# Compact Brownian surfaces II. Orientable surfaces

Jérémie Bettinelli LIX, cnrs, École polytechnique, Institut Polytechnique de
Paris Grégory Miermont UMPA, ENS de Lyon

###### Abstract

Fix an arbitrary compact orientable surface with a boundary and consider a
uniform bipartite random quadrangulation of this surface with $n$ faces and
boundary component lengths of order $\sqrt{n}$ or of lower order. Endow this
quadrangulation with the usual graph metric renormalized by $n^{-1/4}$, mark
it on each boundary component, and endow it with the counting measure on its
vertex set renormalized by $n^{-1}$, as well as the counting measure on each
boundary component renormalized by $n^{-1/2}$. We show that, as $n\to\infty$,
this random marked measured metric space converges in distribution for the
Gromov–Hausdorff–Prokhorov topology, toward a random limiting marked measured
metric space called a _Brownian surface_.

This extends known convergence results of uniform random planar
quadrangulations with at most one boundary component toward the _Brownian
sphere_ and toward the _Brownian disk_ , by considering the case of
quadrangulations on general compact orientable surfaces. Our approach consists
in cutting a Brownian surface into elementary pieces that are naturally
related to the Brownian sphere and the Brownian disk and their noncompact
analogs, the Brownian plane and the Brownian half-plane, and to prove
convergence results for these elementary pieces, which are of independent
interest.

###### Contents

  1. 1 Introduction
    1. 1.1 Context
    2. 1.2 Generalities and terminology on maps
    3. 1.3 The Gromov–Hausdorff–Prokhorov topology
    4. 1.4 The main convergence result
    5. 1.5 Scaling limits of Boltzmann quadrangulations
    6. 1.6 Perspectives
    7. 1.7 Organization of the paper
  2. 2 Variants of the Cori–Vauquelin–Schaeffer bijection
    1. 2.1 Basic construction
    2. 2.2 The generalized Chapuy–Marcus–Schaeffer bijection
    3. 2.3 Composite slices
    4. 2.4 Quadrilaterals with geodesic sides
    5. 2.5 Scaling limits of elementary pieces
  3. 3 Marking and gluing along geodesics
    1. 3.1 Useful facts on the GHP topology and markings
    2. 3.2 Geodesics in metric spaces
    3. 3.3 Gluing along geodesics
  4. 4 Proof of Theorem 1
    1. 4.1 Gluing quadrangulations from elementary pieces
    2. 4.2 Scaling limit of the collection of elementary pieces
    3. 4.3 Gluing pieces together
    4. 4.4 Topology and Hausdorff dimension
  5. 5 Convergence of composite slices
    1. 5.1 Metric spaces coded by real functions
    2. 5.2 Random continuum composite slices
    3. 5.3 The Brownian half-plane, and its embedded slices
    4. 5.4 The uniform infinite half-planar quadrangulation
    5. 5.5 Scaling limit of conditioned slices
  6. 6 Convergence of quadrilaterals with geodesic sides
    1. 6.1 Quadrilaterals coded by two functions
    2. 6.2 Random continuum quadrilaterals
    3. 6.3 The Brownian plane, and its embedded quadrilaterals
    4. 6.4 The uniform infinite planar quadrangulation
    5. 6.5 Discrete quadrilaterals in the UIPQ
    6. 6.6 Scaling limit of conditioned quadrilaterals
  7. 7 Construction from a continuous unicellular map
  8. A Technical lemmas on the Brownian plane
    1. A.1 Equivalence of definitions of the Brownian plane
    2. A.2 Convergence of the UIPQ to the Brownian plane
  9. B Scaling limit of size parameters in labeled maps
    1. B.1 Preliminaries
    2. B.2 Asymptotics of the scheme
    3. B.3 Asymptotics of the size parameters
    4. B.4 Boltzmann quadrangulations

##  1 Introduction

###  1.1 Context

Random maps, seen as discrete models of random $2$-dimensional geometries,
have generated a sustained interest in the last couple of decades. An
important instance of this line of research are the results by Le Gall [LG13]
and the second author [Mie13], showing that a uniform random quadrangulation
of the sphere with $n$ faces, seen as a random finite metric space by endowing
its vertex set with the usual graph metric renormalized by $n^{-1/4}$,
converges in distribution toward the so-called _Brownian sphere_ , or
_Brownian map_. The aim of the present work is to generalize this result to
the case of general compact orientable surfaces. Let us start with some
elements of context.

##### Random surfaces as scaling limits of random maps.

While the idea that continuum random geometries should be obtained as scaling
limits of random maps originates from the physics literature on 2-dimensional
quantum gravity [Pol81, Dav85, KPZ88], this question was first approached in
the mathematical literature in the pioneering work of Chassaing and Schaeffer
[CS04], who studied the model of uniformly chosen random quadrangulations of
the sphere, and found in particular that the proper scaling factor in this
case was $n^{-1/4}$. Marckert and Mokkadem [MM03] then constructed a candidate
limiting space today called the Brownian sphere, and showed the convergence
toward it in another topology than the Gromov–Hausdorff topology. Le Gall
[LG07] later showed that the sequence of rescaled metric spaces associated
with uniform random quadrangulations of the sphere was relatively compact.
Finally, Le Gall [LG13] and the second author [Mie13] showed by two
independent approaches that the previous sequence converges toward the
Brownian sphere.

It is known that the Brownian sphere arises as a universal scaling limit for
many models of planar maps _that are uniformly chosen in a certain class,
given their face degrees_ , and provided that face degrees are typically all
of the same order of magnitude; see [LG13, BLG13, ABA17, BJM14, Abr16, CLG19,
ABA21, Mar22]. See also [LGM11] for models of maps that fall out of this
universality class.

The scaling limits of quadrangulations on surfaces that are more general than
the sphere were considered by the first author in [Bet15, Bet16], who showed
similar results to the above, but only up to extraction of appropriate
subsequences, leaving a gap that amounts to uniquely characterize the limit.
This gap was filled in the particular case of the disk topology in our
previous work [BM17]. In particular, we showed that a uniform quadrangulation
of genus $0$ with one boundary component having $n$ internal faces and
perimeter $2l_{n}$ weakly converges, once scaled by the factor $n^{-1/4}$ and
when $l_{n}\sim L\sqrt{2n}$, toward a random metric space called the _Brownian
disk of perimeter $L$_. Two alternate constructions of Brownian disks were
proposed by Le Gall [LG19a, LG22a], allowing in particular to show that
Brownian disks arise as connected components of the complement of metric balls
in the Brownian sphere, conditionally given their areas and boundary lengths.
See also [MS21a, BCK18, LGR20].

Besides the case of the sphere and the disk, only a few results have been
obtained for maps on compact surfaces. Namely, it has been shown that uniform
quadrangulations of a given compact surface with a boundary exhibit scaling
limits [Bet10, Bet12, Bet16], all of the same topology as the considered
surface, and geodesics to a uniformly chosen points were studied [Bet16]. More
recently, it was shown that uniformly distributed essentially simple toroidal
triangulations (that is, triangulations of the torus without contractible
loops or double edges forming cycles that are homotopic to $0$) also exhibit
scaling limits [BHL19], which are believed to be the same as for random
quadrangulations. See also [ARS22] for a scaling limit result of Boltzmann
random maps with annular topology.

There has also been a growing interest in noncompact versions of these models,
especially as they bridge some Brownian surfaces with so-called _uniform
infinite random maps_ , which are maps with infinitely many faces that first
arose in a work by Angel and Schramm [AS03], as _local limits_ of random
finite maps. Three main models of noncompact Brownian surfaces have been
identified: the _Brownian plane_ [CLG14], the _Brownian half-plane_ [GM17,
BMR19], and the _infinite-volume Brownian disk_ [BMR19], which can be thought
of as noncompact versions of the Brownian sphere and Brownian disks, either
with unbounded or bounded boundary. See [LGR21] for a framework unifying those
objects. The first two of these models will play an important role in the
current work.

This whole line of research crucially depends on strong combinatorial
techniques, and in particular on bijective approaches [Sch98, BDG04, AP15]
that allow to give very detailed quantitative information on the geodesic
paths in random maps and their scaling limits. The present work is no
exception. See for instance [LG10, Mie09, AKM17, MQ21, LG22b] for results
related to the structure of geodesics in the Brownian sphere, [LG19b] for a
recent survey, and [Cur19] for another approach called _peeling_. We note,
however, that, so far, these methods are restricted to models of maps chosen
uniformly, conditionally given their face degrees, as alluded to above.

##### Random surfaces via Liouville quantum gravity metrics.

A line of research parallel to the above consists in building the limiting
spaces directly as continuum random metrics in planar domains or Riemann
surfaces. This approach also finds its roots in the physics theory of
Liouville quantum gravity [Pol81]. In the case of Brownian surfaces, this has
first been implemented by Miller and Sheffield in a series of works [MS21a,
MS20, MS21b, MS21c], where they use a growth model called _Quantum Loewner
Evolution_ (QLE) to define a random metric on the plane, whose metric balls
are described by QLE, and whose law as an abstract metric space is equal to
that of the Brownian plane. Local variants of the construction allow to define
the Brownian sphere in this way. The Miller–Sheffield metric is in fact a
special element of a one-parameter family of _Liouville Quantum Gravity_ (LQG)
metrics, that have been defined as scaling limits of first-passage percolation
models in mollified exponentiated Gaussian free fields landscapes [DDDF20,
GM21b]. See [DDG21] for an overview of LQG metrics.

