\beta_{i}^{1}$, …, $\beta_{i}^{k}$, an area measure $\mu_{i}$,
and boundary measures $\nu_{i}^{1}$, …, $\nu_{i}^{k}$.

We initiate the construction by letting $u_{0}=0$,
$\mathsf{S}_{0}\in\mathbb{M}^{(2,k+1)}$ be the point space with the two marks
$\gamma_{0}^{0}$, $\xi_{0}^{0}$ being the unique point, and measures
$\mu_{0}=0$, ${\boldsymbol{\nu}}_{0}={\boldsymbol{0}}^{k}$. We also let all
the boundary marks be empty.

Next, given $\mathsf{S}_{i-1}$ for some $i\in\\{1,\ldots,\kappa\\}$, consider
the following cases.

  * •

If $e_{i}\in\vec{B}_{r}(S)$ for some $r\in\\{1,\ldots,k\\}$, set

$\displaystyle\mathsf{S}_{i}$
$\displaystyle=G\big{(}\mathsf{S}_{i-1},\mathsf{EP}^{e_{i}};\xi_{i-1}^{0},\gamma(\mathsf{EP}^{e_{i}})\big{)}\,,$
$\displaystyle\beta_{i}^{r}$
$\displaystyle=\beta_{i-1}^{r}\cup\beta(\mathsf{EP}^{e_{i}})\,,$
$\displaystyle\beta_{i}^{r^{\prime}}$ $\displaystyle=\beta_{i-1}^{r^{\prime}}\
\text{ for }r^{\prime}\in\\{1,\ldots,k\\}\setminus\\{r\\}\,,$
$\displaystyle\gamma_{i}^{0}$
$\displaystyle=\gamma_{i-1}^{0}\cup\big{(}\gamma(\mathsf{EP}^{e_{i}})\setminus\xi_{i-1}^{0}\big{)}\,,$
$\displaystyle\xi_{i}^{0}$
$\displaystyle=\xi(\mathsf{EP}^{e_{i}})\cup\big{(}\xi_{i-1}^{0}\setminus\gamma(\mathsf{EP}^{e_{i}})\big{)}\,,$
(20)

and, setting $u_{i}=u_{i-1}$, let $\gamma_{i}^{j}=\gamma_{i-1}^{j}$ and
$\xi_{i}^{j}=\xi_{i-1}^{j}$ for $1\leq j\leq u_{i}$. Finally, we let
$\mu_{i}=\mu_{i-1}+\mu(\mathsf{EP}^{e_{i}})$,
$\nu_{i}^{r}=\nu_{i-1}^{r}+\nu(\mathsf{EP}^{e_{i}})$ and
$\nu_{i}^{r^{\prime}}=\nu_{i-1}^{r^{\prime}}$ for $r^{\prime}\neq r$.

  * •

If $e_{i}\in\vec{I}(S)$, set $\beta_{i}^{r}=\beta_{i-1}^{r}$,
$\nu_{i}^{r}=\nu_{i-1}^{r}$ for $1\leq r\leq k$, and consider the following
two possible situations.

    * –

If $e_{i}\notin\\{\bar{e}_{j},\,1\leq j<i\\}$, let

$\mathsf{S}_{i}=G\big{(}\mathsf{S}_{i-1},\mathsf{EP}^{e_{i}};\xi_{i-1}^{0},\gamma(\mathsf{EP}^{e_{i}})\big{)}\,,$

update the first two geodesic marks by (20), and, setting $u_{i}=u_{i-1}+1$,
let $\gamma_{i}^{j}=\gamma_{i-1}^{j}$ and $\xi_{i}^{j}=\xi_{i-1}^{j}$ for
$1\leq j\leq u_{i}-1$, and
$\gamma_{i}^{u_{i}}=\bar{\gamma}(\mathsf{EP}^{e_{i}})$,
$\xi_{i}^{u_{i}}=\bar{\xi}(\mathsf{EP}^{e_{i}})$. Finally, set
$\mu_{i}=\mu_{i-1}+\mu(\mathsf{EP}^{e_{i}})$.

    * –

if $e_{i}\in\\{\bar{e}_{j},\,1\leq j<i\\}$ let

$\mathsf{S}_{i}=G\big{(}\mathsf{S}_{i-1};\xi_{i-1}^{0},\gamma(\mathsf{EP}^{e_{i}})\big{)}\,,$

update the first two geodesic marks by (20), and, setting $u_{i}=u_{i-1}-1$,
let $(\gamma_{i}^{j},\xi_{i}^{j},1\leq j\leq u_{i})$ be the the sequence
$(\gamma_{i-1}^{j},\xi_{i-1}^{j},1\leq j\leq u_{i-1})$ from which the terms
$\gamma(\mathsf{EP}^{e_{i}})$ and $\xi(\mathsf{EP}^{e_{i}})$ have been
removed. Finally, set $\mu_{i}=\mu_{i-1}$.

  * •

If $e_{i}$ is an external vertex of $S$, set
$\mathsf{S}_{i}=\mathsf{S}_{i-1}$.

Finally, we let
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}=G(\mathsf{S}_{\kappa};\xi_{\kappa}^{0},\gamma_{\kappa}^{0})$,
seen as an element of $\mathbb{M}^{(k,k+1)}$, equipped with the marking
$(\beta_{\kappa}^{1},\dots,\beta_{\kappa}^{k})$ and measures $\mu_{\kappa}$,
${\boldsymbol{\nu}}_{\kappa}$. An application of Proposition 22 and of Lemma
18 at every step of the construction shows that, for every
$i\in\\{1,\ldots,\kappa\\}$, the rescaled marked measured metric space
$\Omega_{n}(Q_{n,i})$ converges to $\mathsf{S}_{i}$ in the marked GHP
topology, and finally $\Omega_{n}(Q_{n})$ converges to
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ by a final application of Proposition 22
and the observation regarding the measure supports in the final gluing
mentioned at the end of Section 4.1. This completes the proof of Theorem 1.

###  4.4 Topology and Hausdorff dimension

In this section, we essentially derive from Proposition 2 in the case
$(g,k)\in\\{(0,0),(0,1)\\}$ an alternate proof of Proposition 2 in the other
cases. In the spherical case, Proposition 2 was obtained by Le Gall and Paulin
[LGP08] thanks to a theorem of Moore by seeing the Brownian sphere as a rather
wild quotient of the sphere by some equivalence relation. The same result was
later obtained in [Mie08] through the theory of regularity of sequences
developed by Begle and studied by Whyburn. The latter approach was generalized
in [Bet12, Bet15, Bet16] in order to obtain the general cases.

Our approach of decomposition into elementary pieces gives a rather direct and
transparent proof of Proposition 2 in the case $(g,k)\notin\\{(0,0),(0,1)\\}$
provided the following lemma, which will be obtained in Sections 5 and 6, and
which amounts to Proposition 2 for the noncompact analogs of the cases
$(g,k)\in\\{(0,0),(0,1)\\}$.

######  Lemma 25.

Almost surely, a slice or a quadrilateral is homeomorphic to a disk and is
locally of Hausdorff dimension $4$. Its boundary as a topological manifold
consists in the union of its marks, the intersection of any two marks being
empty or a singleton. Furthermore, in the case of a slice, the base is locally
of Hausdorff dimension $2$.

Now, a quadrangulation from ${\mathbf{Q}}_{n,{\boldsymbol{l}}_{n}}^{[g]}$ is
“not far” from being homeomorphic to $\mathbf{\Sigma}_{b_{n}}^{[g]}$, where
$b_{n}$ is the number of external faces, in the sense that we can “fill in”
the internal faces with small topological disks and “fill in” the external
faces by thin topological annuli without altering the metric and in such a way
that the resulting object is homeomorphic to $\mathbf{\Sigma}_{b_{n}}^{[g]}$
and at bounded GH distance from the quadrangulation (see [Bet16, Section
4.3.3] for more details about this procedure). The decomposition of the
quadrangulation into elementary pieces gives a decomposition of this surface
into pieces that are non other than the elementary pieces of the
quadrangulation with faces filled in and a thin rectangle added on the bases
of the slices; see Figure 12.

Figure 12: Left. Topological disk corresponding to an elementary piece of a
quadrangulation. Right. Decomposition of the surface associated with a
quadrangulation into surfaces homeomorphic to disks. Here also, we used the
same scheme as in Figure 3 without $\mathlcal{h}_{3}$.

With the notation of the previous section, $Q_{n}$ yields a surface
homeomorphic to $\mathbf{\Sigma}_{b_{n}}^{[g]}$ and its elementary pieces
$\mathrm{EP}^{e}_{n}$, $e\in\vec{E}(S_{n})$, yield surfaces homeomorphic to
disks.

  * •

If $e\in\vec{I}(S_{n})$, then the boundary of the surface associated with
$\mathrm{EP}^{e}_{n}$ consists in the two maximal geodesics and the two
shuttles of $\mathrm{EP}^{e}_{n}$.

  * •

If $e\in\vec{B}(S_{n})$, then the boundary of the surface associated with
$\mathrm{EP}^{e}_{n}$ consists in the maximal geodesic, the shuttle, as well
as three sides of the added thin rectangle.

Then gluing back these disks along their boundaries in the same way as they
were cut gives back the surface $\mathbf{\Sigma}_{b_{n}}^{[g]}$. Forgetting
the slice corresponding to a tadpole topologically amounts to fill in the
corresponding vanishing face. Assuming that $n$ is sufficiently large, all the
vanishing faces correspond to tadpoles in the scheme. So the gluing of the
elementary pieces without the slices corresponding to tadpoles yields
$\mathbf{\Sigma}_{b}^{[g]}$. In the limit, we glue topological disk exactly in
the same way (using markings that are topologically equivalent), so we obtain
the same surface. The result about the topology follows.

The statement about the Hausdorff dimension is even more straightforward as we
glue along geodesics a finite number of objects that are locally of dimension
$4$ and the boundary is the union of the bases of the slices, which are all
locally of Hausdorff dimension $2$.

##  5 Convergence of composite slices

The goal of this section is to prove Theorem 12 on the convergence of slices
to their limiting slices. To this end, we are first going to derive a “free”
version of this result by finding slices with a free area and tilt within the
uniform infinite half-planar quadrangulation. The latter is known to converge
to the Brownian half-plane, which itself contains a “flow” of continuum slices
with free areas and tilts; these are shown to be the scaling limits of the
discrete slices. We conclude by a conditioning argument to pass from free to
fixed area and tilt. First, let us start with deterministic considerations.

###  5.1 Metric spaces coded by real functions

Here we borrow some material from [BMR19, Section 2.1], with however several
slight differences, in order to describe in a unified fashion the various
random metric spaces we will use. Let $\mathcal{C}$ (resp.
$\mathcal{C}^{(2)}$) be the set of continuous functions of one variable (resp.
of two variables) defined on some nonempty closed interval:

$\mathcal{C}=\bigsqcup_{\begin{subarray}{c}I\text{ closed interval}\\\
I\neq\varnothing\end{subarray}}\mathcal{C}(I,\mathbb{R})\qquad\text{ and
}\qquad\mathcal{C}^{(2)}=\bigsqcup_{\begin{subarray}{c}I\text{ closed
interval}\\\ I\neq\varnothing\end{subarray}}\mathcal{C}(I^{2},\mathbb{R})\,.$

For a function $f\in\mathcal{C}(I,\mathbb{R})$ we denote by $I(f)=I$ its
interval of definition and by $\bar{\tau}(f)=\inf I$ and $\tau(f)=\sup I$ its
extremities. The set $\mathcal{C}$ is naturally equipped with the topology of
uniform convergence over compact subsets of $\mathbb{R}$; more precisely, the
topology induced by the following metric:

$\mathrm{dist}_{\mathcal{C}}(f,g)=\big{|}\arctan(\bar{\tau}(f))-\arctan(\bar{\tau}(g))\big{|}+\big{|}\arctan(\tau(f))-\arctan(\tau(g))\big{|}\\\
+\sum_{n\geq
1}\frac{1}{2^{n}}\sup_{t\in[-n,n]}\big{|}f\big{(}\bar{\tau}(f)\vee
t\wedge\tau(f)\big{)}-g\big{(}\bar{\tau}(g)\vee
t\wedge\tau(g)\big{)}\big{|}\,.$

We also equip $\mathcal{C}^{(2)}$ with a straightforward adaptation
$\mathrm{dist}_{\mathcal{C}^{(2)}}$.

####  5.1.1 $\mathbb{R}$-trees coded by functions

#####  $\mathbb{R}$-trees.

For $f\in\mathcal{C}$ and $s$, $t\in I=I(f)$ with $s\leq t$, set

$\mkern 1.0mu\underline{\mkern-1.0muf\mkern-1.0mu}\mkern
1.0mu(s,t)=\inf_{[s,t]}f$ (21)

and, for $s$, $t\in I$, set

$\displaystyle d_{f}(s,t)$ $\displaystyle=f(s)+f(t)-2\mkern
1.0mu\underline{\mkern-1.0muf\mkern-1.0mu}\mkern 1.0mu(s\wedge t,s\vee t)\,.$
(22)

This formula defines a pseudometric on $I$, which is continuous as a function
from $I^{2}$ to $\mathbb{R}_{+}$, since $d_{f}(s,t)\leq 2\omega(f;[s\wedge
t,s\vee t])$, where $\omega(f;J)=\sup_{J}f-\inf_{J}f$. We let
$\mathcal{T}_{f}=(I/\\{d_{f}=0\\},d_{f})$ be the associated quotient space,
and $\mathbf{p}_{f}:I\to\mathcal{T}_{f}$ be the canonical projection, which is
continuous since $d_{f}$ is. The space $\mathcal{T}_{f}$ is a so-called
_$\mathbb{R}$ -tree_, that is, satisfies the following.

  * •

For every two points $a$, $b\in\mathcal{T}_{f}$, there exists a geodesic from
$a$ to $b$, that is, an isometric mapping
$\chi_{a,b}:[0,d_{f}(a,b)]\to\mathcal{T}_{f}$ with $\chi_{a,b}(0)=a$ and
$\chi_{a,b}(d_{f}(a,b))=b$.

  * •

The image of the path $\chi_{a,b}$, which we denote by $\llbracket
a,b\rrbracket_{f}$, is the image of any injective path from $a$ to $b$.

If $I$ is compact, we let $a_{*}(f)=\mathbf{p}_{f}(t_{*})$, where $t_{*}$ is
any point at which $f$ attains its overall minimum. In this case, for $t\in I$
and $a=\mathbf{p}_{f}(t)$, the geodesic segment $\llbracket
a,a_{*}(f)\rrbracket_{f}$ is given by

$\llbracket a,a_{*}(f)\rrbracket_{f}=\mathbf{p}_{f}\big{(}\\{s\in[t\wedge
t_{*},t\vee t_{*}]:f(s)\leq f(u),\ \forall u\in[s\wedge t,s\vee
t]\\}\big{)}\,.$

In the case where $I$ is unbounded, we will systematically make the extra
assumption that

$\begin{cases}\text{when }\bar{\tau}(f)=-\infty\,,&\displaystyle\inf_{t\leq
0}f(t)=-\infty\,\quad\text{ or
}\quad\lim_{t\to-\infty}f(t)=\infty\,;\\\\[11.38109pt] \text{when
}\tau(f)=\infty\,,\qquad&\displaystyle\inf_{t\geq 0}f(t)=-\infty\,\quad\text{
or }\quad\lim_{\hbox
to0.0pt{\scriptsize$t\to\infty$\hss}\hphantom{t\to-\infty}}f(t)=\infty\,.\end{cases}$
(23)

In particular, it holds that

$\forall\,s\in I,\quad\lim_{|t|\to\infty,\,t\in I}d_{f}(s,t)=\infty\,,$ (24)

which implies that $\mathcal{T}_{f}$ is locally compact, as the reader may
easily check.