These constructions operate entirely in the continuum, and naturally ask
whether canonical embeddings of random maps in the sphere are compatible with
the convergence toward the Brownian sphere, in the sense that the metrics
induced by the embedding converge to the random metric of Miller–Sheffield.
Such a result was recently obtained by Holden and Sun [HS19] (which is the
last piece of a vast research project, described in details in this
reference), who showed the joint convergence of the metric and the area
measure generated by a uniform plane triangulation embedded via the Cardy
embedding in an equilateral triangle. We refer to the overview article
[GHS19].

##### Random surfaces and conformal field theories.

While the definition of LQG metrics applies to any field that “locally looks
like” the Gaussian free field, the exact law of the latter is of crucial
importance to obtain the exact law of random surfaces that arise as scaling
limits of maps, and this law can be obtained from Liouville conformal field
theory [Pol81]. Here, rather than dealing with random metrics, one is rather
interested in the computation of partition functions defined from the field,
and it has been shown recently in a rich body of work – see [DKRV16, GRV19,
KRV20] and references therein – that this theory has a probabilistic
interpretation in terms of Gaussian multiplicative chaoses, which are random
measures defined in terms of the Gaussian free field. This approach has
unveiled fundamental integrability properties for planar Gaussian
multiplicative chaoses, which can be used to provide exact distributions for
various quantities related to the LQG metrics, hence to the scaling limits of
random maps. For instance, in [ARS22], the authors compute the law of the
conformal modulus of a Brownian annulus, which is a member of the family of
Brownian surfaces described in the present work.

The interplay between these approaches provides a wealth of methods to prove
various properties of random surfaces [She22], and the geometric properties of
the Brownian surfaces, as well as the other LQG metrics, are the object of
intensive current research.

###  1.2 Generalities and terminology on maps

##### Surface with a boundary.

Recall that a _surface with a boundary_ is a nonempty Hausdorff topological
space in which every point has an open neighborhood homeomorphic to some open
subset of $\mathbb{R}\times\mathbb{R}_{+}$. Its _boundary_ is the set of
points having a neighborhood homeomorphic to a neighborhood of $(0,0)$ in
$\mathbb{R}\times\mathbb{R}_{+}$. When it is nonempty, this set forms a
$1$-dimensional topological manifold. In this work, we will only consider
orientable compact connected surfaces with a (possibly empty) boundary. By the
classification theorem, these are characterized up to homeomorphisms by two
nonnegative integers, the genus $g$ and the number $b$ of connected components
of the boundary. We denote by $\mathbf{\Sigma}_{b}^{[g]}$ the compact
orientable surface of genus $g$ with $b$ boundary components, which is unique
up to homeomorphisms. It can be obtained from the connected sum of $g$ tori,
or from the sphere in the case $g=0$, by removing $b$ disjoint open disks
whose boundaries are pairwise disjoint circles.

##### Map.

A _map_ is a proper cellular embedding of a finite graph, possibly with
multiple edges and loops, into a compact connected orientable surface
_without_ boundary. Here, the word _proper_ means that edges can intersect
only at vertices, and _cellular_ means that the connected components of the
complement of the edges, which are called the _faces_ of the map, are
homeomorphic to $2$-dimensional open disks. Maps will always be considered up
to orientation-preserving homeomorphisms of the surface into which they are
embedded. The _genus_ of a map is defined as the genus of the surface into
which it is embedded; we speak of _plane_ maps when the genus is $0$. We call
_half-edge_ an oriented edge in a map. With every half-edge, we may associate
in a one-to-one way a _corner_ , which is the angular sector lying to its left
at the origin of the half-edge. Note that this makes sense because the
surfaces we are considering are orientable. We say that a corner, or the
corresponding half-edge, is _incident_ to a face $f$ if it lies into $f$. We
also say that the face is incident to the corner or the half-edge in this
case. The number of half-edges (or equivalently, of corners) incident to a
face is called its _degree_.

A map is _rooted_ if it comes with a distinguished corner – or, equivalently,
a half-edge – called the _root_. Rooting is a very useful notion as it allows
to kill the symmetries of a map. In fact, when dealing with nonrooted maps, we
will systematically count them by weighting each map $\mathbf{m}$ by a factor
$1/\operatorname{Aut}(\mathbf{m})$, where $\operatorname{Aut}(\mathbf{m})$
denotes the number of automorphisms of $\mathbf{m}$. The latter is also equal
to $2\,|E(\mathbf{m})|/R(\mathbf{m})$, where $E(\mathbf{m})$ is the edge set
of $\mathbf{m}$, and $R(\mathbf{m})$ is the number of distinct rooted maps
that can be obtained from the nonrooted map $\mathbf{m}$. Therefore, with this
convention, the weighted number of nonrooted maps in a given family of maps
with a given number $e$ of edges is simply the cardinality of the set of
rooted maps from this same family, divided by $2e$.

##### Map with holes.

We will consider maps with pairwise distinct distinguished elements,
generically denoted by $\mathlcal{h}_{1}$, $\mathlcal{h}_{2}$, …,
$\mathlcal{h}_{k}$, that can be either faces or vertices. These distinguished
elements are called the _holes_ of the map, a given hole being called either
an _external face_ or an _external vertex_ , depending on its nature. The
nondistinguished faces and vertices are called the _internal faces_ and
_internal vertices_. The degree of a hole, also called its _perimeter_ , is
defined as $0$ for an external vertex or as the degree of the face for an
external face. Beware that the boundaries of the external faces are in general
neither simple curves, nor pairwise disjoint. As a result, the object obtained
by removing them from the surface in which the map is embedded is not
necessarily a surface. Note that, however, removing from every external face
an open disk whose closure is included in the (open) face results in a surface
with a boundary.

##### Bipartite map.

Finally, we say that a map is _bipartite_ if its vertex set can be partitioned
into two subsets such that no edge links two vertices of the same subset.

##### Tuples.

The many tuples considered in this work will conventionally be denoted by a
boldface font letter (possibly with a subscript) and their coordinates with
the same letter in a normal font, with the index written as a superscript, as
in ${\boldsymbol{x}}=(x^{1},\ldots,x^{r})$ for instance. When
${\boldsymbol{x}}$ is a tuple of real nonnegative numbers, we set
$\|{\boldsymbol{x}}\|=\sum_{i=1}^{r}x^{i}$. We denote by
${\boldsymbol{x}}{\boldsymbol{y}}$ the concatenation of ${\boldsymbol{x}}$
with ${\boldsymbol{y}}$. Finally, when concatenating with a $1$-tuple, we
often identify it with its unique coordinate, writing for instance
${\boldsymbol{x}}0=(x^{1},\ldots,x^{r},0)$.

###  1.3 The Gromov–Hausdorff–Prokhorov topology

In this paper, a _metric measure space_ is a triple
$(\mathcal{X},d_{\mathcal{X}},\mu_{\mathcal{X}})$, where
$(\mathcal{X},d_{\mathcal{X}})$ is a nonempty compact metric space and
$\mu_{\mathcal{X}}$ is a finite Borel measure on $\mathcal{X}$. We say that
two metric measure spaces $(\mathcal{X},d_{\mathcal{X}},\mu_{\mathcal{X}})$
and $(\mathcal{Y},d_{\mathcal{Y}},\mu_{\mathcal{Y}})$ are _isometry-
equivalent_ if there exists an isometry $\phi$ from
$(\mathcal{X},d_{\mathcal{X}})$ onto $(\mathcal{Y},d_{\mathcal{Y}})$ such that
$\mu_{\mathcal{Y}}=\phi_{*}\mu_{\mathcal{X}}$. This defines an equivalence
relation on the class of all metric measure spaces. If
$(\mathcal{X},d_{\mathcal{X}},\mu_{\mathcal{X}})$ and
$(\mathcal{Y},d_{\mathcal{Y}},\mu_{\mathcal{Y}})$ are two metric measure
spaces, the _Gromov–Hausdorff–Prokhorov metric_ (_GHP metric_ for short) is
defined by