##### Gluing two $\mathbb{R}$-trees.

Next, given two functions $f$, $g\in\mathcal{C}$ with common interval of
definition $I=I(f)=I(g)$ both satisfying (23), we define another pseudometric
on $I$ as the quotient pseudometric (defined by (11))

$D_{f,g}(s,t)=d_{g}/\\{d_{f}=0\\}\,,$ (25)

and equip the quotient set ${M}_{f,g}=I/\\{{D}_{f,g}=0\\}$ with the metric
${D}_{f,g}$. Note that $D_{f,g}:I^{2}\to\mathbb{R}_{+}$ is continuous since

$\displaystyle\big{|}D_{f,g}(s,t)-D_{f,g}(s^{\prime},t^{\prime})\big{|}$
$\displaystyle\leq D_{f,g}(s,s^{\prime})+D_{f,g}(t,t^{\prime})$
$\displaystyle\leq 2\omega(g;[s\wedge s^{\prime},s\vee
s^{\prime}])+2\omega(g;[t\wedge t^{\prime},t\vee t^{\prime}])\,.$

For this reason, the canonical projection $\mathbf{p}_{f,g}:I\to M_{f,g}$ is
continuous. We may view $(M_{f,g},D_{f,g})$ as gluing the $\mathbb{R}$-tree
$\mathcal{T}_{g}$ along the equivalence relation defining the
$\mathbb{R}$-tree $\mathcal{T}_{f}$. In fact, since either of $d_{f}(s,t)=0$
or $d_{g}(s,t)=0$ implies $D_{f,g}=0$, the canonical projection
$\mathbf{p}_{f,g}$ factorizes as

$\mathbf{p}_{f,g}=\pi_{f}\circ\mathbf{p}_{f}=\pi_{g}\circ\mathbf{p}_{g}\,,$

where $\pi_{f}:\mathcal{T}_{f}\to M_{f,g}$ and $\pi_{g}:\mathcal{T}_{g}\to
M_{f,g}$ are two surjective maps. Note that these functions are continuous: if
$a_{n}=\mathbf{p}_{f}(t_{n})$ converges to some point $a$, then, up to taking
extractions (and using (24) if $I$ is unbounded), we may assume that $t_{n}$
converges to some limit $t$, and then $\mathbf{p}_{f}(t)=a$ by continuity of
$\mathbf{p}_{f}$, while $\pi_{f}(a_{n})=\mathbf{p}_{f,g}(t_{n})$ converges to
$\mathbf{p}_{f,g}(t)=\pi_{f}(a)$. As a consequence, every geodesic segment
$\llbracket a,b\rrbracket_{f}$ in $\mathcal{T}_{f}$, and every geodesic
$\llbracket c,d\rrbracket_{g}$ in $\mathcal{T}_{g}$ is “immersed” into
$M_{f,g}$ via the mappings $\pi_{f}$, $\pi_{g}$.

####  5.1.2 Composite slices coded by two functions

##### Slice trajectory.

We say that $(f,g)$ is a _slice trajectory_ if $f$, $g\in\mathcal{C}$ have
common interval of definition $I$,

$\displaystyle\forall s,t\in I\,,\qquad d_{f}(s,t)=0\ \implies\ g(s)=g(t)\,,$
(26) if $\inf I=-\infty$, then
$\displaystyle\lim_{t\to-\infty}f(t)=+\infty\qquad\text{ and
}\qquad\inf_{t\leq 0}g(t)=-\infty\,,$ and, if $\sup I=\infty$, then
$\displaystyle\inf_{t\geq 0}f(t)=-\infty\qquad\text{ and }\qquad\inf_{t\geq
0}g(t)=-\infty\,.$

In particular, $f$ and $g$ both satisfy (23) in the case where $I$ is
noncompact, and the quantity $f(\inf I)\in\mathbb{R}\cup\\{+\infty\\}$ is
always well defined.

In the remainder of this section, we fix a slice trajectory $(f,g)$, and call
the metric space

$\mathsf{Sl}_{f,g}=(M_{f,g},D_{f,g})$

the _slice coded by $(f,g)$_. For the moment, we focus on deterministic
considerations; the functions $f$, $g$ will be randomized in the following
section.

##### Marks and measures.

The slice $\mathsf{Sl}_{f,g}$ naturally comes with the following distinguished
elements.

Geodesics sides. For every $t\in I$, we set

$\displaystyle\Gamma_{t}(r)$ $\displaystyle=\inf\\{s\geq t:g(s)=g(t)-r\\}$
$\displaystyle\text{ for $r\in\mathbb{R}_{+}$ such that
}\inf_{\begin{subarray}{c}s\geq t\\\ s\in I\end{subarray}}g(s)\leq g(t)-r\,;$
$\displaystyle\Xi_{t}(r)$ $\displaystyle=\sup\\{s\leq t:g(s)=g(t)-r\\}$
$\displaystyle\text{ for $r\in\mathbb{R}_{+}$ such that
}\inf_{\begin{subarray}{c}s\leq t\\\ s\in I\end{subarray}}g(s)\leq g(t)-r\,.$

In particular, we have $d_{g}(\Gamma_{t}(r),\Xi_{t}(r))=0$ for every $t\in I$
and every $r$ satisfying both inequalities above.

We extend the definition given in Section 3.2 of geodesics to paths
$\chi:[0,\infty)\to\mathcal{X}$ that satisfy (10) for every $s$,
$t\in\mathbb{R}_{+}$. In this case, the point $\chi(0)$ is called the _origin_
of $\chi$, its length is set to $\operatorname{length}(\chi)=\infty$ by
convention, and the range of $\chi$ is called a _geodesic ray_. The geodesic
ray uniquely determines the geodesic $\chi$ by the same argument as for finite
length, since the origin of a geodesic ray is the unique point $a$ such that,
for any $s>0$, the number of points in the ray at distance $s$ from $a$ is
one.

We observe that $\Gamma_{t}$ and $\Xi_{t}$ are geodesics (possibly of infinite
length) from $t$ for the pseudometrics $d_{g}$ and $D_{f,g}$, in the sense
that, for every $r$, $r^{\prime}$ such that $\Gamma_{t}(r)$ and
$\Gamma_{t}(r^{\prime})$ are defined,

$d_{g}(\Gamma_{t}(r),\Gamma_{t}(r^{\prime}))=D_{f,g}(\Gamma_{t}(r),\Gamma_{t}(r^{\prime}))=|r^{\prime}-r|\,,$
(27)

and the same holds with $\Xi_{t}$ in place of $\Gamma_{t}$. This fact is
immediate for $d_{g}$ by definition. In fact, when $I$ is compact, one checks
that the images of $\mathbf{p}_{g}\circ\Gamma_{t}$ and
$\mathbf{p}_{g}\circ\Xi_{t}$ are the geodesic segments
$\llbracket\mathbf{p}_{g}(t),a_{*}(g{\raisebox{-1.49994pt}{$|$}}{}_{[t,\sup
I]})\rrbracket_{g}$ and
$\llbracket\mathbf{p}_{g}(t),a_{*}(g{\raisebox{-1.49994pt}{$|$}}{}_{[\inf
I,t]})\rrbracket_{g}$ in $\mathcal{T}_{g}$.

For $D_{f,g}$, this fact follows from the bound

$|g(s)-g(s^{\prime})|\leq D_{f,g}(s,s^{\prime})\leq
d_{g}(s,s^{\prime})\,,\qquad s,s^{\prime}\in I\,,$

where the first inequality is an easy consequence of the fact that $(f,g)$ is
a slice trajectory.

Therefore, for $t\in I$, the paths defined by

$\displaystyle\gamma_{t}(r)$
$\displaystyle=\mathbf{p}_{f,g}(\Gamma_{t}(r))\,,\qquad 0\leq r\leq
g(t)-\mkern 1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern 1.0mu(t,\sup
I)\,,\ r\in\mathbb{R}\,,$ $\displaystyle\xi_{t}(r)$
$\displaystyle=\mathbf{p}_{f,g}(\Xi_{t}(r))\,,\qquad 0\leq r\leq g(t)-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern 1.0mu(\inf I,t)\,,\
r\in\mathbb{R}\,,$

are two geodesics (possibly of infinite length) from $\mathbf{p}_{f,g}(t)$,
sharing a common initial part. As mentioned in Section 3.2, we will often
identify these paths with the pairs formed by their origins and image sets,
the latter being the projections
$\pi_{g}(\llbracket\mathbf{p}_{g}(t),a_{*}(g{\raisebox{-1.49994pt}{$|$}}{}_{[t,\sup
I]})\rrbracket_{g})$ and
$\pi_{g}(\llbracket\mathbf{p}_{g}(t),a_{*}(g{\raisebox{-1.49994pt}{$|$}}{}_{[\inf
I,t]})\rrbracket_{g})$ when $I$ is compact.

The slice $M_{f,g}$ comes with zero, one, or two geodesic sides. If $\inf
I>-\infty$, then the geodesic $\gamma=\gamma_{\inf I}$ is called the _maximal
geodesic_ of $M_{f,g}$, and, if $\sup I<\infty$, the geodesic $\xi=\xi_{\sup
I}$ is called the _shuttle_ of $M_{f,g}$. If $\inf I=-\infty$ (resp. $\sup
I=\infty$), we let $\gamma_{-\infty}$ (resp. $\xi_{\infty}$) be the empty set.
If $I$ is a bounded interval, then the paths $\gamma_{\inf I}$ and $\xi_{\sup
I}$ have a common endpoint at the _apex_
$x_{*}=\mathbf{p}_{f,g}(s_{*})=\pi_{g}(a_{*}(g))$, where $s_{*}$ denotes any
point $s$ in $I$ such that $g(s)=\inf_{I}g$.

Base. For $x\in\mathbb{R}$, we define

$T_{x}=\inf\\{t\in I:f(t)=-x\\}\in\mathbb{R}\cup\\{\infty\\}\,,$

the hitting time of level $-x$ by the function $f$, with the convention that
$\inf\varnothing=\infty$. Note that, for $x\in\mathbb{R}$, $T_{x}\neq-\infty$
because of the fact that $(f,g)$ is admissible. By convention, we also set
$T_{\infty}=-T_{-\infty}=\infty$. The _base_ of $\mathsf{Sl}_{f,g}$ is the set

$\beta=\mathbf{p}_{f,g}\left(\left\\{T_{x}:-f(\inf I)\leq
x\leq-\inf_{I}f\right\\}\cap\mathbb{R}\right)\,.$

Note that the set inside brackets projects via $\mathbf{p}_{f}$ to a geodesic
in $\mathcal{T}_{f}$. When $I$ is compact, the base is the path
$\pi_{f}(\llbracket\mathbf{p}_{f}(T_{-f(\inf
I)}),\mathbf{p}_{f}(T_{-\inf_{I}f})\rrbracket)$, and in general, it is the
increasing union of the paths

$\pi_{f}(\llbracket\mathbf{p}_{f}(T_{x}),\mathbf{p}_{f}(T_{y})\rrbracket_{f})\,,\qquad-f(\inf
I)\leq x<y\leq-\inf_{I}f\,,\ x,y\in\mathbb{R}\,.$

Measures. Finally, denoting by $\mathrm{Leb}_{J}$ the Lebesgue measure on the
interval $J$, the slice $\mathsf{Sl}_{f,g}$ is endowed with the following
measures:

  * •

the _area measure_ $\mu=(\mathbf{p}_{f,g})_{*}\mathrm{Leb}_{I}$ ;

  * •

the _base measure_ $\nu$, defined as the pushforward of
$\mathrm{Leb}_{[-f(\inf I),-\inf_{I}f]\cap\mathbb{R}}$ by the mapping
$x\mapsto\mathbf{p}_{f,g}(T_{x})$ .

##### Gluing slices.

In what follows, we will make a slight abuse of notation and identify
intervals of the form $[a,\infty]$, $[-\infty,a]$ for $a\in\mathbb{R}$ and
$[-\infty,\infty]$ with the intervals $[a,\infty)$, $(-\infty,a]$ and
$\mathbb{R}$, respectively. For $L$, $L^{\prime}$ in the extended line
$\mathbb{R}\cup\\{\pm\infty\\}$ such that $-f(\inf I)\leq L\leq
L^{\prime}\leq-\inf_{I}f$, we define the restrictions $f^{(L,L^{\prime})}$ and
$g^{(L,L^{\prime})}$ of $f$ and $g$ to the interval
$[T_{L},T_{L^{\prime}}]\cap I$, yielding also a slice trajectory. We may
therefore define the slice coded by $(f^{(L,L^{\prime})},g^{(L,L^{\prime})})$
and denote it by

$\mathsf{Sl}^{(L,L^{\prime})}=\big{(}M^{(L,L^{\prime})},D^{(L,L^{\prime})}\big{)}=\big{(}M_{f^{(L,L^{\prime})},g^{(L,L^{\prime})}},D_{f^{(L,L^{\prime})},g^{(L,L^{\prime})}}\big{)}\,.$

We let $\mathbf{p}^{(L,L^{\prime})}:[T_{L},T_{L^{\prime}}]\to
M^{(L,L^{\prime})}$ be the canonical projection, $\gamma^{(L,L^{\prime})}$,
$\xi^{(L,L^{\prime})}$, $\beta^{(L,L^{\prime})}$ be the maximal geodesic,
shuttle, and base, and $\mu^{(L,L^{\prime})}$, $\nu^{(L,L^{\prime})}$ be the
area and base measures of $\mathsf{Sl}^{(L,L^{\prime})}$.

This family of metric spaces is compatible with the gluing operation in the
following sense, illustrated in Figure 13.

######  Proposition 26.

Let $-f(\inf I)\leq L<L^{\prime}<L^{\prime\prime}\leq-\inf_{I}f$ be in the
extended real line. Then

$\mathsf{Sl}^{(L,L^{\prime\prime})}=G\big{(}\mathsf{Sl}^{(L,L^{\prime})},\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})};\xi^{(L,L^{\prime})},\gamma^{(L^{\prime},L^{\prime\prime})}\big{)}\,.$

Moreover, the marks and measures satisfy

$\displaystyle\gamma^{(L,L^{\prime\prime})}$
$\displaystyle=\gamma^{(L,L^{\prime})}\cup\big{(}\gamma^{(L^{\prime},L^{\prime\prime})}\setminus\xi^{(L,L^{\prime})}\big{)}\,,$
$\displaystyle\xi^{(L,L^{\prime\prime})}$
$\displaystyle=\xi^{(L^{\prime},L^{\prime\prime})}\cup\big{(}\xi^{(L,L^{\prime})}\setminus\gamma^{(L^{\prime},L^{\prime\prime})}\big{)}\,,$
$\displaystyle\beta^{(L,L^{\prime\prime})}$
$\displaystyle=\beta^{(L,L^{\prime})}\cup\beta^{(L^{\prime},L^{\prime\prime})}\,,$
$\displaystyle\mu^{(L,L^{\prime\prime})}$
$\displaystyle=\mu^{(L,L^{\prime})}+\mu^{(L^{\prime},L^{\prime\prime})}\,,$
$\displaystyle\nu^{(L,L^{\prime\prime})}$
$\displaystyle=\nu^{(L,L^{\prime})}+\nu^{(L^{\prime},L^{\prime\prime})}\,,$

with the convention that, in the right hand-side, sets and measures are
identified with their images and pushforwards by the canonical projections in
$\mathsf{Sl}^{(L,L^{\prime\prime})}$ .

Figure 13: Gluing slices encoded by a slice trajectory: the gluing of
$\mathsf{Sl}^{(L,L^{\prime})}$ with
$\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})}$ results in
$\mathsf{Sl}^{(L,L^{\prime\prime})}$. Here, $T_{L}>-\infty$ and
$T_{L^{\prime\prime}}<\infty$. We denoted by
$x_{\smash{*}}^{(L^{\prime},L^{\prime\prime})}$ the apex of
$\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})}$ and
$x_{\smash{*}}^{(L,L^{\prime})}$ the apex of $\mathsf{Sl}^{(L,L^{\prime})}$,
which, on this example, is also the apex of
$\mathsf{Sl}^{(L,L^{\prime\prime})}$. Consequently, the shuttle
$\xi^{(L,L^{\prime\prime})}$ is obtained by the union of
$\xi^{(L^{\prime},L^{\prime\prime})}$ and the part of $\xi^{(L,L^{\prime})}$
that is not glued to $\gamma^{(L^{\prime},L^{\prime\prime})}$, whereas the
maximal geodesic $\gamma^{(L,L^{\prime\prime})}=\gamma^{(L,L^{\prime})}$, as
stated at the end of Proposition 26. The bases and measures simply add up. The
fact that the slices depicted here are topological disks does not hold true in
general; it will, however, be the case for the random processes we will
consider in the upcoming sections.

###### Proof.

In the disjoint union
$[T_{L},T_{L^{\prime}}]\sqcup[T_{L^{\prime}},T_{L^{\prime\prime}}]$, in order
to avoid ambiguities due to the fact that the point $T_{L^{\prime}}$ belongs
to both intervals (thus should be duplicated), we use a superscript $0$ for
points in the first interval and a superscript $1$ for points in the second
interval. We observe that $d_{g^{(L,L^{\prime\prime})}}$ can be seen as a
quotient pseudometric $d/R_{1}$ where $d$ is the disjoint union pseudometric
on $[T_{L},T_{L^{\prime}}]\sqcup[T_{L^{\prime}},T_{L^{\prime\prime}}]$ given
by $d(s,t)=d_{g}(s,t)$ if $s$, $t$ belong to the same of the two intervals
above and $d(s,t)=\infty$ otherwise, and $R_{1}$ is the coarsest equivalence
relation containing

$\big{\\{}\big{(}\Xi_{T_{L^{\prime}}}(r)^{0},\Gamma_{T_{L^{\prime}}}(r)^{1}\big{)},\
0\leq r\leq g(T_{L^{\prime}})-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(T_{L},T_{L^{\prime}})\vee\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(T_{L^{\prime}},T_{L^{\prime\prime}})\big{\\}}\,.$

Note also that, as $T_{L^{\prime}}$ is a hitting time, the equivalence
relation $\\{d_{f^{(L,L^{\prime\prime})}}=0\\}$ factorizes over these two
intervals, in the sense that if $d_{f^{(L,L^{\prime\prime})}}(s,t)=0$ with
$s\neq t$, then $s$, $t$ must belong to the same interval
$[T_{L},T_{L^{\prime}}]$ or $[T_{L^{\prime}},T_{L^{\prime\prime}}]$. So if
$R_{2}$ is the equivalence relation on the above disjoint union given by
$(s^{i},t^{j})\in R_{2}$ if and only if $d_{f}(s,t)=0$ and $i=j\in\\{0,1\\}$,
using (12), we have

$D^{(L,L^{\prime\prime})}=(d/R_{1})/R_{2}=(d/R_{2})/R_{1}=\big{(}D^{(L,L^{\prime})}\sqcup
D^{(L^{\prime},L^{\prime\prime})}\big{)}/R_{1}\,,$

which is precisely the quotient metric of the gluing
$G(\mathsf{Sl}^{(L,L^{\prime})},\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})};\xi^{(L,L^{\prime})},\gamma^{(L^{\prime},L^{\prime\prime})})$.

Checking the claimed formulas for the marks and measures of
$\mathsf{Sl}^{(L,L^{\prime\prime})}$ is straightforward. ∎

We finish this paragraph with a very strong identity, saying that the
distances in a slice $\mathsf{Sl}^{(L,L^{\prime})}$ encoded by a restriction
of the slice trajectory $(f,g)$ are in fact the restrictions of the distances
in the “whole” slice $\mathsf{Sl}_{f,g}$.

######  Corollary 27.

Let $(f,g)$ be a slice trajectory on the interval $I$, and $-f(\inf I)\leq
L\leq L^{\prime}\leq-\inf_{I}f$. Then $D^{(L,L^{\prime})}$ is the restriction
of the function $D_{f,g}$ to $[T_{L},T_{L^{\prime}}]$.

###### Proof.

This is a direct consequence of the preceding proposition, which entails that
$D_{f,g}$ is the pseudometric obtained by gluing
$\mathsf{Sl}^{(L,L^{\prime})}$ with $\mathsf{Sl}^{(L^{\prime},\sup I)}$ along
$\xi^{(L,L^{\prime})}$ and $\gamma^{(L^{\prime},\sup I)}$, and then by gluing
the resulting space $\mathsf{Sl}^{(L,\sup I)}$ with $\mathsf{Sl}^{(\inf I,L)}$
along $\gamma^{(L,\sup I)}$ and $\xi^{(\inf I,L)}$. Since at each stage, the
spaces that are glued together are isometrically embedded in the resulting
gluing, we obtain that $\mathsf{Sl}^{(L,L^{\prime})}$ is isometrically
embedded in $\mathsf{Sl}^{(\inf I,\sup I)}=\mathsf{Sl}_{f,g}$. ∎

###  5.2 Random continuum composite slices

We now randomize the functions $f$, $g$ considered in the preceding section in
various ways to construct random spaces of interest. For a fixed continuous
function $f\in\mathcal{C}$ with $0\in I(f)$, the _snake_ 999Literally, this is
rather called the “head of the snake driven by $f$”; see [LG99]. driven by $f$
is a random centered Gaussian process $(Z^{f}_{t},t\in I(f))$ with
$Z^{f}_{0}=0$ and with covariance function specified by

$\mathbb{E}\big{[}(Z^{f}_{t}-Z^{f}_{s})^{2}\big{]}={d}_{f}(s,t)\,,\qquad
s,t\in I(f)\,.$ (28)

As soon as $f$ is Hölder continuous, which will always be the case in this
paper, this process admits a continuous modification; we systematically
consider this continuous modification of $Z^{f}$. If now $Y$ is a (a.s. Hölder
continuous) random function, then the random snake driven by $Y$ is defined as
the Gaussian process $Z^{Y}$ conditionally given $Y$.

By (28), it holds that $Z_{s}^{f}=Z_{t}^{f}$ whenever $d_{f}(s,t)=0$, so that,
provided $f$ satisfies the required limit conditions if $I(f)$ is noncompact,
the pair $(f,Z^{f})$ is a slice trajectory. In what follows, we will let
$(X,W):(f,g)\mapsto(f,g)$ be the canonical process on $\mathcal{C}^{2}$.

Below and throughout this work, we use, for any process $Y$ defined on an
interval $I$, the piece of notation $\mkern
1.0mu\underline{\mkern-1.0muY\mkern-1.0mu}\mkern 1.0mu_{t}=\inf_{s\leq
t,\,s\in I}Y_{s}$.

Let us proceed to the definition of continuum slices, which arise in Theorem
12. Fix $A$, $L\in(0,\infty)$ and $\Delta\in\mathbb{R}$. We let
$\mathbf{Slice}_{A,L,\Delta}$ be the probability distribution under which

  * •

the process $X$ is a first-passage bridge101010A _first-passage bridge of
Brownian motion_ can be defined from Brownian motion stopped when first
hitting $-L$ by absolute continuity; see [BM17]. of standard Brownian motion
from $0$ to $-L$ with duration $A$;

  * •

conditionally given $X$, the process $W$ has the same law as
$(Z_{t}+\zeta_{-\mkern 1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern
1.0mu_{t}},0\leq t\leq A)$, where $Z$ is the random snake driven by $X-\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern 1.0mu$, and $\zeta/\sqrt{3}$
is a standard Brownian bridge of duration $L$ and terminal value
$\Delta/\sqrt{3}$, independent of $X$ and $Z$.

To be more precise, the process $\zeta=(\zeta_{t},0\leq t\leq L)$ is Gaussian,
with $\mathbb{E}[\zeta_{t}]=t\Delta/L$ for $t\in[0,L]$ and

$\operatorname{Cov}(\zeta_{s},\zeta_{t})=3\,\frac{s(L-t)}{L}\,,\qquad 0\leq
s\leq t\leq L\,.$

With this definition, it is simple to see that $\mathbf{Slice}_{A,L,\Delta}$
is indeed carried by slice trajectories defined on the interval $I=[0,A]$.

######  Definition 28.

The _(composite) slice with area $A$, width $L$ and tilt $\Delta$_,
generically denoted by $\mathsf{Sl}_{A,L,\Delta}$, is the $5$-marked
$2$-measured metric space $\mathsf{Sl}_{X,W}$ under the law
$\mathbf{Slice}_{A,L,\Delta}$, endowed with the marking

$\partial\mathsf{Sl}_{A,L,\Delta}=(\beta,\gamma_{0},\xi_{A})$

comprising its base, its maximal geodesic and its shuttle, as well as its area
and base measures $\mu$, $\nu$.

The piece of notation $\partial\mathsf{Sl}_{A,L,\Delta}$ for the marking comes
from the fact that the union of the three marks gives the topological boundary
of $\mathsf{Sl}_{A,L,\Delta}$, as stated in Lemma 25.

###  5.3 The Brownian half-plane, and its embedded slices

There is a natural relation between slices and the Brownian half-plane [GM17,
BMR19], which we now introduce. Let $(B_{t},t\geq 0)$, $(B^{\prime}_{t},t\geq
0)$ be two independent standard Brownian motions, and let
$(\Pi_{t}=B^{\prime}_{t}-2\inf_{\\{0\leq s\leq t\\}}B_{s}^{\prime},t\geq 0)$
be the so-called _Pitman transform_ of $B^{\prime}$, which is a three-
dimensional Bessel process. Recall the piece of notation $\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern 1.0mu_{t}=\inf_{s\leq
t}X_{s}$. We let $\mathbf{Half}$ be the probability distribution on
$\mathcal{C}^{2}$ under which

  * •

the process $X$ has same distribution as $(B_{t}\mathbf{1}_{\\{t\geq
0\\}}+\Pi_{-t}\mathbf{1}_{\\{t<0\\}},t\in\mathbb{R})$, and

  * •

conditionally given $X$, the process $W$ has same distribution as
$(Z_{t}+\zeta_{-\mkern 1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern
1.0mu_{t}},t\in\mathbb{R})$, where $Z$ is the random snake driven by $X-\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern 1.0mu$, and $\zeta/\sqrt{3}$
is a two-sided standard Brownian motion111111This means that
$(\zeta_{x}/\sqrt{3},x\geq 0)$ and $(\zeta_{-x}/\sqrt{3},x\geq 0)$ are
independent (one-dimensional) standard Brownian motions issued from $0$.,
independent of $X$ and $Z$.

The measure $\mathbf{Half}$ is carried by slice trajectories defined on the
interval $\mathbb{R}$. We note that we can make this definition more symmetric
using standard excursion theory, in a way similar to the encoding triples of
[LGR21]. For this, we denote by

$\displaystyle X^{(L)}$ $\displaystyle=(L+X_{T_{L-}+t},0\leq t\leq
T_{L}-T_{L-})\,,$ $\displaystyle W^{(L)}$ $\displaystyle=(W_{T_{L-}+t},0\leq
t\leq T_{L}-T_{L-})\,,$

the excursion of $X$ above its past infimum at level $-L$, and the
corresponding piece of $W$. Note first that the process $\zeta_{L}=W_{T_{L}}$,
$L\in\mathbb{R}$, is under $\mathbf{Half}$ a standard two-sided Brownian
motion multiplied by $\sqrt{3}$. Then, conditionally given $\zeta$, the point
measure on $\mathbb{R}\times\mathcal{C}\times\mathcal{C}$ given by

$\mathcal{M}(\mathrm{d}L\,\mathrm{d}X\,\mathrm{d}W)=\sum_{L\in\mathbb{R}:\,T_{L}\neq
T_{L-}}\delta_{(L,X^{(L)},W^{(L)})}$ (29)

is a Poisson measure with intensity
$2\mathrm{d}L\,\mathbb{N}_{\zeta_{L}}(\mathrm{d}(X,W))$, where
$\mathbb{N}_{x}$ is the $\sigma$-finite “law” of the lifetime process and head
of the Brownian snake (started at $x$) driven by the Itô measure of positive
excursions of Brownian motion. The process $(X,W)$ is then a measurable
function of $\zeta$ and $\mathcal{M}$ by Itô’s reconstruction theory of
Brownian motion from its excursions.

######  Definition 29.

The _Brownian half-plane_ , generically denoted by $\mathsf{BHP}$, is the
$1$-marked $2$-measured metric space $\mathsf{Sl}_{X,W}$ considered under
$\mathbf{Half}$, endowed with the mark $\partial\mathsf{BHP}=\beta$, its area
measure $\mu$ and its base measure $\nu$.

There is only one mark here, the base; there is no maximal geodesic nor
shuttle since the interval of definition is $\mathbb{R}$. The name comes from
the fact that $\mathsf{BHP}$ is homeomorphic to the half-plane
$\mathbb{R}\times\mathbb{R}_{+}$, its boundary as a topological manifold being
equal to the base; see [BMR19, Corollary 3.8].

In the light of Proposition 26, the Brownian half-plane can be seen to have a
natural Markov property. First, let $\theta_{t}:f\mapsto f(t+\cdot)-f(t)$ be
the translation operator on $\mathcal{C}$. We claim that $\mathbf{Half}$ is
invariant under $\theta_{T_{L}}$, since its action simply consists in
translating by $L$ the time in process $\zeta$, and the first coordinate of
$\mathcal{M}$, which leaves their laws invariant. For similar reasons, for
every $L\in\mathbb{R}$, the processes $\big{(}X^{(0,L)},W^{(0,L)}\big{)}$,
$(X^{(-\infty,0)},W^{(-\infty,0)})$ and
$(\theta_{T_{L}}X^{(L,+\infty)},\theta_{T_{L}}W^{(L,+\infty)})$ are
independent under $\mathbf{Half}$, since they are respectively functionals of
the independent random elements

$(\zeta_{x},0\leq x\leq
L),\mathcal{M}((0,L]\times\mathcal{C}\times\mathcal{C})\,,\qquad(\zeta_{x},x\leq
0),\mathcal{M}((-\infty,0]\times\mathcal{C}\times\mathcal{C})\,,\\\ \text{and
}\quad(\zeta_{L+x}-\zeta_{L},x\geq
0),\mathcal{M}((L,\infty)\times\mathcal{C}\times\mathcal{C})\,.$

##### Free slices.

Note that, under $\mathbf{Half}$, the process $X^{(0,L)}$ is simply a standard
Brownian motion killed at its first hitting time of $-L$, while the process
$(W^{(0,L)}_{T_{x}}/\sqrt{3},0\leq x\leq L)$ is a standard Brownian motion
killed at time $L$. For this reason, the law of $(X^{(0,L)},W^{(0,L)})$ under
$\mathbf{Half}$ is the mixture

$\mathbf{FSlice}_{L}=\int_{0}^{\infty}q_{L}(A)\mathrm{d}A\int_{\mathbb{R}}p_{3L}(\Delta)\mathrm{d}\Delta\,\mathbf{Slice}_{A,L,\Delta}\,,$
(30)

where $p_{t}(x)=e^{-x^{2}/2t}/\sqrt{2\pi t}$ is the Gaussian density, and
$q_{x}(t)=(x/t)\,p_{t}(x)\mathbf{1}_{\\{t>0\\}}$ is the (stable $1/2$) density
for the hitting time of level $-x$ by standard Brownian motion. In what
follows, a random metric space with same law as $\mathsf{Sl}^{(0,L)}$ under
$\mathbf{FSlice}_{L}$ will be referred to as a _free (composite) slice_ of
width $L$. This, together with Proposition 26, yields the following result.