$\mathrm{d}_{\mathrm{GHP}}\big{(}(\mathcal{X},d_{\mathcal{X}},\mu_{\mathcal{X}}),(\mathcal{Y},d_{\mathcal{Y}},\mu_{\mathcal{Y}})\big{)}=\inf_{\begin{subarray}{c}\hphantom{\psi}\hbox
to0.0pt{\hss\scriptsize$\phi$}\,:\,\mathcal{X}\to\mathcal{Z}\\\ \psi\,:\,\hbox
to0.0pt{\scriptsize$\mathcal{Y}$\hss}\hphantom{\mathcal{X}}\to\mathcal{Z}\end{subarray}}\left\\{d_{\mathcal{Z}}^{\mathrm{H}}(\phi(\mathcal{X}),\psi(\mathcal{Y}))\vee
d_{\mathcal{Z}}^{\mathrm{P}}(\phi_{*}\mu_{\mathcal{X}},\psi_{*}\mu_{\mathcal{Y}})\right\\},$
(1)

where the infimum is taken over all choices of compact metric spaces
$(\mathcal{Z},d_{\mathcal{Z}})$, and all isometric maps $\phi$, $\psi$ from
$\mathcal{X}$, $\mathcal{Y}$ to $\mathcal{Z}$, where
$d_{\mathcal{Z}}^{\mathrm{H}}$ is the _Hausdorff metric_ on compact subsets of
$\mathcal{Z}$, and $d_{\mathcal{Z}}^{\mathrm{P}}$ is the _Prokhorov metric_ on
finite positive measures on $\mathcal{Z}$, defined as follows. First, for any
$\varepsilon>0$ and any closed subset $A\subseteq\mathcal{Z}$, we denote by

$A^{\varepsilon}=\Big{\\{}z\in\mathcal{Z}\,:\,\inf_{y\in
A}d_{\mathcal{Z}}(z,y)<\varepsilon\Big{\\}}$

its $\varepsilon$-enlargement. Then, for any compact subsets $A$,
$B\subseteq\mathcal{Z}$,

$d_{\mathcal{Z}}^{\mathrm{H}}(A,B)=\inf\\{\varepsilon>0\,:\,A\subseteq
B^{\varepsilon}\text{ and }B\subseteq A^{\varepsilon}\\},$

and, for any finite Borel measures $\mu$, $\nu$ on $\mathcal{Z}$,

$d_{\mathcal{Z}}^{\mathrm{P}}(\mu,\nu)=\inf\big{\\{}\varepsilon>0\,:\,\text{for
all closed
}A\subseteq\mathcal{Z},\,\mu(A)\leq\nu(A^{\varepsilon})+\varepsilon\text{ and
}\nu(A)\leq\mu(A^{\varepsilon})+\varepsilon\big{\\}}\,.$

Equation (1) defines a metric on the set $\mathbb{M}$ of isometry-equivalence
classes of metric measure spaces, making it a complete and separable metric
space. The references [Vil09, Chapter 27] as well as [ADH13, LG19a] discuss
relevant aspects of the GHP topology, with some variations, as the exact
definition of the metric may differ from place to place.

More generally, for $\ell$, $m\geq 0$, we will consider _$\ell$ -marked,
$m$-measured metric spaces_ of the form
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$,
where

  * •

$(\mathcal{X},d_{\mathcal{X}})$ is a nonempty compact metric space,

  * •

$\mathbf{A}$ is an $\ell$-tuple, called _marking_ , of nonempty compact
subsets of $\mathcal{X}$, called _marks_ ,

  * •

${\boldsymbol{\mu}}_{\mathcal{X}}$ is an $m$-tuple of finite Borel measures on
$\mathcal{X}$.

We often consider marks that are singletons; in this case, we identify the
singleton with the point it contains. We define the _$\ell$ -marked,
$m$-measured Gromov–Hausdorff–Prokhorov metric_ (still _GHP metric_ for short)
on such spaces by

$\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}\big{(}(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})\big{)}\\\
=\inf_{\begin{subarray}{c}\hphantom{\psi}\hbox
to0.0pt{\hss\scriptsize$\phi$}\,:\,\mathcal{X}\to\mathcal{Z}\\\ \psi\,:\,\hbox
to0.0pt{\scriptsize$\mathcal{Y}$\hss}\hphantom{\mathcal{X}}\to\mathcal{Z}\end{subarray}}\Big{\\{}d_{\mathcal{Z}}^{\mathrm{H}}(\phi(\mathcal{X}),\psi(\mathcal{Y}))\vee\max_{1\leq
i\leq\ell}d_{\mathcal{Z}}^{\mathrm{H}}(\phi(A^{i}),\psi(B^{i}))\vee\max_{1\leq
j\leq
m}d_{\mathcal{Z}}^{\mathrm{P}}(\phi_{*}\mu^{j}_{\mathcal{X}},\psi_{*}\mu^{j}_{\mathcal{Y}})\Big{\\}}\,,$

where the infimum is taken over the same family as in (1). Again, this defines
a complete and separable metric on the set $\mathbb{M}^{(\ell,m)}$ of
isometry-equivalence classes of $\ell$-marked, $m$-measured metric spaces,
where
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$
and
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})$
are isometry-equivalent if there exists an isometry $\phi$ from $\mathcal{X}$
onto $\mathcal{Y}$ such that $\phi(A^{i})=B^{i}$ for $1\leq i\leq\ell$ and
$\phi_{*}\mu^{j}_{\mathcal{X}}=\mu^{j}_{\mathcal{Y}}$ for $1\leq j\leq m$.
Note that we have
$\mathrm{d}_{\mathrm{GHP}}^{(0,1)}=\mathrm{d}_{\mathrm{GHP}}$. Finally, the
space
$(\mathbb{M}^{(\ell)},\mathrm{d}_{\mathrm{GH}}^{(\ell)})=(\mathbb{M}^{(\ell,0)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,0)})$
of $\ell$-marked compact metric spaces without measures is the so-called
_$\ell$ -marked Gromov–Hausdorff metric_ (_GH metric_ for short).

As a first example, we will sometimes use in the present work the _point
space_ $\\{\varrho\\}$ consisting of a single point, seen as the element
$(\\{\varrho\\},(\varrho,\dots,\varrho),(0,\dots,0)\\}\in\mathbb{M}^{(\ell,m)}$
for any values of $\ell$ and $m$.

In what follows, we will often simply use the terminology “marked” or
“measured” instead of “$\ell$-marked” or “$m$-measured” if the numbers $\ell$
or $m$ are clear from the context. Furthermore, when $m\geq 2$, we will often
single out the first measure by writing it as a separate coordinate, writing
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\mu_{\mathcal{X}},{\boldsymbol{\nu}}_{\mathcal{X}})$
for instance. The reason is that this first measure will often be an _area_
measure whereas the other will be _boundary_ measures, and these have
different natural scales, as we will see shortly.

###  1.4 The main convergence result

##### Brownian surfaces.

For $k\geq 0$, a _quadrangulation with $k$ holes_ is a bipartite map having
$k$ holes $\mathlcal{h}_{1}$, …, $\mathlcal{h}_{k}$ and whose internal faces
are all of degree $4$. For111We will write $\mathbb{Z}_{+}=\\{0,1,2,\dots\\}$
the set of nonnegative integers, as well as $\mathbb{N}=\\{1,2,3,\dots\\}$ the
set of positive integers. $n\in\mathbb{Z}_{+}$ and
${\boldsymbol{l}}=(l^{1},\dots,l^{k})\in(\mathbb{Z}_{+})^{k}$ (with the
convention that $(\mathbb{Z}_{+})^{0}=\\{\varnothing\\}$), we define the set
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}}^{[g]}$
of all genus $g$ _rooted_ quadrangulations with $k$ holes having $n$ internal
faces, and whose holes $\mathlcal{h}_{1}$, …, $\mathlcal{h}_{k}$ are of
respective degrees $2l^{1}$, …, $2l^{k}$; see Figure 1 for an example.

Figure 1: A quadrangulation from
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}_{19,(4,1,2,0,0)}^{[1]}$.
The root is the corner $c_{*}$. Here, $\mathlcal{h}_{1}$, $\mathlcal{h}_{2}$,
and $\mathlcal{h}_{3}$ are external faces, while $\mathlcal{h}_{4}$ and
$\mathlcal{h}_{5}$ are external vertices.

Likewise, we denote by ${\mathbf{Q}}_{n,{\boldsymbol{l}}}^{[g]}$ the set of
_nonrooted_ quadrangulations of genus $g$ with $n$ internal faces and half-
perimeters given by ${\boldsymbol{l}}$. Since maps are counted with an inverse
factor given by the number of automorphisms, the weighted cardinality of this
set is

$\sum_{\mathbf{q}\in{\mathbf{Q}}_{n,{\boldsymbol{l}}}^{[g]}}\frac{1}{\operatorname{Aut}(\mathbf{q})}=\frac{\big{|}{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}}^{[g]}\big{|}}{4n+2\|{\boldsymbol{l}}\|}\,,$
(2)

where
$|{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}}^{[g]}|$
is the cardinality of
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}}^{[g]}$,
and $4n+2\|{\boldsymbol{l}}\|$ is the number of oriented edges, hence of
potential roots, in any element of ${\mathbf{Q}}_{n,{\boldsymbol{l}}}^{[g]}$.

It will be useful to notice for further reference that the quadrangulations
with $k$ holes in
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}}^{[g]}$
or in ${\mathbf{Q}}_{n,{\boldsymbol{l}}}^{[g]}$ all have the same number of
internal vertices, namely

$n+\|{\boldsymbol{l}}\|+2-2g-k$ (3)

Indeed, let us consider such a map, and denote by $v$, $e$, $f$, its number of
vertices, edges, faces. The number of external faces is thus $f-n$ so that the
desired number is $v-k+f-n$. Furthermore, counting the corners yields
$2e=4n+2\|{\boldsymbol{l}}\|$, and the result follows from the Euler
characteristic formula $v-e+f=2-2g$.