######  Proposition 30.

Fix $L<L^{\prime}<L^{\prime\prime}$ in the extended line. Then, under
$\mathbf{Half}$, it holds that
$\mathsf{Sl}^{(L,L^{\prime\prime})}=G(\mathsf{Sl}^{(L,L^{\prime})},\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})};\xi^{(L,L^{\prime})},\gamma^{(L^{\prime},L^{\prime\prime})})$,
where the spaces $\mathsf{Sl}^{(L,L^{\prime})}$,
$\mathsf{Sl}^{(L^{\prime},L^{\prime\prime})}$ are independent. Moreover, if
$L$ and $L^{\prime}$ are finite, then $\mathsf{Sl}^{(L,L^{\prime})}$ is a free
slice of width $L^{\prime}-L$.

Recall that this result is illustrated in Figure 13, which can be completed by
extending the brown segment into a line, letting the half-plane above be
$\mathsf{BHP}$, the line being its base $\beta=\beta^{(-\infty,\infty)}$. This
also suggests that $\mathsf{Sl}^{(L,L^{\prime})}$ is the bounded connected
component of the complement of
$\gamma^{(L,L^{\prime})}\cup\xi^{(L,L^{\prime})}$ in $\mathsf{BHP}$. More
precisely, the following holds.

######  Proposition 31.

For every $L<L^{\prime}$ in $\mathbb{R}$, almost surely under $\mathbf{Half}$,
the geodesics $\gamma^{(L,L^{\prime})}$ and $\xi^{(L,L^{\prime})}$ meet only
at the apex $x_{\smash{*}}^{(L,L^{\prime})}$, and meet the base $\beta$ only
at their respective origins $\mathbf{p}_{X,W}(T_{L})$ and
$\mathbf{p}_{X,W}(T_{L^{\prime}})$. Moreover, $\mathsf{Sl}^{(L,L^{\prime})}$
is the closure of the bounded connected component of the complement of the
union of these two paths in $\mathsf{BHP}$. It is therefore homeomorphic to
the closed unit disk, with boundary given by the union of the three sets
$\beta^{(L,L^{\prime})}$, $\gamma^{(L,L^{\prime})}$ and
$\xi^{(L,L^{\prime})}$, which meet only at $\mathbf{p}_{X,W}(T_{L})$,
$\mathbf{p}_{X,W}(T_{L^{\prime}})$ and $x_{\smash{*}}^{(L,L^{\prime})}$.

###### Proof.

This proposition is proved in the same way as Lemma 6.15 in [BMR19]. Let us
recall briefly the ideas. For any point $t\in\mathbb{R}$, we let

$\Sigma_{t}(r)=\inf\\{s\geq t:X_{s}=X_{t}-r\\}\qquad\text{ for }\qquad 0\leq
r\leq X_{t}-\mkern 1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern
1.0mu_{t}\,,$

so that the range of $\mathbf{p}_{X}\circ\Sigma_{t}$ is the geodesic path
$\llbracket\mathbf{p}_{X}(t),\mathbf{p}_{X}(T_{-\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern 1.0mu_{t}})\rrbracket_{X}$ in
$\mathcal{T}_{X}$. Its image by $\pi_{X}$ defines a path $\sigma_{t}$ starting
at $\mathbf{p}_{X,W}(t)$ and ending at the point $\mathbf{p}_{X,W}(T_{-\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern 1.0mu_{t}})$ of the base.
Moreover, almost surely, any path $\sigma_{t}$, $t\in\mathbb{R}$, do not
intersect a geodesic $\gamma_{s}$, $s\in\mathbb{R}$, except possibly at the
starting point of either $\mathbf{p}_{X,W}(s)$ or $\mathbf{p}_{X,W}(t)$. This
implies that any point $\mathbf{p}_{X,W}(t)$ of $\mathsf{Sl}^{(L,L^{\prime})}$
that is not in the union $\gamma^{(L,L^{\prime})}\cup\xi^{(L,L^{\prime})}$ can
be linked to the bounded segment $\beta^{(L,L^{\prime})}$ of the base of
$\mathsf{BHP}$ by the path $\sigma_{t}$ without intersecting
$\gamma^{(L,L^{\prime})}\cup\xi^{(L,L^{\prime})}$ except perhaps at its
endpoint. This latest possibility can be discarded by noting that, with
probability $1$, we have $T_{-\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern
1.0mu_{t}}\notin\\{T_{L},T_{L^{\prime}}\\}$. Similarly, a point
$\mathbf{p}_{X,W}(t)$ of $\mathsf{BHP}$ outside of
$\mathsf{Sl}^{(L,L^{\prime})}$ is linked to the unbounded set
$\beta\setminus\beta^{(L,L^{\prime})}$ of the base of $\mathsf{BHP}$ by the
path $\sigma_{t}$, which does not intersect
$\gamma^{(L,L^{\prime})}\cup\xi^{(L,L^{\prime})}$. This means that
$\mathsf{Sl}^{(L,L^{\prime})}$ is the closure of the bounded connected
component of $\mathsf{BHP}$ minus
$\gamma^{(L,L^{\prime})}\cup\xi^{(L,L^{\prime})}$. ∎

The above discussion shows that the Brownian half-plane contains a natural
“flow” of free slices. We can also link directly the slices of Section 5.2
with the Brownian half-plane via an absolute continuity argument. Recall the
definitions of $p_{t}$ and $q_{x}$ after (30).

######  Lemma 32.

Fix $0<K<L$, as well as $A>0$ and $\Delta\in\mathbb{R}$. For every nonnegative
function $G$ that is measurable with respect to the $\sigma$-algebra generated
by $(X^{(0,K)},W^{(0,K)})$, we have

$\mathbf{Slice}_{A,L,\Delta}[G]=\mathbf{Half}\big{[}\varphi_{A,L,\Delta}\big{(}T_{K},K,W_{T_{K}}\big{)}\cdot
G\big{]}\,,$

where

$\varphi_{A,L,\Delta}(A^{\prime},L^{\prime},\Delta^{\prime})=\frac{q_{L-L^{\prime}}(A-A^{\prime})}{q_{L}(A)}\frac{p_{3(L-L^{\prime})}(\Delta-\Delta^{\prime})}{p_{3L}(\Delta)}\,.$
(31)

###### Proof.

This comes from similar statements for Brownian bridges and first-passage
bridges; see for instance [Bet10, Equations (18) and (19)]. For bounded
measurable functions $f$, $g$ on $\mathcal{C}$, for $0<A^{\prime}<A$ and
$0<K<L$,

$\mathbf{Slice}_{A,L,\Delta}\Big{[}f\big{(}X{\raisebox{-1.49994pt}{$|$}}{}_{[0,A^{\prime}]}\big{)}\cdot
g\big{(}\zeta{\raisebox{-1.49994pt}{$|$}}{}_{[0,K]}\big{)}\Big{]}\\\
=\mathbf{Half}\Bigg{[}f\big{(}X{\raisebox{-1.49994pt}{$|$}}{}_{[0,A^{\prime}]}\big{)}\frac{q_{L-X_{A^{\prime}}}(A-A^{\prime})}{q_{L}(A)}\mathbf{1}_{\\{\mkern
1.0mu\underline{\mkern-1.0muX\mkern-1.0mu}\mkern
1.0mu_{A^{\prime}}>-L\\}}\cdot
g\big{(}\zeta{\raisebox{-1.49994pt}{$|$}}{}_{[0,K]}\big{)}\frac{p_{3(L-K)}(\Delta-\zeta_{K})}{p_{3L}(\Delta)}\Bigg{]}.$
(32)

Here, the factor $3$ in the index of the Gaussian density function comes from
the fact that $\zeta/\sqrt{3}$ is a bridge of standard Brownian motion. We
replace $A^{\prime}$ with $T_{K}$ by a standard argument, writing

$f\big{(}X^{(0,K)}\big{)}=\lim_{n\to\infty}\sum_{i\geq
0}\mathbf{1}_{\\{(i-1)2^{-n}<T_{K}\leq i\,2^{-n}\\}}\leavevmode\nobreak\
f\big{(}X{\raisebox{-1.49994pt}{$|$}}{}_{[0,i\,2^{-n}]}\big{)},$

using dominated convergence and applying the above equality (32) to
$A^{\prime}=i\,2^{-n}$, noting that $\mathbf{1}_{\\{(i-1)2^{-n}<T_{K}\leq
i\,2^{-n}\\}}\leavevmode\nobreak\
f\big{(}X{\raisebox{-1.49994pt}{$|$}}{}_{[0,i\,2^{-n}]}\big{)}$ is a function
of $X{\raisebox{-1.49994pt}{$|$}}{}_{[0,i\,2^{-n}]}$.

The result follows by noting that $W^{(0,K)}$ is built in the same way from
$X^{(0,K)}$ and $\zeta{\raisebox{-1.49994pt}{$|$}}{}_{[0,K]}$ under
$\mathbf{Slice}_{A,L,\Delta}$ as from $X^{(0,K)}$ and
$\zeta{\raisebox{-1.49994pt}{$|$}}{}_{[0,K]}$ under $\mathbf{Half}$. ∎

We may now prove the statement about the topology and Hausdorff dimension of a
slice.

###### Proof of Lemma 25 for slices.

First, almost surely, the Brownian half-plane is homeomorphic to the half-
plane [BMR19, Corollary 3.8], is locally of Hausdorff dimension $4$ and its
boundary is locally of Hausdorff dimension $2$. The latter facts are obtained
from similar statements for Brownian disks [Bet15] thanks to [BMR19, Theorem
3.7] allowing to couple arbitrary balls of $\mathsf{BHP}$ centered at the root
$\mathbf{p}_{X,W}(0)$ with balls of large enough Brownian disks, centered at a
point on the boundary.

Consequently, under the probability distribution $\mathbf{Half}$, for any
$L<L^{\prime}$, by Lemma 27, the metric space $\mathsf{Sl}^{(L,L^{\prime})}$
is a.s. locally of Hausdorff dimension $4$ and its base
$\beta^{(L,L^{\prime})}$ is locally of Hausdorff dimension $2$. Furthermore,
it is homeomorphic to the disk by Proposition 31 and its boundary is the union
of its three marks $\beta^{(L,L^{\prime})}$, $\gamma^{(L,L^{\prime})}$ and
$\xi^{(L,L^{\prime})}$, whose pairwise intersections are identified
singletons.

Now, arguing under $\mathbf{Slice}_{A,L,\Delta}$, we use the fact from
Proposition 26 that
$\mathsf{Sl}^{(0,L)}=G(\mathsf{Sl}^{(0,L/2)},\mathsf{Sl}^{(L/2,L)};\xi^{(0,L/2)},\gamma^{(L/2,L)})$.
Lemma 32 entails that, almost surely, under this probability distribution, the
law of $\mathsf{Sl}^{(0,L/2)}$ is absolutely continuous with respect to that
of the same random variable under $\mathbf{Half}$, and so is homeomorphic to a
disk. Now, we observe that, under $\mathbf{Slice}_{A,L,\Delta}$, the process
$\theta_{T_{L/2}}(X^{(L/2,L)},W^{(L/2,L)})$ has same distribution as
$(X^{(0,L/2)},W^{(0,L/2)})$, which we leave as an exercise to the reader.
Therefore, under this law, $\mathsf{Sl}^{(L/2,L)}$ has same distribution as
$\mathsf{Sl}^{(0,L/2)}$ and both are homeomorphic to a disk. We conclude that
the same is true for $\mathsf{Sl}^{(0,L)}$ since it is obtained by gluing two
topological disks along two segments of their boundaries. The identification
of the marks given in Proposition 26 easily yields the desired property on the
marks of $\mathsf{Sl}^{(0,L)}$. The facts on the local Hausdorff dimension are
obtained similarly. ∎

###  5.4 The uniform infinite half-planar quadrangulation

##### The UIHPQ.

We now define a slight variant of the classical UIHPQ [CM15, CC18, BMR19,
BR18], the half-planar version of the uniform infinite random planar
quadrangulation, in the following way. Let
$F_{\infty}=(\mathbf{T}^{k},k\in\mathbb{Z})$ be a two-sided sequence of
independent Bienaymé–Galton–Watson trees with a geometric offspring
distribution of parameter $1/2$. Conditionally on $F_{\infty}$, we let
$\lambda^{0}_{\infty}$ be a uniformly chosen well labeling function, meaning
that every tree $\mathbf{T}^{k}$ is assigned a well labeling function giving
label $0$ to its root vertex, independently, uniformly at random. Lastly, and
independently of $F_{\infty}$ and $\lambda^{0}_{\infty}$, we let
$(b_{k},k\in\mathbb{Z})$ be a doubly-infinite walk with shifted geometric
steps, meaning that $b_{0}=0$ a.s., and that $b_{k}-b_{k-1}$,
$k\in\mathbb{Z}$, are independent and identically distributed random variables
with $\mathbb{P}(b_{1}=r)=2^{-r-2}$ for every $r\in\\{-1,0,1,2,\ldots\\}$. For
a vertex $v\in\mathbf{T}^{k}$ we let
$\lambda_{\infty}(v)=b_{k}+\lambda^{0}_{\infty}(v)$, and call
$(F_{\infty},\lambda_{\infty})$ the _infinite random well-labeled forest_. We
then embed $F_{\infty}$ in the plane in such a way that all trees are
contained in the upper half-plane, and the root $\rho^{k}$ of $\mathbf{T}^{k}$
is located at the point $(k,0)\in\mathbb{R}^{2}$. We also link consecutive
roots $\rho^{k}$, $\rho^{k+1}$ by a line segment. We then let
$(c_{i},i\in\mathbb{Z})$ be the sequence of corners of the upper half-plane
part of the resulting map, in contour order from left to right, with origin
the first corner $c_{0}$ incident to $\rho^{0}$. The _uniform infinite half-
planar quadrangulation_ (_UIHPQ_ for short) is then the infinite map
$Q_{\infty}$ obtained by applying the CVS construction to
$(F_{\infty},\lambda_{\infty})$, that is, by linking every corner to its
successor as defined in Section 2.1, and removing all edges of the forest
afterward. The root of $Q_{\infty}$ is defined as the corner preceding the arc
from $c_{0}$ to its successor. Note that, in this case, there is no need to
add an extra vertex with a corner $c_{\infty}$.

######  Remark 33.

The difference between this definition of the UIHPQ and the one appearing in
the mentioned references is a slight rooting bias. Indeed, the simplest way to
obtain the usual definition is to consider a two-sided simple random walk
$(z_{i},i\in\mathbb{Z})$ and construct the sequence $(b_{k},k\in\mathbb{Z})$
from it as follows. Let
$S^{\downarrow}=\\{i\in\mathbb{Z}\,:\,z_{i+1}-z_{i}=-1\\}$ be the set of
descending steps of $(z_{i},i\in\mathbb{Z})$ and
$i_{0}=\sup(S^{\downarrow}\cap\mathbb{Z}_{-})$ the index of the descending
step preceding $0$. Then we define the sequence $(b_{k},k\in\mathbb{Z})$ by
reindexing $(z_{i}-z_{i_{0}},i\in S^{\downarrow})$ with $\mathbb{Z}$ in such a
way that $i_{0}$ corresponds to the index $0$. The UIHPQ is then constructed
as above with this bridge but rooted at the corner preceding the arc linking
$s^{-i_{0}}(c_{0})$ to its successor $s^{-i_{0}+1}(c_{0})$ instead of the
convention we presented. Apart from this slight root shift, the resulting law
of $(b_{k},k\in\mathbb{Z})$ is not exactly that of a doubly-infinite bridge
with shifted geometric steps. The first step gets a size-biased distribution
$\mathbb{P}(b_{1}=r)=(r+2)2^{-r-3}$, $r\geq-1$, whereas all other steps get
the desired shifted geometric distribution. See the discussion in [BMR19,
Section 4.5.2] for more information.

The construction we use here has the advantage of making the law of the slices
invariant by translation.

##### Convergence toward the Brownian half-plane.

Denoting by $v_{i}$ the vertex of $F_{\infty}$ incident to $c_{i}$ and by
$\Upsilon(i)\in\mathbb{Z}$ the index of the tree to which $v_{i}$ belongs, we
define the _contour_ and _label processes_ on $\mathbb{R}$ by

$C(i)=d_{\mathbf{T}^{\Upsilon(i)}}\big{(}v_{i},\rho^{\Upsilon(i)}\big{)}-\Upsilon(i)\qquad\text{
and }\quad\Lambda(i)=\lambda_{\infty}(v_{i})\,,\qquad i\in\mathbb{Z}\,,$

and by linear interpolation between integer values; see Figure 14. As is well
known, the part of the contour process corresponding to $\mathbf{T}^{k}$
(counting the edge linking $\rho^{k}$ to $\rho^{k+1}$) has the same
distribution as a simple random walk started at $-k$ and killed when first
hitting $-k-1$. Finally, for $k\geq 1$, we denote by $\tau_{k}$ the hitting
time of $-k$ by $C$; its value is thus also equal to $k$ plus twice the number
of edges in the first $k$ trees $\mathbf{T}^{0}$, …, $\mathbf{T}^{k-1}$.

Figure 14: Contour and label processes associated with
$(F_{\infty},\lambda_{\infty})$. The edges of the floor are represented with
dashed lines. The tree $\mathbf{T}^{0}$ and the corresponding part of the
encoding processes are highlighted (the corresponding floor edge and the final
descending step of the contour function are also highlighted). For instance,
$\tau_{3}=17$ on this example. The contour process can be thought of as
recording the height of a particle moving at speed one around the forest. In
this point of view, the root $\rho^{k}$ should be at height $-k$ for each
$k\in\mathbb{Z}$; this can be achieved for instance by vertically translating
each tree in such a way that $\rho^{k}$ is mapped to location $(k,-k)$ instead
of $(k,0)$.

As the vertices of the encoding objects are preserved through the CVS
bijection, the vertex $v_{i}$ can also be seen as a vertex of $Q_{\infty}$.
Let us define

$\displaystyle D_{\infty}(i,j)$
$\displaystyle=d_{Q_{\infty}}(v_{i},v_{j})\,,\qquad i,j\in\mathbb{Z}\,.$

We extend ${D}_{\infty}$ to a continuous function on $\mathbb{R}^{2}$ by
“bilinear interpolation,” writing $\\{s\\}=s-\lfloor s\rfloor$ for the
fractional part of $s$ and then setting

$D_{\infty}(s,t)=(1-\\{s\\})(1-\\{t\\})D_{\infty}(\lfloor s\rfloor,\lfloor
t\rfloor)+\\{s\\}(1-\\{t\\})D_{\infty}(\lfloor s\rfloor+1,\lfloor t\rfloor)\\\
+(1-\\{s\\})\\{t\\}D_{\infty}(\lfloor s\rfloor,\lfloor
t\rfloor+1)+\\{s\\}\\{t\\}D_{\infty}(\lfloor s\rfloor+1,\lfloor t\rfloor+1).$
(33)

We then define the renormalized versions of $C$, $\Lambda$, and $D_{\infty}$ :
for every $s$, $t\in\mathbb{R}$, we set

$C_{(n)}(s)=\frac{C(2ns)}{\sqrt{2n}}\,,\qquad\Lambda_{(n)}(s)=\frac{\Lambda(2ns)}{(8n/9)^{1/4}}\,,\qquad
D_{(n)}(s,t)=\frac{D_{\infty}(2ns,2nt)}{(8n/9)^{1/4}}\,.$ (34)

The next result can be seen as a reformulation of [GM17, Theorem 1.11] or
[BMR19, Theorem 3.6], proving the convergence of the UIHPQ to the Brownian
half-plane defined in Section 5.3.

######  Proposition 34.

On $\mathcal{C}\times\mathcal{C}\times\mathcal{C}^{(2)}$, it holds that

$\big{(}C_{(n)},\Lambda_{(n)},D_{(n)}\big{)}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\big{(}X,W,D_{X,W}\big{)}\,,$
(35)

where the limiting triple is understood under $\mathbf{Half}$.

This statement does not appear in this exact form in the aforementioned
references, which do not explicitly focus on the processes $C_{(n)}$,
$\Lambda_{(n)}$, $X$, $W$. In [BMR19, Remark 6.16], it was however mentioned
how to extend the results therein in order to take into account these
processes, so we will follow the line of reasoning sketched in that work.

###### Proof.

The proof proceeds via established convergence results for random
quadrangulations with one external face to Brownian disks. Fix some number
$K>0$. We will sample a quadrangulation with one external face, whose areas
and perimeters are so large that, in a neighborhood of $0$ of amplitude $K$,
this rescaled large quadrangulation and its limit, a Brownian disk of large
area and perimeter, are indistinguishable from the rescaled UIHPQ and the
Brownian half-plane, in a sense to be made precise. In the following, we will
use for all the objects related to the quadrangulation with one external face
or the limiting Brownian disk a similar notation as for those related to the
UIHPQ or the Brownian half-plane, only with a superscript prime symbol
$\prime$.

Fix $L>0$, which should be thought of as being large. For $n\geq 1$, we sample
the aforementioned quadrangulation $Q^{\prime}_{n}$ with one external face as
follows. First, consider a uniform random element
$(M^{\prime}_{n},\lambda^{\prime}_{n})$ of
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{M}}}}^{[0]}_{a_{n},(l_{n})}$,
where $a_{n}=\lfloor nL\rfloor$ and $l_{n}=\lfloor L\sqrt{2n}\rfloor$. We can
view this as a labeled forest $(F^{\prime}_{n},\lambda^{\prime}_{n})$ with
$l_{n}$ trees arranged in a circle, and rooted at $\rho^{0}$, …,
$\rho^{l_{n}-1}$ where $\rho^{0}$ is the root of the tree containing the root
corner of $f_{*}$. We let $C^{\prime}_{n}$, $\Lambda^{\prime}_{n}$ be the
contour and label process of this forest, defined as above, starting from the
tree rooted at $\rho^{0}$. We let
$Q^{\prime}_{n}=\operatorname{CVS}(M^{\prime}_{n},\lambda^{\prime}_{n};f_{*})$
be the rooted quadrangulation encoded by
$(M^{\prime}_{n},\lambda^{\prime}_{n})$, and we let
$D^{\prime}_{n}(i,j)=d_{Q_{n}^{\prime}}(v^{\prime}_{i},v^{\prime}_{j})$ for
$0\leq i,j\leq 2a_{n}+l_{n}$, where $v^{\prime}_{i}$ is the $i$-th visited
vertex of $F_{n}^{\prime}$ in contour order, viewed as a vertex of
$Q^{\prime}_{n}$. As usual, we extend $D^{\prime}_{n}$ into a continuous
function on $[0,2a_{n}+l_{n}]^{2}$. Finally, we extend the definition of these
processes to the interval $[-2a_{n}-l_{n},2a_{n}+l_{n}]$ by the simple
translation formulas

$C^{\prime}_{n}(t)=C^{\prime}_{n}(t+2a_{n}+l_{n})+l_{n}\,,\quad\Lambda^{\prime}_{n}(t)=\Lambda^{\prime}_{n}(t+2a_{n}+l_{n})\,,\quad
t\in[-2a_{n}-l_{n},0]\,,$ (36)

and

$D^{\prime}_{n}(s,t)=D^{\prime}_{n}(s+(2a_{n}+l_{n})\mathbf{1}_{\\{s<0\\}},t+(2a_{n}+l_{n})\mathbf{1}_{\\{t<0\\}})\,,\quad
s,t\in[-2a_{n}-l_{n},2a_{n}+l_{n}]\,.$ (37)

The idea behind this extension is that we are going to consider these
processes in neighborhoods of $0$, so that we are really interested in the
behavior of these processes when the argument is close from $0$ or from
$2a_{n}+l_{n}$.

Define their rescaled versions: for $s$, $t\in[-2a_{n}-l_{n},2a_{n}+l_{n}]$,

$C^{\prime}_{(n)}(s)=\frac{C^{\prime}_{n}(2ns)}{\sqrt{2n}}\,,\qquad\Lambda^{\prime}_{(n)}(s)=\frac{\Lambda^{\prime}_{n}(2ns)}{(8n/9)^{1/4}}\,,\qquad
D^{\prime}_{(n)}(s,t)=\frac{D^{\prime}_{n}(2ns,2nt)}{(8n/9)^{1/4}}\,.$ (38)

Then by [BM17, Equation (26) and Theorem 20], one has the joint convergence

$\big{(}C^{\prime}_{(n)},\Lambda^{\prime}_{(n)},D^{\prime}_{(n)}\big{)}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}(X^{\prime},W^{\prime},D^{\prime})$
(39)

in distribution in
$\mathcal{C}([0,L])\times\mathcal{C}([0,L])\times\mathcal{C}([0,L]^{2})$,
where $(X^{\prime},W^{\prime},D^{\prime})$ is an explicit limiting process,
which is the encoding process of the Brownian disk of area $L$ and width
$\sqrt{L}$. In particular, the process $D^{\prime}$ is a measurable function
of the pair $(X^{\prime},W^{\prime})$. Due to the formulas in (36) and (37),
this easily implies the convergence of these processes on
$\mathcal{C}([-L,L])\times\mathcal{C}([-L,L])\times\mathcal{C}([-L,L]^{2})$,
where $(X^{\prime},W^{\prime},D^{\prime})$ are extended to functions on
$[-L,L]$ or $[-L,L]^{2}$ in a similar way as above. Note that we choose to
omit the dependence of $(X^{\prime},W^{\prime},D^{\prime})$ on $L$ for lighter
notation, but we will need later to choose $L$ appropriately.

Now recall that $K>0$ is a fixed number. The first crucial observation is that
we may choose $L$ large enough, so that with high probability, the laws of the
restrictions of $(X^{\prime},W^{\prime},D^{\prime})$ and $(X,W,D_{X,W})$ to
the interval $[-K,K]$ are very close. More precisely, given
$\varepsilon\in(0,1)$, fix $r>0$ and $A>0$ such that

$\mathbb{P}\left(\max_{-K\leq t\leq
K}D_{X,W}(0,t)>r\right)<\varepsilon/3\,,\qquad\mathbb{P}(T_{-A}<-K<K<T_{A})\geq
1-\varepsilon/3\,.$

Then [BMR19, Proposition 6.6] and its proof (Lemmas 6.7 and 6.8) show that
there exists $L_{0}>0$ such that, for $L>L_{0}$, the two processes $(X,W)$ and
$(X^{\prime},W^{\prime})$ can be coupled in such a way that on some event
$\mathcal{F}$ of probability $\mathbb{P}(\mathcal{F})\geq 1-\varepsilon/3$, we
have

$X_{t}=X^{\prime}_{t}\,,\quad W_{t}=W^{\prime}_{t}\,,\quad
D_{X,W}(s,t)=D^{\prime}(s,t)\,,$ (40)

for every $s$, $t\in[T_{-A},T_{A}]$ such that
$\max(D_{X,W}(0,t),D_{X,W}(0,s))\leq r$. Given our choice of $r$, $A$, we see
that with probability at least $1-\varepsilon$, (40) holds for every
$s,t\in[-K,K]$.

Our second important observation is that, still with $K$ and $\varepsilon$
fixed, and possibly up to choosing $L$ even larger than the above, albeit in a
way that does not depend on $n$, the laws of $(C_{(n)},\Lambda_{(n)},D_{(n)})$
and $(C^{\prime}_{(n)},\Lambda^{\prime}_{(n)},D^{\prime}_{(n)})$ in
restriction to the interval $[-K,K]$ are also close, in the sense that they
can be coupled in such a way that these restrictions coincide with probability
at least $1-\varepsilon$. This follows from the proof of [BMR19, Theorem 3.6],
a minor difference being that this proposition establishes that the balls of
radius $(8n/9)^{1/4}r$ centered at the root in $Q_{\infty}$ and
$Q^{\prime}_{n}$ are isometric, rather than giving a statement on $D_{(n)}$
and $D^{\prime}_{(n)}$. Therefore, in order to show that the latter coincide
on $[-K,K]$, one again has to choose in the first place a radius $r>0$ so that
uniformly over $n$, with probability at least $1-\varepsilon/3$, the vertices
$v^{\prime}_{i}$ for integers $i$ lying in $[-2Kn,2Kn]$ (where we naturally
let $v^{\prime}_{i}=v^{\prime}_{i+2a_{n}+l_{n}}$ for $i\leq 0$), all belong to
this ball. The existence of such an $r$ is guaranteed by the convergence (39)
and the continuity of $D^{\prime}$. Finally, we see that both sides of (39)
can be coupled in such a way that with probability at least $1-\varepsilon$,
they coincide with both sides of (35). Since $\varepsilon$ was arbitrary, we
conclude that (35) holds in restriction to $[-K,K]$. Since $K$ was arbitrary,
this concludes the proof. ∎

##### Seeing a slice as part of the UIHPQ.

We consider a fixed $L>0$ and a sequence $(l_{n})\in\mathbb{N}^{\mathbb{N}}$
such that

$\frac{l_{n}}{\sqrt{2n}}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}L$

and, for each $n$, we let $(F_{n},\lambda_{n})$ be the random well-labeled
forest obtained by keeping only the labeled trees $\mathbf{T}^{0}$, …,
$\mathbf{T}^{l_{n}-1}$ of the infinite random well-labeled forest
$(F_{\infty},\lambda_{\infty})$, as well as the root $\rho^{l_{n}}$ of the
tree $\mathbf{T}^{l_{n}}$. In particular, the forest $F_{n}$ has $l_{n}$
independent Bienaymé–Galton–Watson trees with Geometric($1/2$) offspring
distribution, and the labels of the root vertices of the trees (including
$\rho^{l_{n}}$) follow a random walk of length $l_{n}$ whose step distribution
is a shifted Geometric($1/2$) given by $\mathbb{P}(\,\cdot=r)=2^{-r-2}$ for
$r\geq-1$.