If $\mathbf{q}$ is a quadrangulation with $k$ holes, we can view it as a
$k$-marked, $k+1$-measured metric space, in the following way. We let
$V(\mathbf{q})$ be the vertex set of $\mathbf{q}$, and $d_{\mathbf{q}}$ the
graph metric on this set. We let

$\partial\mathbf{q}=\big{(}V(\mathlcal{h}_{1}),\ldots,V(\mathlcal{h}_{k})\big{)}\,,$

where for $1\leq i\leq k$, $V(\mathlcal{h}_{i})$ is either
$\\{\mathlcal{h}_{i}\\}$ if $\mathlcal{h}_{i}$ is an external vertex, or the
set of vertices incident to $\mathlcal{h}_{i}$ if it is an external face. We
let $\mu_{\mathbf{q}}$ and ${\boldsymbol{\nu}}_{\partial\mathbf{q}}$ be the
counting measures on $V(\mathbf{q})$ and the elements of $\partial\mathbf{q}$:

$\mu_{\mathbf{q}}=\sum_{v\in
V(\mathbf{q})}\delta_{v}\,,\qquad\nu_{\partial\mathbf{q}}^{i}=\sum_{v\in
V(\mathlcal{h}_{i})}\delta_{v}\,.$

These are respectively called the _area measure_ and _boundary measures_. We
associate with the quadrangulation $\mathbf{q}$ the space

$\big{(}V(\mathbf{q}),d_{\mathbf{q}},\partial\mathbf{q},\mu_{\mathbf{q}},{\boldsymbol{\nu}}_{\partial\mathbf{q}}\big{)}\in\mathbb{M}^{(k,k+1)}\,.$

Our main result exhibits a family

$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}\,,\qquad g\geq
0,\quad{\boldsymbol{L}}\in\bigsqcup_{k\geq 0}[0,\infty)^{k},$

of random marked measured metric spaces, where
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ will be called the _Brownian surface of
genus $g$ with boundary perimeter vector ${\boldsymbol{L}}$ and unit area_.
The latter family describes the scaling limits of uniform random elements of
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}_{n}}^{[g]}$,
in the following sense. Define the scaling operator $\Omega_{n}$ by

$\Omega_{n}(\mathbf{q})=\Bigg{(}V(\mathbf{q}),\Big{(}\frac{9}{8n}\Big{)}^{1/4}d_{\mathbf{q}},\partial\mathbf{q},\frac{1}{n}\mu_{\mathbf{q}},\frac{1}{\sqrt{8n}}{\boldsymbol{\nu}}_{\partial\mathbf{q}}\Bigg{)}\,.$
(4)

The scaling constants $(8/9)^{1/4}$ and $\sqrt{8}$ are here to make the
upcoming description of $\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ simpler in
Sections 4 to 7. Our main result is the following.

######  Theorem 1.

Fix $g$, $k\geq 0$. Let ${\boldsymbol{L}}=(L^{1},\dots,L^{k})$ be a $k$-tuple
of nonnegative real numbers and, for $n\geq 1$, let
${\boldsymbol{l}}_{n}=(l_{n}^{1},\dots,l_{n}^{k})\in(\mathbb{Z}_{+})^{k}$ be
such that $l_{n}^{i}/{\sqrt{2n}}\to L^{i}$ as $n\to\infty$, for $1\leq i\leq
k$. Let $Q_{n}$ be a random variable that is uniformly distributed over
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}_{n}}^{[g]}$.
Then

$\Omega_{n}(Q_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$

where the convergence holds in distribution in the space
$\big{(}\mathbb{M}^{(k,k+1)},\mathrm{d}_{\mathrm{GHP}}^{(k,k+1)}\big{)}$.

By our discussion on nonrooted maps, note that the same statement holds if
$Q_{n}$ is rather distributed over the set
${\mathbf{Q}}_{n,{\boldsymbol{l}}_{n}}^{[g]}$ of nonrooted maps, with a
probability proportional to the inverse of the number of automorphisms. Note
however that this automorphism number is equal to $1$ for the vast majority of
maps [RW95], so we expect that our results also hold for genuine uniform
random nonrooted maps.

If
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}=(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\mu_{\mathcal{X}},{\boldsymbol{\nu}}_{\mathcal{X}})$,
we will call $\mu_{\mathcal{X}}$ the _area measure_ , and
${\boldsymbol{\nu}}_{\mathcal{X}}$ the _boundary measures_. Note that
$\mu_{\mathcal{X}}$ is a probability measure, since (3) implies that
$|V(Q_{n})|\sim n$ as $n\to\infty$, while $\nu^{i}_{\mathcal{X}}$ has total
mass $L^{i}$ for $1\leq i\leq k$, so $\nu^{i}_{\mathcal{X}}$ is the trivial
zero measure if $L^{i}=0$.

Note that, for $(g,k)=(0,0)$, the above result amounts to the aforementioned
convergence of plane quadrangulations to the Brownian sphere [LG13, Mie13],
while for $(g,k)=(0,1)$ with $L^{1}>0$, it corresponds to the convergence of
quadrangulations with a boundary to the Brownian disk [BM17]. Note however
that the statement of the present paper is slightly stronger, since it is
formulated in terms of the marked GHP topology rather than the weaker GH
topology. In the case $(g,k)=(0,0)$ of the Brownian sphere, it amounts to the
GHP topology since there are no marks and only one measure; this stronger
forms appears for instance in [ABW17, Theorem 1.2] and [LG19a, Theorem 7].

##### Topology and Hausdorff dimension.

Let us also list some basic properties of the limiting metric spaces, which
justify the terminology of _Brownian surfaces_. We say that a metric space is
_locally of Hausdorff dimension $d$_ if any nontrivial ball has Hausdorff
dimension $d$.

######  Proposition 2.

Let ${\boldsymbol{L}}=(L^{1},\dots,L^{k})$ be fixed and let $b$ denote the
number of positive coordinates of ${\boldsymbol{L}}$. Almost surely, the
random metric space $\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ is homeomorphic to
$\mathbf{\Sigma}_{b}^{[g]}$, is locally of Hausdorff dimension $4$, and, if
$b>0$, each of the $b$ connected components of its boundary, considered as a
metric space by restriction of the metric on
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$, is locally of Hausdorff dimension $2$.

This statement is an immediate corollary of a result from [Bet16], showing
that, in the case $b=k$, any subsequential limit in distribution of
$n^{-1/4}Q_{n}$ satisfies the stated properties. The case $b<k$ is easily
obtained from there by the observation concerning null perimeters at the end
of this section. However, our method of proof of Theorem 1 will also provide
an alternative and rather transparent proof of Proposition 2, once an
analogous statement has been established for the noncompact analogs of the
cases of the sphere and disk, namely, the Brownian plane and the Brownian
half-plane [CLG14, GM17, BMR19] (see Section 4.4). We also mention that the
case $(g,k,b)=(0,1,0)$ was obtained in [Bet15] (see also [BG09]).

##### A comment on notation.

Throughout this paper, we will often work in fixed topology and consistently
use the following pieces of notation, as in the above statement:

  * •

$g$ for the genus of the surface;

  * •

$k$ for the size of the boundary perimeter vector, that is the number of holes
in the discrete maps;

  * •

$b$, as in _boundary_ , for the number of nonzero coordinates in the boundary
perimeter vector;

  * •

$p$, as in _puncture_ , for the number of null coordinates in the boundary
perimeter vector.

Beware that the latter two numbers do not always correspond to the numbers of
external faces and external vertices in the discrete maps, since we only
require that ${\boldsymbol{l}}_{n}/\sqrt{2n}\to{\boldsymbol{L}}$. However, for
$n$ sufficiently large, the $b$ holes corresponding to the $b$ nonzero
coordinates in the boundary perimeter vector are external faces; each of the
$p$ remaining holes can be either a vertex or a face but, in the latter case,
it should be thought of as a “small face” in the sense that its perimeter is
of order $\scalebox{0.5}[0.62]{$\mathbfcal{O}$}(\sqrt{n})$, and we will see
that this implies a diameter of order
$\scalebox{0.5}[0.62]{$\mathbfcal{O}$}(n^{1/4})$.

##### Method of proof.

We prove Theorem 1 by some surgical methods, and from the known cases $g=0$
and $k\in\\{0,1\\}$. Heuristically, we will cut $Q_{n}$ along well-chosen
geodesics into a _finite number_ of elementary pieces of planar topology, to
which we can apply a variant of the cases $(g,k)\in\\{(0,0),(0,1)\\}$ of
Theorem 1. The idea of cutting quadrangulations along geodesics into so-called
slices appears in Bouttier and Guitter [BG09, BG12]. The use of these slices
and the study of their scaling limits play an important role in Le Gall’s
proof [LG13] of the uniqueness of the Brownian sphere (they are called _maps
with a piecewise geodesic boundary_ in this reference) and are crucial to our
study [BM17] in the case of the disk. More specifically, in the latter
reference, we view Brownian disks as a continuum version of the slice
decomposition.

The proof of Theorem 1 relies on similar but yet different ideas, and will
require the introduction of other types of surgeries on objects that we call
_(composite) slices_ and _quadrilaterals (with geodesic sides)_. The core of
the proof of Theorem 1 consists in showing scaling limit results for these
elementary pieces, as stated in Theorems 12 and 14. We believe that these
results are of independent interest, as elementary pieces and their scaling
limits might serve as building blocks in other models of random surfaces. In
order to prove this result, it turns out that it is simpler to view the
discrete and continuum elementary pieces as embedded into non-compact vertion
of the Brownian sphere and disk, namely the Brownian plane and half-plane
defined in [CLG14, GM17, BMR19]. We stress that the description of the
Brownian half-plane in terms of gluing of composite slices considered in
Section 5.3 below is related to the _slice decomposition of metric bands_
property used by Miller and Qian [MQ21] for studying geodesic stars in the
Brownian sphere.

Theorem 1 generalizes the case of the sphere at two different levels, one
given by the positive genus and one given by the addition of a boundary.
Although these two levels of generalization rely to some extent on similar
ideas, the difficulties that they generate are of quite different nature. The
case of the disk, which was the focus of [BM17], relied on relatively well-
understood objects, but required gluing an infinite number of such objects,
which in principle could create problems in the limit. On the other hand, the
surgery involved in the general case consists in gluing a bounded number of
objects, but the objects themselves will turn out to be of a more complicated
nature.

##### Null perimeter coordinates.

We end this section by the following observation relating Brownian surfaces in
case of null perimeter coordinates. The operations of adding or removing a
mark used in the following proposition are given by Lemmas 15 and 18 in
Section 3.

######  Proposition 3.

Let ${\boldsymbol{L}}=(L^{1},\dots,L^{k})\in[0,\infty)^{k}$, and
${\boldsymbol{L}}0$ be the sequence $(L^{1},\ldots,L^{k},0)$. Then
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}0}$ has same distribution as the space
$\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$, where, denoting by $\mu$ the area
measure of the latter space, a random $\mu$-distributed point has been added
to the set of marks of $\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ in $(k+1)$-th
position (and the zero measure has been added as a trivial $(k+1)$-th boundary
measure).

Consequently, if $L^{i}=0$ for some given $i\in\\{1,2,\ldots,k\\}$, and if
$\hat{{\boldsymbol{L}}}$ denotes the vector ${\boldsymbol{L}}$ with $i$-th
coordinate removed, then $\mathsf{S}^{[g]}_{\hat{{\boldsymbol{L}}}}$ has same
distribution as the space $\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$ with its
$i$-th mark and (trivial) $i$-th boundary measure removed.

###### Proof.

Let us fix
${\boldsymbol{l}}_{n}=(l_{n}^{1},\dots,l_{n}^{k})\in(\mathbb{Z}_{+})^{k}$ such
that $l_{n}^{j}\sim\sqrt{2n}\,L^{j}$ for $1\leq j\leq k$, and let $Q_{n}$ be
uniformly distributed over
${\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}_{n}}^{[g]}$.
Setting ${\boldsymbol{l}}_{n}0=(l_{n}^{1},\dots,l_{n}^{k},0)$, a uniformly
distributed random variable $Q_{n}^{\prime}$ in
$\smash{{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}}_{n,{\boldsymbol{l}}_{n}0}^{[g]}}$
may be obtained by choosing an extra distinguished external vertex
$\mathlcal{h}_{k+1}$ uniformly at random among the internal vertices of
$Q_{n}$, that is, according to the measure $\mu_{Q_{n}}$ conditioned on the
set of internal vertices. Since the number of distinguished vertices in
$Q_{n}$ is at most $k$, while the total number of vertices is asymptotically
equivalent to $n$, the GHP limit of the quadrangulation $Q_{n}^{\prime}$
rescaled as in Theorem 1 is the same as if we had chosen $\mathlcal{h}_{k+1}$
uniformly at random among the set of all vertices of $Q_{n}$. By Theorem 1
applied to $Q_{n}$ and Lemma 15 below, we obtain the result. The second part
of the statement is obtained by permuting or removing marks and measures
appropriately, as discussed in Lemma 18 below. ∎

As an example, the Brownian sphere $\mathsf{S}^{[0]}_{\varnothing}$ can be
seen as $\mathsf{S}^{[0]}_{(0,0)}$ by forgetting its two marks. Anticipating
on the construction of the Brownian surfaces in Section 4, this provides a
nontrivially equivalent construction of the Brownian sphere as the gluing of
one quadrilateral with geodesic sides, rather than the one from [LG13, Mie13].

###  1.5 Scaling limits of Boltzmann quadrangulations

We may also consider scaling limits for models of quadrangulations with holes
in which the area and perimeters are not fixed, but rather weighted by
Boltzmann factors. We introduce the following sets of nonrooted maps:

$\displaystyle{\mathbf{Q}}_{{\boldsymbol{l}}}^{[g]}$
$\displaystyle=\bigsqcup_{n\geq 0}{\mathbf{Q}}_{n,{\boldsymbol{l}}}^{[g]}\,,$
$\displaystyle\text{ for }g\geq 0,\ {\boldsymbol{l}}\in\bigsqcup_{k\geq
0}(\mathbb{Z}_{+})^{k}\,,$ and $\displaystyle\mathbf{Q}^{[g]}(b,p)$
$\displaystyle=\bigsqcup_{{\boldsymbol{l}}\in\mathbb{N}^{b}}\mathbf{Q}^{[g]}_{{\boldsymbol{l}}{\boldsymbol{0}}^{p}}\,,$
$\displaystyle\text{ for $g$, $b$, $p\geq 0$}\,.$

We then let $\mathcal{W}$ be the $\sigma$-finite measure on the set of
nonrooted quadrangulations with an arbitrary number of holes and arbitrary
genus, given by

$\mathcal{W}(\mathbf{q})=\frac{1}{\operatorname{Aut}(\mathbf{q})}12^{-|\mathbf{q}|}\,8^{-\|\partial\mathbf{q}\|}\,,$

where $|\mathbf{q}|$ is the number of internal faces of $\mathbf{q}$, and
$\|\partial\mathbf{q}\|$ is the sum of the perimeters of its holes. The reason
for the choice of the weights $1/12$ and $1/8$ for the internal faces and
perimeters comes from the following enumeration result, which will be proved
in an extended form in Proposition 61, in Appendix B.

######  Proposition 4.

Fix $b\geq 0$ and ${\boldsymbol{L}}\in(0,\infty)^{b}$. Let
$({\boldsymbol{l}}_{n},n\geq 0)$ be a sequence of integers such that
$l_{n}^{i}\sim\sqrt{2n}\,L^{i}$ as $n\to\infty$ for $1\leq i\leq b$. Then
there exist a continuous function $t_{g}$ of ${\boldsymbol{L}}$, such that

$\mathcal{W}\big{(}\mathbf{Q}^{[g]}_{n,{\boldsymbol{l}}_{n}}\big{)}\mathrel{\mathop{\kern
0.0pt\sim}\limits_{n\to\infty}}t_{g}({\boldsymbol{L}})\,n^{\frac{5g-7}{2}+\frac{3b}{4}}\,.$

The function $t_{g}({\boldsymbol{L}})$ is related to the so-called _double
scaling limit_ of maps, as described in [Eyn16, Chapter 5], and its Laplace
transform can be computed by solving Eynard and Orantin’s topological
recursion. The method presented in Appendix B is based on the bijections
presented in Section 2.

For any $g\geq 0$, $p\geq 0$, ${\boldsymbol{L}}\in[0,\infty)^{p}$ and $A>0$,
if $(\mathcal{X},d,\mathbf{A},\mu)$ is a random variable with same law as
$\smash{\mathsf{S}^{[g]}_{{\boldsymbol{L}}/\sqrt{A}}}$, we define the Brownian
surface of genus $g$ with boundary perimeter vector ${\boldsymbol{L}}$ and
area $A$ as a random variable $\mathsf{S}^{[g]}_{A,{\boldsymbol{L}}}$ with
same law as $(\mathcal{X},A^{1/4}d,\mathbf{A},A\mu)$. If
${\boldsymbol{L}}\in\bigsqcup_{b\geq 0}(0,\infty)^{b}$ and $p\geq 0$, we let
${\boldsymbol{L}}{\boldsymbol{0}}^{p}\in[0,\infty)^{b+p}$ be the sequence
${\boldsymbol{L}}$ to which we append $p$ terms equal to $0$.