Recall that $(c_{i},i\in\mathbb{Z})$ denotes the sequence of corners of the
infinite random well-labeled forest $(F_{\infty},\lambda_{\infty})$ and that
$v_{i}$ is the vertex of $F_{\infty}$ incident to $c_{i}$. According to the
construction of Section 2.3, $(F_{n},\lambda_{n})$ encodes a slice $Q_{n}$,
which is part of the UIHPQ $Q_{\infty}$ constructed from the whole infinite
forest $(F_{\infty},\lambda_{\infty})$. More precisely, the maximal geodesic
(resp. shuttle) can be read inside the UIHPQ as the chain of arcs linking
$c_{0}$ (resp. $c_{\tau_{l_{n}}}$) to its subsequent successors121212Recall
Section 2.1. and the edges of the slice are given by the arcs from $c_{i}$ to
$s(c_{i})$, for $0\leq i<\tau_{l_{n}}$. As a consequence, the vertex $v_{i}$
can be seen both as a vertex of $Q_{\infty}$ and as a vertex of $Q_{n}$, for
$0\leq i\leq\tau_{l_{n}}$.

Furthermore, we can check that $Q_{n}$ is in fact isometrically embedded in
$Q_{\infty}$ in the sense that, whenever $0\leq i,j\leq\tau_{l_{n}}$, it holds
that $d_{Q_{n}}(v_{i},v_{j})=D_{\infty}(i,j)$. Indeed, similarly to
Proposition 26, $Q_{\infty}$ can be obtained as the gluing of the infinite
quadrangulation corresponding to the trees $\mathbf{T}^{k}$, $k<0$, of
$(F_{\infty},\lambda_{\infty})$, with $Q_{n}$ and then with the infinite
quadrangulation corresponding to the trees $\mathbf{T}^{k}$, $k\geq l_{n}$, of
$(F_{\infty},\lambda_{\infty})$ along the proper shuttles and maximal
geodesics. Alternatively, one may also argue that there are no shortcuts
outside $Q_{n}$ : for $0\leq i,j\leq\tau_{l_{n}}$, any path linking $v_{i}$ to
$v_{j}$ in $Q_{\infty}$ may be shorten to a path that stays within $Q_{n}$
since the maximal geodesic and shuttle are geodesics and since the path
$c_{0}\to s(c_{0})\to s^{2}(c_{0})\to\dots$ is a geodesic ray that disconnects
$Q_{\infty}$.

The contour function, label function and pseudometric corresponding to $Q_{n}$
are thus obtained by restricting to $[0,\tau_{l_{n}}]$ the analog functions
corresponding to $Q_{\infty}$. After rescaling, their joint limit is a direct
consequence of Proposition 34.

######  Corollary 35.

On $\mathcal{C}\times\mathcal{C}\times\mathcal{C}^{(2)}$, it holds that

$\big{(}C_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]},\Lambda_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]},D_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]^{2}}\big{)}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\big{(}X^{(0,L)},W^{(0,L)},D^{(0,L)}\big{)}\,,$
(41)

where we used the notation of Section 5.1, that is,

  1. (a)

the pair $(X^{(0,L)},W^{(0,L)})$ is the restriction to the interval
$[0,T_{L}]$ of $(X,W)$ distributed under $\mathbf{Half}$,

  2. (b)

$D^{(0,L)}=D_{X^{(0,L)},W^{(0,L)}}$ is the random pseudometric on $\mathbb{R}$
defined by (25).

###### Proof.

By the Skorokhod representation theorem, we may and will assume that the
convergence (35) holds almost surely. Classically, the a.s. path properties of
$X$ at time $T_{L}$, namely, the fact that $X$ immediately visits the interval
$(L-\varepsilon,L)$ after time $T_{L}$, yield that $\tau_{l_{n}}/2n$ a.s.
converges to $T_{L}$. Proposition 35 and Corollary 27 then yield the result. ∎

###  5.5 Scaling limit of conditioned slices

We now derive Theorem 12 from the results of the previous section by standard
conditioning arguments.

##### Convergence of the encoding processes.

First, without loss of generality, we may assume that the contour and label
processes $(C,\Lambda)$ of the infinite random well-labeled forest defined in
Section 5.4 are the canonical processes, considered under the probability
distribution $\mathbb{P}_{\infty}$ on the canonical space. Next, for $a$,
$l\in\mathbb{N}$ and $\delta\in\mathbb{Z}$, we denote by
$\mathbb{P}_{a,l,\delta}$ the distribution of
$\big{(}C{\raisebox{-1.49994pt}{$|$}}{}_{[0,2a+l]},\Lambda{\raisebox{-1.49994pt}{$|$}}{}_{[0,2a+l]}\big{)}$
where $(C,\Lambda)$ is distributed under
$\mathbb{P}_{\infty}[\;\cdot\mid\tau_{l}=2a+l,\,\Lambda(\tau_{l})=\delta]$.
The corresponding forest encoded by this random process is thus composed of
$l$ Bienaymé–Galton–Watson trees with Geometric($1/2$) offspring distribution
and uniform admissible labels, conditioned on the fact that the total number
of edges in the trees is $a$ and the label of the root of the last vertex-tree
is $\delta$. Similarly to the slice it encodes, we will say that the forest
has _tilt_ $\delta$.

For every measurable nonnegative functional $G$, it thus holds that

$\mathbb{E}_{a,l,\delta}[G]=\mathbb{E}_{\infty}\big{[}G\big{(}(C(k),\Lambda(k)),0\leq
k\leq\tau_{l}\big{)}\bigm{|}\tau_{l}=2a+l,\,\Lambda(\tau_{l})=\delta\big{]}\,.$

Let $(\mathcal{F}_{k},k\geq 0)$ be the natural filtration associated with the
canonical process $(C,\Lambda)$. Note that $((C(k),\Lambda(k)),0\leq
k\leq\tau_{l})$ is the pair of contour and label processes of the first $l$
trees in the forest, and that $\mathcal{F}_{\tau_{l}}$ is the $\sigma$-algebra
generated by these first $l$ trees (with their labels and that of the root
$\rho^{l}$). Recall from Proposition 9 the definitions of $Q_{\ell}$,
$P_{\ell}$.

######  Lemma 36.

Fix $0<k<l$, as well as $a\in\mathbb{N}$ and $\delta\in\mathbb{Z}$. For every
nonnegative functional $G$ that is $\mathcal{F}_{\tau_{k}}$-measurable, we
have

$\mathbb{E}_{a,l,\delta}[G]=\mathbb{E}_{\infty}\big{[}\Phi_{a,l,\delta}(\tau_{k},k,\Lambda(\tau_{k}))\cdot
G\big{]}\,,$

where

$\Phi_{a,l,\delta}(t,l^{\prime},j)=\frac{Q_{l-l^{\prime}}(2a+l-t)}{Q_{l}(2a+l)}\frac{P_{l-l^{\prime}}(\delta-j)}{P_{l}(\delta)}\,.$

###### Proof.

It suffices to prove the result when $G$ is the indicator of the contour and
label processes of a given well-labeled forest with $l^{\prime}$ trees,
$(t-l^{\prime})/2$ edges, and tilt $j$. In this case,
$\mathbb{E}_{a,l,\delta}[G]$ is equal to the number of ways in which one can
complete this labeled forest into a well-labeled forest with $l$ trees, $a$
edges and tilt $\delta$, which is the number of forests with $l-l^{\prime}$
trees, $a+(l^{\prime}-t)/2$ edges and tilt $\delta-j$, divided by the number
of well-labeled forests with $l$ trees, $a$ edges and tilt $\delta$. We
conclude by Proposition 9. ∎

In addition to the already fixed sequence $(l_{n})$, we consider two more
sequences $(a_{n})$, $(\delta_{n})$ satisfying (8). We will need the following
direct consequence of the local limit theorem [BGT89, Theorem 8.4.1]. Recall
the definition of $\varphi_{A,L,\Delta}$ given in (31).

######  Lemma 37.

If the integer-valued sequence $(l^{\prime}_{n})$ satisfies
$l^{\prime}_{n}/\sqrt{2n}\to L^{\prime}\in(0,L)$, it holds that

$\sup_{0\leq t\leq
a_{n},\,j\in\mathbb{Z}}\left|\Phi_{a_{n},l_{n},\delta_{n}}(t,l^{\prime}_{n},j)-\varphi_{A,L,\Delta}\Big{(}\frac{t}{n},L^{\prime},\Big{(}\frac{9}{8n}\Big{)}^{\frac{1}{4}}j\Big{)}\right|\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}0\,.$ (42)

We start with the following conditioned version of Corollary 35.

######  Proposition 38.

On $\mathcal{C}\times\mathcal{C}\times\mathcal{C}^{(2)}$, the triple
$\big{(}C_{(n)},\Lambda_{(n)},D_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]^{2}}\big{)}$
considered under $\mathbb{P}_{a_{n},l_{n},\delta_{n}}$ converges in
distribution to $\big{(}X,W,D_{X,W}\big{)}$, considered under
$\mathbf{Slice}_{A,L,\Delta}$.

###### Proof.

The joint convergence of the first two coordinates is standard; see e.g.
[Bet10, Corollary 16]. Let us fix $\varepsilon\in(0,L)$, define
$l_{n}^{\varepsilon}=l_{n}-\lfloor\varepsilon\sqrt{2n}\rfloor$, so that
$l_{n}^{\varepsilon}/\sqrt{2n}\to L-\varepsilon$, and set

$D_{(n)}^{\varepsilon}=D_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}^{\varepsilon}}/2n]^{2}}\qquad\text{
and }\qquad
D_{(n)}^{0}=D_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]^{2}}\,.$

By the usual bound (5), for every $i$, $j\in[0,\tau_{l_{n}}]$,

$\displaystyle|D_{\infty}(i\wedge\tau_{l_{n}^{\varepsilon}},j\wedge\tau_{l_{n}^{\varepsilon}})-D_{\infty}(i,j)|$
$\displaystyle\leq
D_{\infty}(i,i\wedge\tau_{l_{n}^{\varepsilon}})+D_{\infty}(j,j\wedge\tau_{l_{n}^{\varepsilon}})$
$\displaystyle\leq
4\big{(}\omega(\Lambda_{n};\tau_{l_{n}}-\tau_{l_{n}^{\varepsilon}})+1\big{)}\,,$

where $\omega(f;\cdot)$ denotes the modulus of continuity of $f$. This implies
that

$\mathrm{dist}_{\mathcal{C}^{(2)}}\big{(}D_{(n)}^{\varepsilon},D_{(n)}^{0}\big{)}\leq\frac{\tau_{l_{n}}-\tau_{l_{n}^{\varepsilon}}}{2n}+4\,\omega\big{(}\Lambda_{(n)},(\tau_{l_{n}}-\tau_{l_{n}^{\varepsilon}})/2n\big{)}+\mathcal{O}\big{(}n^{-1/4}\big{)}\,.$

From the joint convergence of the first two coordinates, we have, for every
$\eta>0$,

$\limsup_{n\to\infty}\mathbb{P}_{a_{n},l_{n},\delta_{n}}\Big{(}\mathrm{dist}_{\mathcal{C}^{(2)}}\big{(}D_{(n)}^{\varepsilon},D_{(n)}^{0}\big{)}\geq\eta\Big{)}\\\
\leq\mathbf{Slice}_{A,L,\Delta}\big{(}A-T_{L-\varepsilon}+4\,\omega(W;A-T_{L-\varepsilon})\geq\eta\big{)}\,,$

which tends to $0$ as $\varepsilon\to 0$ since $T_{L-\varepsilon}\to T_{L}=A$
a.s. under $\mathbf{Slice}_{A,L,\Delta}$. We next show that
$D^{\varepsilon}_{(n)}$ under $\mathbb{P}_{a_{n},l_{n},\delta_{n}}$ converges
in distribution to $D^{(0,L-\varepsilon)}$ under
$\mathbf{Slice}_{A,L,\Delta}$, and use the principle of accompanying laws
[Str11, Theorem 9.1.13] to conclude that, jointly with the convergence of
$(C_{(n)},\Lambda_{(n)})$ to $(X,W)$, the process $D_{(n)}^{0}$ converges to
the distributional limit of $D^{(0,L-\varepsilon)}$ as $\varepsilon\to 0$,
which is none other than $D^{(0,L)}$, due to Corollary 27.

To prove the claimed convergence of $D^{\varepsilon}_{(n)}$ to
$D^{(0,L-\varepsilon)}$, we denote by $C_{(n)}^{\varepsilon}$ and
$\Lambda_{(n)}^{\varepsilon}$ the restrictions of $C_{(n)}$ and
$\Lambda_{(n)}$ to $[0,\tau_{l_{n}^{\varepsilon}}/2n]$ and let $F$ be a
nonnegative bounded continuous function. Using Lemma 36, then Corollary 35
(for the choice of $L-\varepsilon$ instead of $L$) and Lemma 37 gives

$\mathbb{E}_{a_{n},l_{n},\delta_{n}}\big{[}F\big{(}C_{(n)}^{\varepsilon},\Lambda_{(n)}^{\varepsilon},D^{\varepsilon}_{(n)}\big{)}\big{]}=\mathbb{E}_{\infty}\left[\Phi_{a_{n},l_{n},\delta_{n}}\big{(}\tau_{l_{n}^{\varepsilon}},l_{n}^{\varepsilon},\Lambda(\tau_{l_{n}^{\varepsilon}})\big{)}F\big{(}C_{(n)}^{\varepsilon},\Lambda_{(n)}^{\varepsilon},D^{\varepsilon}_{(n)}\big{)}\right]\\\
\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}\mathbf{Half}\big{[}\varphi_{A,L,\Delta}\big{(}T_{L-\varepsilon},L-\varepsilon,W_{T_{L-\varepsilon}}\big{)}F\big{(}X^{(0,L-\varepsilon)},W^{(0,L-\varepsilon)},D^{(0,L-\varepsilon)}\big{)}\big{]}\,,$

the latter being equal to
$\mathbf{Slice}_{A,L,\Delta}\big{[}F\big{(}X^{(0,L-\varepsilon)},W^{(0,L-\varepsilon)},D^{(0,L-\varepsilon)}\big{)}\big{]}$
by Lemma 32. ∎

##### GHP convergence.

We infer from Proposition 38 the GHP convergence of Theorem 12 by a standard
method. First, by Skorokhod’s representation theorem, we may assume that we
are working on a probability space on which the convergence of Proposition 38
is almost sure. We let $\mathsf{Sl}_{A,L,\Delta}$ be the continuum slice coded
by the limiting process, and $\mathrm{Sl}_{n}$ be the slice encoded by the
forest whose rescaled contour and label processes make up the pair
$(C_{(n)},\Lambda_{(n)})$. As mentioned before Corollary 35, $\mathrm{Sl}_{n}$
is isometrically embedded in $Q_{\infty}$, so that the process
$D_{(n)}{\raisebox{-1.49994pt}{$|$}}{}_{[0,\tau_{l_{n}}/2n]^{2}}$ under
$\mathbb{P}_{a_{n},l_{n},\delta_{n}}$ projects into the metric of
$\Omega_{n}(\mathrm{Sl}_{n})$.