For integers $g$, $b$, $p\geq 0$, and for ${\boldsymbol{L}}\in(0,\infty)^{b}$,
setting $k=b+p$, we define a $\sigma$-finite measure on $\mathbb{M}^{(k,k+1)}$
by the formula

$\mathscr{S}^{[g]}_{{\boldsymbol{L}},p}(\cdot)=\int_{(0,\infty)}\mathrm{d}A\,A^{\frac{5g-7}{2}+\frac{3b}{4}+p}\,t_{g}\big{(}{\boldsymbol{L}}/\sqrt{A}\big{)}\,\mathbb{P}\big{(}\mathsf{S}^{[g]}_{A,{\boldsymbol{L}}{\boldsymbol{0}}^{p}}\in\cdot\,\big{)}\,.$

The measure $\mathscr{S}^{[g]}_{{\boldsymbol{L}},p}$ is a $\sigma$-finite
measure that “randomizes” the area measure of the Brownian surface of genus
$g$ with $b$ boundary components of lengths given by ${\boldsymbol{L}}$, as
well as $p$ marked vertices, in the sense that its conditional law given
having total area $A$ is that of
$\mathsf{S}^{[g]}_{A,{\boldsymbol{L}}{\boldsymbol{0}}^{p}}$.

Recall that the scaling operator $\Omega_{n}$ is defined by (4); here, we use
it for any $n\in(0,\infty)$.

######  Theorem 5.

Let $g$, $b$, $p\in\mathbb{Z}_{+}$, $k=b+p$,
${\boldsymbol{L}}\in(0,\infty)^{b}$, $K>0$, and
$F:\mathbb{M}^{(k,k+1)}\to\mathbb{R}$ be a continuous and bounded function
that is supported on the set of spaces
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\mu_{\mathcal{X}},{\boldsymbol{\nu}}_{\mathcal{X}})$
such that $\mu_{\mathcal{X}}(\mathcal{X})\in[1/K,K]$. Let
$({\boldsymbol{l}}_{a},a>0)$ be a family where
${\boldsymbol{l}}_{a}\in\mathbb{N}^{b}$ is such that
$l_{a}^{i}\sim\sqrt{2/a}\,L^{i}$ for $1\leq i\leq b$. Then, it holds that

$a^{\frac{5(g-1)}{2}+\frac{3b}{4}+p}\,\mathcal{W}\left(F\left(\Omega_{a^{-1}}(Q)\right)\mathbf{1}_{\mathbf{Q}^{[g]}_{{\boldsymbol{l}}_{a}{\boldsymbol{0}}^{p}}}\right)\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{a\downarrow
0}}\mathscr{S}^{[g]}_{{\boldsymbol{L}},p}(F)\,.$

Note that our main result, Theorem 1, can be seen as a “local limit” version
of Theorem 5, in the sense that it gives the conditional statement of this
last result given
$\smash{\mathbf{Q}^{[g]}_{a^{-1},{\boldsymbol{l}}_{a}{\boldsymbol{0}}^{p}}}$,
taking $a=1/n$. There is also a version of this theorem where the perimeters
given by ${\boldsymbol{L}}$ are left free as well. For $g$, $b$, $p\geq 0$, we
define the $\sigma$-finite measure

$\mathscr{S}^{[g]}_{b,p}(\cdot)=\int_{(0,\infty)^{b}}\mathrm{d}{\boldsymbol{L}}\,\mathscr{S}^{[g]}_{{\boldsymbol{L}},p}(\cdot)\,.$

######  Corollary 6.

Let $g$, $b$, $p\in\mathbb{Z}_{+}$, $k=b+p$, $K>0$, and
$F:\mathbb{M}^{(k,k+1)}\to\mathbb{R}$ be a continuous and bounded function
that is supported on the set of spaces
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\mu_{\mathcal{X}},{\boldsymbol{\nu}}_{\mathcal{X}})$
such that $\mu_{\mathcal{X}}(\mathcal{X})$ and
$\nu^{i}_{\mathcal{X}}(A^{i}),1\leq i\leq b$, all lie in $[1/K,K]$. Then it
holds that

$2^{\frac{b}{2}}a^{\frac{5(g-1)}{2}+\frac{5b}{4}+p}\,\mathcal{W}\left(F\left(\Omega_{a^{-1}}(Q)\right)\mathbf{1}_{\mathbf{Q}^{[g]}(b,p)}\right)\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{a\downarrow 0}}\mathscr{S}^{[g]}_{b,p}(F)\,.$

Interestingly, the measure $\mathscr{S}^{[g]}_{{\boldsymbol{L}},p}$ is finite
in the particular cases $g=0$, $b=1$ and $p\in\\{0,1\\}$, or $g=0$, $b=2$ and
$p=0$; it can be checked that it is infinite in all other cases. By computing
the functions $t_{0}({\boldsymbol{L}})$ in the case $b\in\\{1,2\\}$, we obtain
three probability distributions by normalizing the measures
$\mathscr{S}^{[0]}_{(L),0}$, $\mathscr{S}^{[0]}_{(L),1}$,
$\mathscr{S}^{[0]}_{(L,L^{\prime}),0}$. Those are the law of the _free
Brownian disk_ of perimeter $L\in(0,\infty)$:

$\mathrm{FBD}_{L}=\int_{0}^{\infty}\mathrm{d}A\,\frac{L^{3}}{\sqrt{2\pi
A^{5}}}\exp\left(-\frac{L^{2}}{2A}\right)\,\mathbb{P}\big{(}\mathsf{S}^{[0]}_{(L),A}\in\cdot\,\big{)}\,,$

the law of the _free pointed Brownian disk_ of perimeter $L\in(0,\infty)$:

$\mathrm{FBD}^{\bullet}_{L}=\int_{0}^{\infty}\mathrm{d}A\,\frac{L}{\sqrt{2\pi
A^{3}}}\exp\left(-\frac{L^{2}}{2A}\right)\,\mathbb{P}\big{(}\mathsf{S}^{[0]}_{(L,0),A}\in\cdot\,\big{)}\,,$

and the law of the _free Brownian annulus_ of boundary perimeters $L$,
$L^{\prime}$ :

$\mathrm{FBA}_{L,L^{\prime}}=\int_{0}^{\infty}\mathrm{d}A\,\frac{(L+L^{\prime})}{\sqrt{2\pi
A^{3}}}\exp\left(-\frac{(L+L^{\prime})^{2}}{2A}\right)\,\mathbb{P}\big{(}\mathsf{S}^{[0]}_{(L,L^{\prime}),A}\in\cdot\,\big{)}\,.$

Note in particular that
$\mathrm{FBD}^{\bullet}_{L}=\lim_{\varepsilon\downarrow
0}\mathrm{FBA}_{L,\varepsilon}$. These laws, as well as the associated
$\sigma$-finite measures $\mathscr{S}^{[0]}_{1,0}$, $\mathscr{S}^{[0]}_{1,1}$,
$\mathscr{S}^{[0]}_{2,0}$, play an important role in [ARS22].

In the case $b=0$, the two previous statements are in fact the same, since
$\mathscr{S}^{[g]}_{\varnothing,p}=\mathscr{S}^{[g]}_{0,p}$. This measure
describes the scaling limit of quadrangulations with no boundary, $p$ marked
vertices, and free area measure. In this case, the quantity
$t_{g}(\varnothing)$ is equal to the classical universal constant $t_{g}$
arising in map enumeration; see [BC86, LZ04]. Explicitly, the numbers
$\tau_{g}=2^{5g-2}\Gamma(\frac{5g-1}{2})t_{g}$ satisfy $\tau_{0}=-1$ and the
recursion

$\tau_{g+1}=\frac{(5g+1)(5g-1)}{3}\tau_{g}+\frac{1}{2}\sum_{h=1}^{g}\tau_{h}\tau_{g+1-h}\,,\qquad
g\geq 0\,.$

In this case, we thus have the following formula

$\mathscr{S}^{[g]}_{0,p}(\cdot)=t_{g}\int_{(0,\infty)}\mathrm{d}A\,A^{\frac{5g-7}{2}+p}\,\mathbb{P}(\mathsf{S}^{[g]}_{A,{\boldsymbol{0}}^{p}}\in\cdot\,)\,.$

###  1.6 Perspectives

A natural question, which we plan to investigate in future works, is to derive
the analog of Theorem 1 for bipartite quadrangulations on _nonorientable
compact surfaces_ , using the bijective techniques developed in [CD17, Bet22].
The first step of showing the existence of subsequential limits for
nonorientable quadrangulations without boundary has been taken in [CD17].
Addressing this question would complete the catalog of compact Brownian
surfaces.

As mentioned in the first section of this introduction, an important aspect is
that of _universality_ of the spaces $\mathsf{S}^{[g]}_{{\boldsymbol{L}}}$. In
fact, we expect these spaces to be the scaling limits of many other models of
random maps on surfaces. In the case of the Brownian sphere
$\mathsf{S}^{[0]}_{\varnothing}$, this was indeed verified for several models;
see the references mentioned above. In the case of Brownian disks, we showed
in [BM17] that the spaces $\mathsf{S}^{[0]}_{(L^{1})}$ appear as scaling
limits of many conditioned Boltzmann models. This approach to universality
should generalize to our context, at the price of some specific
technicalities. We will not address this question here, but will comment more
on this in Section 7.

It would be interesting to complete the bridge between Brownian surfaces and
LQG metrics and CFT. See in particular [DKRV16, GRV19] for precise conjectures
linking Liouville CFT with scaling limits of the area measure of random maps
(without boundary) after suitable uniformization. In a nutshell, the LQG
metrics are local objects that can be defined globally by using charts and
atlases on general Riemann surfaces. However, fixing a surface amounts to
fixing the conformal modulus of the LQG metric, while Brownian surfaces have a
random modulus. Hence, the computation of the law of this modulus is an
important question, which has been solved by [ARS22] in the case of the
annular topology. It seems that the case of general compact surfaces should be
approachable as well given the recent developments on conformal bootstrap in
Liouville CFT [GKRV21, Wu22].

We also mention that random surfaces with boundaries of the type studied in
this paper are related to the study of self-avoiding paths in random
geometries. See [GM19, GM21a] for more on this in the case of the gluing of
two Brownian half-planes or disks. It would be interesting to explicitly
describe the scaling limits of self-avoiding paths and loops on maps of fixed
topologies as gluings of Brownian surfaces along boundaries.

###  1.7 Organization of the paper

In Section 2, we present the extension of the famous Cori–Vauquelin–Schaeffer
bijection allowing to encode a quadrangulation with a simpler tree-like
structure carrying integer labels on its vertices. We also present a variant
of the bijection, which leads to the definition of the elementary pieces into
which we decompose a quadrangulation. We finally state the relevant scaling
limit results for these elementary pieces. In Section 3, we present the
surgical operation we need in order to reconstruct a metric space from its
elementary pieces, namely gluing along geodesic segments. The proof of Theorem
1 and Proposition 2 are given in Section 4. In Sections 5 and 6, we present
the metric spaces forming the continuum elementary pieces into consideration
and explain how they are natural building blocks of the Brownian plane and
half-plane, which are the noncompact analogs of the Brownian sphere and disk,
and we tweak known convergence results to these noncompact Brownian surfaces
to prove that the continuum elementary pieces are the scaling limits of the
discrete elementary pieces. Finally, we give in Section 7 an alternate
description of Brownian surfaces that does not involve gluing operations and
that is closer to the usual definition of the Brownian sphere and disks.

######  Acknowledgment.

We thank G. Chapuy for stimulating discussions during the elaboration of this
work, and in particular for his encouragements to deal with general compact
surfaces with a boundary. We thank J. Bouttier for bringing our attention to
the fact that the semigroup property of discrete slices might have an
interesting continuum counterpart. Thanks are also due to X. Sun for
discussions around the problem of defining Brownian surfaces in the context of
LQG and the question of the random modulus of these surfaces, and also for
sharing the work [ARS22].

##  2 Variants of the Cori–Vauquelin–Schaeffer bijection

As is customary when studying on scaling limits of maps, this work strongly
relies on powerful encodings of discrete maps by tree-like objects. We now
present variants of the famous Cori–Vauquelin–Schaeffer (CVS) bijection [CV81,
Sch98] between plane quadrangulations and so-called _well-labeled trees_ , and
its generalizations by Chapuy–Marcus–Schaeffer [CMS09] for higher genera and
by Bouttier–Di Francesco–Guitter [BDG04] for plane maps with faces of
arbitrary degrees. We only give the constructions from the encoding objects to
the considered maps and refer the reader to the aforementioned works for
converse constructions and proofs.

###  2.1 Basic construction

Let $\mathbf{m}$ be a map, rooted or not, and $f$ be a face of $\mathbf{m}$.
Starting from a choice of a corner $c_{0}$ in $f$, we index the subsequent
corners of $f$ in counterclockwise order as $(c_{i},i\in\mathbb{Z})$ (forming
a periodic sequence). Let $\lambda:V(\mathbf{m})\to\mathbb{Z}$ be a labeling
of the vertices of $\mathbf{m}$ by integers. We extend the definition of
$\lambda$ to the corners of the map by setting $\lambda(c)=\lambda(v)$ if $v$
is the vertex incident to the corner $c$. In what follows, we will either
consider that $\lambda$ is defined up to addition of a constant, or that the
value of $\lambda$ at some corner is fixed, for instance that
$\lambda(c_{0})=0$.

We say that $(\mathbf{m},\lambda)$ is _well labeled_ inside $f$ if
$\lambda(c_{i+1})\geq\lambda(c_{i})-1$ for every $i\geq 0$. In particular, if
$(\mathbf{m},\lambda)$ is well labeled inside $f$ and $e$ is a half-edge of
$\mathbf{m}$ such that both $e$ and its reverse $\bar{e}$ are incident to $f$,
then $|\lambda(e^{+})-\lambda(e^{-})|\leq 1$, where $e^{-}$, $e^{+}$ denote
the origin and end of $e$. Note that this will be the case for every edge when
$\mathbf{m}$ is a map with a single face.

Let $(\mathbf{m},\lambda)$ be well labeled inside $f$. With the above
notation, let us define
$s(i)=\inf\\{j>i:\lambda(c_{j})=\lambda(c_{i})-1\\}\in\mathbb{Z}\cup\\{\infty\\}$
and the _successor_ of $c_{i}$ as $s(c_{i})=c_{s(i)}$, where $c_{\infty}$ is
by convention the unique corner incident to a vertex $v_{*}$ that is added in
the interior of $f$, and which naturally carries the label
$\lambda(v_{*})=\min\\{\lambda(c_{i}),i\in\mathbb{Z}\\}-1$. Clearly,
$s(c_{i})$ is then well defined for all corners (distinct from $c_{\infty}$),
and only depends on the corner $c_{i}$ and not on the particular choice of the
index $i$. The _CVS construction inside the face $f$_ consists in

  * •

linking by an arc every corner $c$ incident to $f$ to its successor $s(c)$, in
such a way that arcs do not cross, which is always possible due to the well
labeling condition,

  * •

deleting all the edges of $\mathbf{m}$.

This construction results in an embedded graph222In general, this embedded
graph is not a map of the surface into consideration. In all the constructions
we will use in this work, it will, however, always turn out to be a map.
denoted by $\operatorname{CVS}(\mathbf{m},\lambda;f)$, whose vertex set
consists of $v_{*}$ and the vertices of $\mathbf{m}$ incident to $f$, and
whose edges are the arcs between the corners of $f$ and their successors. By
construction, the edges of $\operatorname{CVS}(\mathbf{m},\lambda;f)$ are in
bijection with the corners of $\mathbf{m}$ incident to $f$. If $\mathbf{m}$ is
rooted inside $f$, say at the corner $c_{i}$, then
$\operatorname{CVS}(\mathbf{m},\lambda;f)$ naturally inherits a root at the
corner preceding the arc linking $c_{i}$ to $s(c_{i})$. Note that the well
labeling condition, as well as the output
$\operatorname{CVS}(\mathbf{m},\lambda;f)$, are invariant under addition of a
constant to $\lambda$, as they should.

We will also need an _interval variant_ of this construction, where we fix a
sequence, also referred to as an _interval_ , of subsequent corners
$I=\\{c_{0},c_{1},\ldots,c_{r}\\}$ of a face $f$ of $\mathbf{m}$, and only ask
that $(\mathbf{m},\lambda)$ is _well labeled_ on $I$ in the sense that
$\lambda(c_{i+1})\geq\lambda(c_{i})-1$ for $0\leq i\leq r-1$. In this case, we
set $\lambda_{*}=\min\\{\lambda(c_{i}),0\leq i\leq r\\}-1$ and
$\ell=\lambda(c_{r})-\lambda_{*}$. Instead of a single extra corner
$c_{\infty}$, we introduce inside $f$ a sequence of distinct consecutive
corners $c_{r+1}$, $c_{r+2}$, …, $c_{r+\ell}$, incident to new vertices
$v_{r+1}$, $v_{r+2}$, …, $v_{r+\ell}$ with labels $\lambda(c_{r})-1$,
$\lambda(c_{r})-2$, …, $\lambda_{*}$. The successor mapping $s$ is then
defined for all corners except $c_{r+\ell}$. We let
$\operatorname{CVS}(\mathbf{m},\lambda;I)$ be the resulting (nonrooted)
embedded graph whose edges are the arcs. In this embedded graph, the following
are of particular interest:

  1. (1)

the _apex_ $v_{r+\ell}$, which will usually be denoted with a subscript $*$;

  2. (2)

the _maximal geodesic_ , which is the chain of arcs linking $c_{0}$,
$s(c_{0})$, $s(s(c_{0}))$, …, $c_{r+\ell}$, and which will always be denoted
with the letter $\gamma$ and depicted in red in the figures;

  3. (3)

the _shuttle_ , which is the chain of arcs linking $c_{r}$, $c_{r+1}$, …,
$c_{r+\ell}$, and which will always be denoted with the letter $\xi$ and
depicted in green in the figures.

Note that the two latter are paths from the first and last corners of $I$ to
the apex.

The construction generalizes to several intervals $I$, $J$, … that pairwise
share at most one extremity. In the case of a shared extremity, say
$I=\\{c_{0},c_{1},\ldots,c_{r}\\}$ and
$J=\\{c^{\prime}_{0},c^{\prime}_{1},\ldots,c^{\prime}_{r^{\prime}}\\}$ with
$c_{r}=c^{\prime}_{0}$, one first duplicates the common corner before applying
the construction, in the sense that the copy $c_{r}$ is used in the shuttle of
$I$ and $c^{\prime}_{0}$ is used in the maximal geodesic of $J$; see Figure 2.
In this construction, each interval yields a distinct apex, maximal geodesic,
and shuttle, and the construction results in an embedded graph denoted by
$\operatorname{CVS}(\mathbf{m},\lambda;I,J,\ldots)$. Plainly, the ordering of
the intervals does not affect the construction.