Then, from this almost sure convergence, we easily deduce that the distortion
of the correspondence $\mathcal{R}_{n}$ given by

$\mathcal{R}_{n}=\big{\\{}\big{(}v_{\lfloor(2a_{n}+l_{n})s\rfloor},\mathbf{p}_{X,W}(As)\big{)}\,:\,s\in[0,1]\big{\\}}$
(43)

between $\Omega_{n}(\mathrm{Sl}_{n})$ minus its shuttle and
$\mathsf{Sl}_{A,L,\Delta}$ tends to $0$ as $n\to\infty$. Forgetting the marks
and measures, this already gives the desired convergence in the $0$-marked
Gromov–Hausdorff topology.

In order to include the marking and measures, we use the technique of
enlargement of correspondences already used in the proof of Lemma 23. Namely,
we fix $\varepsilon>0$ and let $\mathcal{R}_{n}^{\varepsilon}$ be the set of
points of the form $(v,x)$ in $\mathrm{Sl}_{n}\times\mathsf{Sl}_{A,L,\Delta}$
such that there exists $(w,y)\in\mathcal{R}_{n}$ satisfying
$d_{\mathrm{Sl}_{n}}(v,w)<(8n/9)^{1/4}\varepsilon$ and $D(x,y)<\varepsilon$.
As before, the distortion of $\mathcal{R}_{n}^{\varepsilon}$ is at most
$\operatorname{dis}(\mathcal{R}_{n})+4\varepsilon$. Let us start with the
marks.

##### Marks.

For a function $f\in\mathcal{C}$ defined over the interval $I$, we say that
$s\in I$ is a _left-minimum_ of $f$ if $f(t)\geq f(s)$ for every $t\leq s$ in
$I$, and we call it strict if $f(t)>f(s)$ for $t<s$ in $I$. Note that the
points of the form $v_{i}$ and $\mathbf{p}_{X,W}(s)$ where $i$ and $s$ are
left-minimums of $\Lambda_{n}$ and $W$ respectively belong to the maximal
geodesics of $\mathrm{Sl}_{n}$ and $\mathsf{Sl}_{A,L,\Delta}$, and that all
points in these sets are in fact of this form, where we can even require the
stronger property that $i$ and $s$ are strict left-minimums.

By the uniform convergence of $\Lambda_{(n)}$ to $W$, for every $\eta>0$, the
following holds provided $n\geq n_{0}$ for some $n_{0}$ : every strict left-
minimum of $\Lambda_{(n)}$ is at distance at most $\eta/2$ from some (not
necessarily strict) left-minimum of $W$, and vice-versa, exchanging the roles
of $\Lambda_{(n)}$ and $W$. Up to increasing $n_{0}$, we furthermore assume
that $|({2a_{n}+l_{n}})/{2n}-A|<\eta/2$ as soon as $n\geq n_{0}$. Choosing
$\eta$ small enough so that
$|D_{(n)}(s,t)-D_{(n)}(s^{\prime},t^{\prime})|\leq\varepsilon$ for every $n$
and $|s-s^{\prime}|\leq\eta$, $|t-t^{\prime}|\leq\eta$, we deduce that the
extended correspondence $\mathcal{R}_{n}^{\varepsilon}$ is compatible with the
maximal geodesics for $n\geq n_{0}$.

The argument is similar for the shuttles. This time, we note that elements of
the shuttle of $\mathsf{Sl}_{A,L,\Delta}$ are of the form
$\mathbf{p}_{X,W}(s)$ where $s$ is a right-minimum of the function $W$ (with
an obvious definition), while elements of the shuttle of $\mathrm{Sl}_{n}$ are
at distance $1$ away from points of the form $v_{i}$ where $i$ is a right-
minimum of the function $\Lambda_{n}$.

The mark corresponding to the base is also treated similarly. Recall from
Section 2.3 that vertices of the base are at distance at most
$B_{n}=\max_{1\leq i\leq
l_{n}}|\Lambda_{n}(\rho^{i})-\Lambda_{n}(\rho^{i-1})|+1$ from some element of
the floor $\\{\rho^{0},\ldots,\rho^{l_{n}}\\}$ of the forest coding the slice.
The process of labels $(\Lambda_{n}(\rho^{i}),0\leq i\leq l_{n})$ forms a
random walk with shifted geometric(1/2) increments conditioned to be equal to
$\delta_{n}$ at time $l_{n}$, so, under our assumptions, it converges, after
rescaling by $\sqrt{2n}$ in time and $(8n/9)^{1/4}$ in space, to a continuous
process (which is easily checked to be the Brownian bridge
$\zeta=(W_{T_{x}},0\leq x\leq L)$), so that
$B_{n}=\scalebox{0.5}[0.62]{$\mathbfcal{O}$}(n^{1/4})$ a.s. Therefore, the
base of $\mathrm{Sl}_{n}$ is at Hausdorff distance
$\scalebox{0.5}[0.62]{$\mathbfcal{O}$}(n^{1/4})$ from the floor
$\\{\rho^{i},0\leq i\leq l_{n}\\}$. In turn, these vertices are exactly those
of the form $v_{i}$ where $i$ is a left-minimum of the contour process
$C_{n}$. Moreover, by definition, the base of $\mathsf{Sl}_{A,L,\Delta}$
consists of the points $\mathbf{p}_{X,W}(s)$ where $s$ is a left-minimum of
the process $X$. Therefore, the same argument as for the maximal geodesic –
replacing the processes $\Lambda_{n}$ and $W$ by $C_{n}$ and $X$ – shows that,
a.s., for every $n$ large enough, the correspondence
$\mathcal{R}_{n}^{\varepsilon}$ is also compatible with the bases of
$\mathrm{Sl}_{n}$ and of $\mathsf{Sl}_{A,L,\Delta}$.

##### Measures.

Finally, let us deal with the convergence of the measures, starting with the
area measure. To this end, note that, for $t$ in $[0,2a_{n}+l_{n}]$, the
contour process $C_{n}$ at time $t$ has either a left derivative equal to $+1$
or to $-1$. Letting $i^{n}_{t}=\lceil t\rceil$ in the former case and
$i^{n}_{t}=\lfloor t\rfloor$ in the latter case, the image of
$\mathrm{Leb}_{[0,(2a_{n}+l_{n})/2n]}$ by $t\mapsto v_{i^{n}_{2nt}}$ is the
counting measure on the set of all non-floor vertices of the encoding forest,
divided by $n$. Since the number of floor vertices is $\mathcal{O}(\sqrt{n})$,
the counting measure on all vertices of $\mathrm{Sl}_{n}$ (except on the
shuttle) divided by $n$ is at vanishing Prokhorov distance from the counting
measure on non-floor vertices of the forest, divided by $n$. Let $\omega_{n}$
be the image of the Lebesgue measure on $[0,A\wedge((2a_{n}+l_{n})/{2n})]$ by
the mapping $t\mapsto(v_{i^{n}_{2nt}},\mathbf{p}_{X,W}(t))$. Then $\omega_{n}$
is carried by the correspondence $\mathcal{R}_{n}^{\varepsilon}$ for every $n$
large enough, and its image measures on $\mathrm{Sl}_{n}$ and
$\mathsf{Sl}_{A,L,\Delta}$ by the coordinate projections are at vanishing
Prokhorov distances from $\mu_{\mathrm{Sl}_{n}}/n$ and $\mu^{(0,L)}$
respectively.

For the base measure, we let $\omega^{\prime}_{n}$ be the image of the
Lebesgue measure on $[0,L\wedge l_{n}/\sqrt{2n}]$ by the mapping
$t\mapsto(v_{\tau_{\lfloor\sqrt{2n}\,t\rfloor}},\mathbf{p}_{X,W}(T_{t}))$.
Then $\omega^{\prime}_{n}$ is carried by $\mathcal{R}_{n}^{\varepsilon}$, by
the above discussion on the mark corresponding to the base. Moreover, the
coordinate projections of $\omega^{\prime}_{n}$ are at vanishing Prokhorov
distance, respectively, from the counting measure on
$\\{\rho^{0},\ldots,\rho^{l_{n}}\\}$ divided by $\sqrt{2n}$, and
$\nu^{(0,L)}$. We now observe that, in turn, the counting measure on
$\\{\rho^{0},\ldots,\rho^{l_{n}}\\}$ divided by $\sqrt{2n}$, is at vanishing
Prokhorov distance from the renormalized counting measure
$\nu_{\beta_{n}}/\sqrt{8n}$ of the base. To justify this, observe from Section
2.3 and the definition of the interval CVS bijection that the sequence
$\Lambda_{n}(w_{0})$, …, $\Lambda_{n}(w_{2l_{n}+\delta_{n}})$ of labels of the
vertices $w_{0}$, …, $w_{2l_{n}+\delta_{n}}$ of the base, taken in contour
order, forms a simple random walk starting with a $-1$ step, and conditioned
on hitting $\delta_{n}$ at time $2l_{n}+\delta_{n}$. Moreover, if we write the
set
$\\{j\in\\{0,\ldots,2l_{n}+\delta_{n}-1\\}:\Lambda_{n}(w_{j+1})-\Lambda_{n}(w_{j})=-1\\}$
of down steps of this walk as $\\{j_{0},j_{1},\ldots,j_{l_{n}-1}\\}$ with
$0=j_{0}<j_{1}<j_{2}<\dots<j_{l_{n}-1}$, then the $i$-th root $\rho^{i}$ is
equal to $w_{j_{i}}$ for $0\leq i<l_{n}$. Now consider a uniform random
variable $U$ in $[0,1)$. Then $w_{j_{\lfloor l_{n}U\rfloor}}$ is a uniformly
chosen forest root, while $w_{\lfloor(2l_{n}+\delta_{n})U\rfloor}$ is a
uniformly chosen vertex of the base (excluding $\rho^{l_{n}}$ in both cases).
Moreover, a standard large deviation estimate entails that $\max_{0\leq
k<l_{n}}|j_{k}-2k|=\mathcal{O}(\log n)$ in probability. In turn, this easily
implies that $d_{\mathrm{Sl}_{n}}(w_{j_{\lfloor
l_{n}U\rfloor}},w_{\lfloor(2l_{n}+\delta_{n})U\rfloor})=\mathcal{O}(\log n)$
in probability, showing that the uniform measure on the $l_{n}\sim L\sqrt{2n}$
elements of $\\{\rho^{0},\ldots,\rho^{l_{n}-1}\\}$ is at vanishing Prokhorov
distance from the uniform measure on the $2l_{n}+\delta_{n}\sim L\sqrt{8n}$
vertices of the base.

##### Conclusion.

By Lemma 17, we finally obtain that

$\limsup_{n\to\infty}\mathrm{d}_{\mathrm{GHP}}^{(5,2)}\big{(}\Omega_{n}(\mathrm{Sl}_{n}),\mathsf{Sl}_{A,L,\Delta}\big{)}\leq\varepsilon\,.$

Since $\varepsilon>0$ was arbitrary, this concludes the proof of Theorem 12.

##  6 Convergence of quadrilaterals with geodesic sides

The general method to prove Theorem 14 is the same as for slices. We start by
seeing a discrete quadrilateral as part of a discrete map that is known to
converge to a Brownian surface, which in this case is the Brownian plane
rather than the Brownian half-plane. However, the lack of an analog of
Corollary 27, namely that quadrilaterals are only _locally_ isometrically
embedded in the Brownian plane, makes matters considerably more delicate. For
this reason, we adapt the strategy we used in [BM17, Section 4] when treating
the case of noncomposite slices. Beware that, in this section, part of the
notation we will be using is slightly conflicting with that of Section 5: in
particular, the random times $T_{x}$ will be re-defined.

###  6.1 Quadrilaterals coded by two functions

In contrast with slices, which were coded by a pair of functions defined on a
common interval $I$, a quadrilateral will be coded by a pair of functions
defined on a common union of two intervals $I_{-}\cup I_{+}$, each interval
accounting for one “half” of the quadrilateral. This leads to similar but
slightly more intricate definitions. We start with the most convenient
setting, asking that $\sup I_{-}=\inf I_{+}=0$.

Recall the notation of Section 5.1.1. We adapt Section 5.1.2 to quadrilaterals
instead of slices as follows. We now say that a pair $(f,g)\in\mathcal{C}^{2}$
of functions with common closed interval of definition $I$ is a _quadrilateral
trajectory_ if they satisfy (26) and the following:

  * •

the interval $I$ contains $0$ in its interior and is either bounded or equal
to the whole real line $\mathbb{R}$, and, letting $I_{+}=I\cap\mathbb{R}_{+}$
and $I_{-}=I\cap\mathbb{R}_{-}$,

  * •

we have $\inf_{I_{-}}f=\inf_{I_{+}}f$, and,

  * •

if $I=\mathbb{R}$, then $\inf_{t\geq 0}f(t)=\inf_{t\leq 0}f(t)=\inf_{t\geq
0}g(t)=\inf_{t\leq 0}g(t)=-\infty$.

We may observe that
$(f{\raisebox{-1.49994pt}{$|$}}{}_{I_{+}},g{\raisebox{-1.49994pt}{$|$}}{}_{I_{+}})$
is a slice trajectory, a fact that will not be used here. For a quadrilateral
trajectory $(f,g)$, we set

$\mkern 1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern
1.0mu_{g}(s,t)=\begin{cases}d_{g}(s,t)&\text{ for }\ s,t\in I_{+}\mbox{ or
}s,t\in I_{-}\\\ \infty&\text{ for }\ st<0\end{cases},$ (44)

and

${\mkern 1.0mu\widehat{\mkern-1.0mu{D}\mkern-1.0mu}\mkern 1.0mu}_{f,g}=\mkern
1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern 1.0mu_{g}/\\{d_{f}=0\\}\,.$
(45)

Note that $\mkern 1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern 1.0mu_{g}$ is
the disjoint union pseudometric of the two $\mathbb{R}$-tree pseudometrics
$d_{g{\text{\scriptsize\raisebox{-1.04996pt}{$|$}}}{}_{I_{+}}}$ and
$d_{g{\text{\scriptsize\raisebox{-1.04996pt}{$|$}}}{}_{I_{-}}}$. Let
$\big{(}\mkern 1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern
1.0mu_{f,g},{\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu}_{f,g}\big{)}$ be the quotient space $I/\\{\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern 1.0mu_{f,g}=0\\}$ equipped with
the metric induced by ${\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu}_{f,g}$, still denoted by the same symbol. We call the metric space

$\mathsf{Qd}_{f,g}=(\mkern 1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern
1.0mu_{f,g},\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu_{f,g})$

the _quadrilateral_ coded by $(f,g)$.

We extend the above constructions to unions of two closed intervals
$I=I_{-}\cup I_{+}$ where $I_{-}\subseteq\mathbb{R}_{-}$ and
$I_{+}\subseteq\mathbb{R}_{+}$, as follows. First, a pair of functions $(f,g)$
defined on $I$ is a _quadrilateral trajectory_ if the pair
$(f^{\prime},g^{\prime})$ defined by

$f^{\prime}(t)=\begin{cases}f(t+\inf I_{+})&\text{ for }t\in I_{+}-\inf
I_{+}\\\ f(t+\sup I_{-})&\text{ for }t\in I_{-}-\sup I_{-}\end{cases},$

and similarly for $g^{\prime}$, is a quadrilateral trajectory as defined
above. Note that the continuity hypothesis on $f^{\prime}$ implies in
particular that $f(\sup I_{-})=f(\inf I_{+})$, and similarly for $g$. We then
define the quadrilateral coded by $(f,g)$ using the exact same definitions as
above. Note that the mapping $t\mapsto(t-\inf I_{+})\mathbf{1}_{t\in
I_{+}}+(t-\sup I_{-})\mathbf{1}_{t\in I_{-}}$ induces an isometry from
$(\mkern 1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern 1.0mu_{f,g},\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern 1.0mu_{f,g})$ onto $(\mkern
1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern
1.0mu_{f^{\prime},g^{\prime}},\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu_{f^{\prime},g^{\prime}})$.

From now on, we work in this extended framework and consider a fixed
quadrilateral trajectory $(f,g)$.

##### Geodesic sides and area measure.

For every $t\in I\setminus\\{0\\}$, we let $I_{t}=I_{-}$ if $t<0$ or
$I_{t}=I_{+}$ if $t>0$, and set

$\displaystyle\Gamma_{t}(r)$ $\displaystyle=\inf\\{s\geq t:g(s)=g(t)-r\\}$
$\displaystyle\text{ for $r\in\mathbb{R}_{+}$ such that
}\inf_{\begin{subarray}{c}s\geq t\\\ s\in I_{t}\end{subarray}}g(s)\leq
g(t)-r\,;$ $\displaystyle\Xi_{t}(r)$ $\displaystyle=\sup\\{s\leq
t:g(s)=g(t)-r\\}$ $\displaystyle\text{ for $r\in\mathbb{R}_{+}$ such that
}\inf_{\begin{subarray}{c}s\leq t\\\ s\in I_{t}\end{subarray}}g(s)\leq
g(t)-r\,.$

If $0\in I$, we also define $\Gamma_{0}$ and $\Xi_{0}$ with the same
definition, using $I_{0}=I_{+}$ in the definition of $\Gamma_{0}$, while using
$I_{0}=I_{-}$ in the definition of $\Xi_{0}$. Observe that, in contrast with
the definition for slices, the infimum of $g$ is now taken on a subset of
$I_{t}$. In particular, this implies that the ranges of $\Gamma_{t}$,
$\Xi_{t}$ are included in $I_{t}$. From the same discussion as the one around
(27), we see that $\Gamma_{t}$, $\Xi_{t}$ are geodesics for the pseudometrics
$\mkern 1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern 1.0mu_{g}$ and $\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern 1.0mu_{f,g}$. In the case where
$\sup I_{+}=\infty$, then, for every $t\in I_{+}$, the range of the path
$\Gamma_{t}$ is a geodesic ray, and, in the case where $\inf I_{-}=-\infty$,
then the same goes for $\Xi_{t}$ for every $t\in I_{-}$. This allows to define
geodesic paths in $\mathsf{Qd}_{f,g}$ by the formulas

$\displaystyle\gamma_{t}(r)$ $\displaystyle=\mkern
1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu_{f,g}(\Gamma_{t}(r))\,,\qquad 0\leq r\leq g(t)-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern 1.0mu(t,\sup I_{t})\,,\
r\in\mathbb{R}\,,$ $\displaystyle\xi_{t}(r)$ $\displaystyle=\mkern
1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu_{f,g}(\Xi_{t}(r))\,,\qquad 0\leq r\leq g(t)-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern 1.0mu(\inf I_{t},t)\,,\
r\in\mathbb{R}\,,$

where $\mkern 1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu_{f,g}:I\to\mkern 1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern
1.0mu_{f,g}$ is the canonical projection and, as above, if $0\in I$,
$I_{0}=I_{+}$ in the definition of $\gamma_{0}$ and $I_{0}=I_{-}$ in that of
$\xi_{0}$. Note that the geodesics $\gamma_{t}$, $\xi_{t}$ share a common
initial part.

The quadrilateral $\mathsf{Qd}_{f,g}$ comes with four or two geodesic sides,
defined as follows. If $I$ is bounded, the particular geodesics
$\gamma=\gamma_{\inf I_{+}}$ and $\bar{\gamma}=\gamma_{\inf I_{-}}$ are called
the _maximal geodesics_ of $\mathsf{Qd}_{f,g}$, while $\xi=\xi_{\sup I_{+}}$
and $\bar{\xi}=\xi_{\sup I_{-}}$ are called the _shuttles_ of
$\mathsf{Qd}_{f,g}$. In this case, $\gamma$, $\xi$ (resp. $\bar{\gamma}$,
$\bar{\xi}$) have a common endpoint $x_{*}=\mkern
1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern 1.0mu_{f,g}(s_{*})$
(resp. $\bar{x}_{*}=\mkern
1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu_{f,g}(\bar{s}_{*})$) where $s_{*}\in I_{+}$ is such that
$g(s)=\inf_{I_{+}}g$ (resp. $\bar{s}_{*}\in I_{-}$ is such that
$g(\bar{s}_{*})=\inf_{I_{-}}g$). The points $x_{*}$, $\bar{x}_{*}$ are called
the _apexes_ of $\mathsf{Qd}_{f,g}$. If $I$ is unbounded, then
$\mathsf{Qd}_{f,g}$ has one _maximal geodesic_ $\gamma=\gamma_{\inf I_{+}}$
and one _shuttle_ $\bar{\xi}=\xi_{\sup I_{-}}$; we set
$\xi_{\infty}=\gamma_{-\infty}=\varnothing$.

Finally, the _area measure_ is defined as $\mu=(\mkern
1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu_{f,g})_{*}\mathrm{Leb}_{I}$.

##### Gluing quadrilaterals.

For $x\in\mathbb{R}$, we let

$\displaystyle T_{x}$ $\displaystyle=\inf\\{t\in
I_{+}:f(t)=-x\\}\in\mathbb{R}_{+}\cup\\{+\infty\\}\,,$
$\displaystyle\bar{T}_{x}$ $\displaystyle=\sup\\{t\in
I_{-}:f(t)=-x\\}\in\mathbb{R}_{-}\cup\\{-\infty\\}\,,$

as well as $T_{\infty}=-\bar{T}_{\infty}=\infty$. Note that, here again, there
is a slight difference with the definition of Section 5 since, now,
$\mathbb{R}_{-}$ and $\mathbb{R}_{+}$ play different roles. Recall that
$\inf_{I_{-}}f=\inf_{I_{+}}f=\inf_{I}f$, and let $H$,
$H^{\prime}\in\mathbb{R}_{+}\cup\\{\infty\\}$ be such that $0\leq H\leq
H^{\prime}\leq-\inf_{I}f$. We may define the restrictions
$f^{(H,H^{\prime})}$, $g^{(H,H^{\prime})}$ of $f$ and $g$ to the union of
intervals
$I^{(H,H^{\prime})}=[\bar{T}_{H^{\prime}},\bar{T}_{H}]\cup[T_{H},T_{H^{\prime}}]$,
which is a subset of $I$. The pair $(f^{(H,H^{\prime})},g^{(H,H^{\prime})})$
is another quadrilateral trajectory. The associated quadrilateral is defined
as

$\mathsf{Qd}^{(H,H^{\prime})}=\big{(}\mkern
1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern 1.0mu^{(H,H^{\prime})},\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime})}\big{)}=\mathsf{Qd}_{f^{(H,H^{\prime})},g^{(H,H^{\prime})}}\,.$

We let $\mkern 1.0mu\widehat{\mkern-1.0mu\mathbf{p}\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime})}:I^{(H,H^{\prime})}\to\mkern
1.0mu\widehat{\mkern-1.0muM\mkern-1.0mu}\mkern 1.0mu^{(H,H^{\prime})}$ be the
canonical projection, $\mu^{(H,H^{\prime})}$ be the area measure of
$\mathsf{Qd}^{(H,H^{\prime})}$, and, whenever they exist,
$\gamma^{(H,H^{\prime})}$, $\bar{\gamma}^{(H,H^{\prime})}$ be the maximal
geodesics, $\xi^{(H,H^{\prime})}$, $\bar{\xi}^{(H,H^{\prime})}$ be the
shuttles.

We refer to Figure 15 for an illustration of the following proposition in the
upcoming context of random quadrilaterals in the Brownian plane.

######  Proposition 39.

Let $0\leq H<H^{\prime}<H^{\prime\prime}\leq-\inf_{I}f$ be in the extended
positive real line. Then

$\mathsf{Qd}^{(H,H^{\prime\prime})}=G\big{(}G\big{(}\mathsf{Qd}^{(H,H^{\prime})},\mathsf{Qd}^{(H^{\prime},H^{\prime\prime})};\xi^{(H,H^{\prime})},\gamma^{(H^{\prime},H^{\prime\prime})}\big{)};\bar{\gamma}^{(H,H^{\prime})},\bar{\xi}^{(H^{\prime},H^{\prime\prime})}\big{)}\,,$
(46)

and it holds that

$\displaystyle\gamma^{(H,H^{\prime\prime})}$
$\displaystyle=\gamma^{(H,H^{\prime})}\cup\big{(}\gamma^{(H^{\prime},H^{\prime\prime})}\setminus\xi^{(H,H^{\prime})}\big{)}\,,$
$\displaystyle\xi^{(H,H^{\prime\prime})}$
$\displaystyle=\xi^{(H^{\prime},H^{\prime\prime})}\cup\big{(}\xi^{(H,H^{\prime})}\setminus\gamma^{(H^{\prime},H^{\prime\prime})}\big{)}\,,$
$\displaystyle\bar{\gamma}^{(H,H^{\prime\prime})}$
$\displaystyle=\bar{\gamma}^{(H^{\prime},H^{\prime\prime})}\cup\big{(}\bar{\gamma}^{(H,H^{\prime})}\setminus\bar{\xi}^{(H^{\prime},H^{\prime\prime})}\big{)}\,,$
$\displaystyle\bar{\xi}^{(H,H^{\prime\prime})}$
$\displaystyle=\bar{\xi}^{(H,H^{\prime})}\cup\big{(}\bar{\xi}^{(H^{\prime},H^{\prime\prime})}\setminus\bar{\gamma}^{(H,H^{\prime})}\big{)}\,.$

Observe that, after the first gluing operation is performed, the marks
$\bar{\gamma}$ and $\bar{\xi}$ remain geodesic, as observed in Section 3.3.
Note also that the order of the gluings in (46) is not important, due to (12).

###### Proof.

The proof is similar to that of Proposition 26, and we only sketch the
argument. Again, we view $I^{(H,H^{\prime\prime})}$ as a disjoint union
$I^{(H,H^{\prime})}\sqcup I^{(H^{\prime},H^{\prime\prime})}$ (denoting
elements of these sets with superscripts $0$, $1$ respectively) where the
extremities $T_{H^{\prime}}^{0}$, $T_{H^{\prime}}^{1}$ and
$\bar{T}_{H^{\prime}}^{0}$, $\bar{T}_{H^{\prime}}^{1}$ are identified. We then
observe that the pseudometric $\mkern
1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern 1.0mu_{g}$ can be viewed as a
quotient $d/R_{1}$, where $d$ is the disjoint union metric on
$I^{(H,H^{\prime})}\sqcup I^{(H^{\prime},H^{\prime\prime})}$ whose restriction
to each interval composing this set equals the restriction of $d_{g}$ to that
interval, and $R_{1}$ is the coarsest equivalence relation containing

$\big{\\{}\big{(}\Xi_{T_{H^{\prime}}}(r)^{0},\Gamma_{T_{H^{\prime}}}(r)^{1}\big{)},\
0\leq r\leq g(T_{H^{\prime}})-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(T_{H},T_{H^{\prime}})\vee\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(T_{H^{\prime}},T_{H^{\prime\prime}})\big{\\}}\\\ \text{ and
}\big{\\{}\big{(}\Gamma_{\bar{T}_{H^{\prime}}}(r)^{0},\Xi_{\bar{T}_{H^{\prime}}}(r)^{1}\big{)},\
0\leq r\leq g(\bar{T}_{H^{\prime}})-\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(\bar{T}_{H},\bar{T}_{H^{\prime}})\vee\mkern
1.0mu\underline{\mkern-1.0mug\mkern-1.0mu}\mkern
1.0mu(\bar{T}_{H^{\prime}},\bar{T}_{H^{\prime\prime}})\big{\\}}.$

Moreover, the equivalence relation $\\{d_{f}=0\\}$ factorizes in the sense
that, if $d_{f}(s,t)=0$ with $s$, $t\in I^{(H,H^{\prime\prime})}$, then it
must hold that $s$, $t$ belong either both to $I^{(H,H^{\prime})}$ or both to
$I^{(H^{\prime},H^{\prime\prime})}$. Therefore, setting $R_{2}$ as the
equivalence relation on $I^{(H,H^{\prime})}\sqcup
I^{(H^{\prime},H^{\prime\prime})}$ defined by $s^{i}\mathbin{R}_{2}t^{j}$ if
and only if $d_{f}(s,t)=0$ and $i=j\in\\{0,1\\}$, we obtain

$\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime\prime})}=(d/R_{1})/R_{2}=(d/R_{2})/R_{1}\,.$

We now recognize that $d/R_{2}$ is the pseudometric of the disjoint union of
$(I^{(H,H^{\prime})},\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime})})$ and $(I^{(H^{\prime},H^{\prime\prime})},\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H^{\prime},H^{\prime\prime})})$, while $R_{1}$ can be seen as the
coarsest equivalence relation obtained by first gluing $\xi^{(H,H^{\prime})}$
with $\gamma^{(H^{\prime},H^{\prime\prime})}$, and then
$\bar{\gamma}^{(H,H^{\prime})}$ with
$\bar{\xi}^{(H^{\prime},H^{\prime\prime})}$. ∎

Due to the fact that the second gluing operation in Proposition 39 involves
two geodesics belonging to the same space, there is no direct analog of
Corollary 27. However, we have the following alternative, which is an
immediate consequence of Lemma 21 and a crude estimate of the length of the
path $\bar{\xi}^{(H^{\prime},H^{\prime\prime})}$.

######  Proposition 40.

Under the same assumptions as in Proposition 39, it holds that for every $s$,
$t\in I^{(H,H^{\prime})}$,

$\mkern 1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime\prime})}(s,t)\leq\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime})}(s,t)\leq\mkern
1.0mu\widehat{\mkern-1.0muD\mkern-1.0mu}\mkern
1.0mu^{(H,H^{\prime\prime})}(s,t)+\omega(g;I^{(H^{\prime},H^{\prime\prime})})\,.$

Finally, we observe that the metric space $(M_{f,g},D_{f,g})$ obtained by
metric gluing of the pseudometric $d_{g}$ along the relation $\\{d_{f}=0\\}$,
rather than using $\mkern 1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern
1.0mu_{g}$ as in the definition of $\mathsf{Qd}_{f,g}$, is related to the
latter by a final gluing operation. The proof is analog to that of Proposition
39, noting that $d_{g}$ is the gluing of $\mkern
1.0mu\widehat{\mkern-1.0mud\mkern-1.0mu}\mkern 1.0mu_{g}$ along the coarsest
equivalence relation containing $\\{(\Gamma_{0}(r),\Xi_{0}(r)),r\geq 0\\}$.

######  Lemma 41.

One has
$\big{(}M_{f,g},D_{f,g}\big{)}=G\big{(}\mathsf{Qd}_{f,g};\gamma,\bar{\xi}\big{)}$.

###  6.2 Random continuum quadrilaterals

Let us now describe the limiting continuum quadrilaterals that appear in