Figure 2: Performing the interval variant of the Cori–Vauquelin–Schaeffer
bijection with two intervals sharing an extremity. The interval $I$ consists
of the corners in the purple area, starting with $c_{0}$ and ending with
$c_{r}$, while $J$ consists of the corners in the red area, starting with
$c^{\prime}_{0}$ and ending with $c^{\prime}_{r^{\prime}}$. The interval $I$
yields the apex $v_{*}$, maximal geodesic $\gamma$, and shuttle $\xi$, while
$J$ yields respectively $v^{\prime}_{*}$, $\gamma^{\prime}$, and
$\xi^{\prime}$. As will be the case in all the figures, the maximal geodesics
are in red and the shuttles in green.

We make the important observation that any chain $c$, $s(c)$, …, $s^{i}(c)$ of
consecutive successors induces a geodesic chain for the graph metric in the
resulting embedded graph $\operatorname{CVS}(\mathbf{m},\lambda;I,J,\dots)$,
that is, a path of minimal length between its extremities. This is simply
because, by construction, any arc of the resulting embedded graph links two
vertices $u$ and $v$ such that $|\lambda(u)-\lambda(v)|=1$, and because
$\lambda$ decreases by $1$ at every step on a chain of consecutive successors.
In particular, the maximal geodesics and shuttles of
$\operatorname{CVS}(\mathbf{m},\lambda;I,J,\ldots)$ are geodesic chains.

###  2.2 The generalized Chapuy–Marcus–Schaeffer bijection

##### Encoding quadrangulations.

As a first example, let us perform this construction on a particular class of
maps. For $n\in\mathbb{Z}_{+}$ and
${\boldsymbol{l}}=(l^{1},\dots,l^{k})\in\mathbb{Z}_{+}^{k}$, we let
$\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{M}}}}^{[g]}_{n,{\boldsymbol{l}}}$
be the set of labeled rooted maps $(\mathbf{m},\lambda)$ satisfying the
following properties:

  * •

$\mathbf{m}$ is a map of genus $g$ with $n+\sum_{i=1}^{k}l^{i}$ edges, one
internal face $f_{*}$ and $k$ holes $\mathlcal{h}_{1}$, …, $\mathlcal{h}_{k}$,
rooted at a corner of its internal face $f_{*}$;

  * •

for all $i$, the hole $\mathlcal{h}_{i}$ is of degree $l^{i}$; if it is an
external face, then it has a simple boundary333A face has a _simple boundary_
if it is incident to as many vertices as its degree.;

  * •

for any $i\neq j$, if $\mathlcal{h}_{i}$ and $\mathlcal{h}_{j}$ are faces,
then they are not incident to any common edge;

  * •

$(\mathbf{m},\lambda)$ is well labeled inside $f_{*}$.

We similarly define the set $\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}$ of labeled
nonrooted maps. Setting ${\boldsymbol{l}}0=(l^{1},\dots,l^{k},0)$, the CVS
construction applied to the internal face $f_{*}$ provides a bijection between
$\smash{\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}}$ and
$\smash{\mathbf{Q}^{[g]}_{n,{\boldsymbol{l}}0}}$, through which the $k$ first
holes correspond, while the extra hole $\mathlcal{h}_{k+1}$ of the
quadrangulation is the extra vertex $v_{*}$ of the construction. In case of
rooted maps, it yields a one-to-two correspondence444The factor $2$ comes from
the fact that the corners of $f_{*}$ correspond to the edges of the resulting
map, each edge corresponding to $2$ half-edges. We refer the interested reader
to [Bet16, Section 3.1] for a presentation of the reverse mapping. between
$\smash{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{M}}}}^{[g]}_{n,{\boldsymbol{l}}}}$
and
$\smash{\smash{\overset{\rightarrow}{\vphantom{o}\smash{\mathbf{Q}}}}^{[g]}_{n,{\boldsymbol{l}}0}}$.

##### Decomposition into elementary pieces.

Let us now perform the construction on the same set of maps
$\smash{\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}}$ but with well-chosen
intervals. We will decompose a map of $\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}$
into a collection of labeled forests indexed by an underlying structure called
the _scheme_. For the remainder of this section, we exclude the cases
$(g,k)\in\\{(0,0),(0,1)\\}$ leading to encoding objects not entering the
upcoming framework. We fix
$(\mathbf{m},\lambda)\in\mathbf{M}^{[g]}_{n,{\boldsymbol{l}}}$.

Figure 3: Left. A labeled map from $\mathbf{M}^{[1]}_{63,(6,3,0)}$. The
outlined vertices are its nodes and the thicker edges correspond to the map
$\mkern 1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$.
Right. The corresponding scheme.

Let $\mkern 1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$
be the nonrooted map obtained from $\mathbf{m}$ by iteratively removing all
its vertices of degree $1$ that are not holes. The resulting map $\mkern
1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$ may be seen
as a submap of $\mathbf{m}$: the map $\mathbf{m}$ is obtained from $\mkern
1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$ by appending
rooted labeled trees at its corners. We call _nodes_ of $\mathbf{m}$ the
following vertices:

  * •

the external vertices of $\mathbf{m}$;

  * •

the vertices of $\mathbf{m}$ having degree $3$ or more in $\mkern
1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$.

These nodes are linked in $\mkern
1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern 1.0mu$ by maximal
chains of edges not containing any nodes other than their extremities.
Replacing every such chain with a single edge yields a nonrooted map
$\mathbf{s}$, called the _scheme_ of $\mathbf{m}$. It has one internal face,
still denoted by $f_{*}$, and $k$ holes, still denoted by $\mathlcal{h}_{1}$,
…, $\mathlcal{h}_{k}$; see Figure 3.

We denote by $\vec{E}(\mathbf{s})$ the set of half-edges incident to the
internal face of $\mathbf{s}$; this set is partitioned into the set
$\vec{I}(\mathbf{s})$ of half-edges whose reverses belong to
$\vec{E}(\mathbf{s})$ as well, and the set $\vec{B}(\mathbf{s})$ of half-edges
whose reverses do not belong to $\vec{E}(\mathbf{s})$. (We used the letter $I$
for _internal_ and $B$ for _boundary_.) The set $\vec{B}(\mathbf{s})$ is
further partitioned as

$\vec{B}(\mathbf{s})=\bigsqcup_{1\leq r\leq k}\vec{B}_{r}(\mathbf{s})$

where $\vec{B}_{r}(\mathbf{s})$ is either empty if $\mathlcal{h}_{r}$ is a
vertex, or the set of half-edges of $\vec{E}(\mathbf{s})$ whose reverse are
incident to $\mathlcal{h}_{r}$ if it is a face. We consider
$e\in\vec{E}(\mathbf{s})$. It corresponds to a chain $e_{1}$, …, $e_{j}$ of
half-edges in $\mathbf{m}$. Let us denote by $c_{e}$ and $c^{\prime}_{e}$ the
corners of $\mkern 1.0mu\widetilde{\mkern-1.0mu\mathbf{m}\mkern-1.0mu}\mkern
1.0mu$ preceding $e_{1}$ and succeeding $e_{j}$ in the contour order. In
$\mathbf{m}$, there are several corners that make up $c_{e}$ and
$c^{\prime}_{e}$. The corner interval $I_{e}$ is the interval of corners of
$\mathbf{m}$ from the first corner corresponding to $c_{e}$ to the first
corner corresponding to $c^{\prime}_{e}$. Observe that, in $\mathbf{m}$, the
tree grafted at $c_{e}$ is thus covered by $I_{e}$, whereas the tree grafted
at $c^{\prime}_{e}$ is not.

By construction, $\bigcup_{e\in\vec{E}(\mathbf{s})}I_{e}$ is equal to the set
of corners of $f_{*}$ and each extremity of these intervals is shared by
exactly two such intervals. More precisely, the intervals $I_{e}$,
$e\in\vec{E}(\mathbf{s})$, with their last corner removed give a partition of
the corners of $f_{*}$. Applying the interval CVS construction
$\operatorname{CVS}(\mathbf{m},\lambda;\\{I_{e},e\in\vec{E}(\mathbf{s})\\})$
gives a natural decomposition of the quadrangulation
$(\mathbf{q},v_{*})=\operatorname{CVS}(\mathbf{m},\lambda;f_{*})$ into
submaps, whose study, starting in the next section, are the key to this work;
see Figure 4. These submaps are called the _elementary pieces_ of
$(\mathbf{q},v_{*})$ and are of two types: the ones corresponding to half-
edges of $\vec{B}(\mathbf{s})$ are called _(composite) slices_ and the ones
corresponding to half-edges of $\vec{I}(\mathbf{s})$ are called
_quadrilaterals (with geodesic sides)_. They are not rooted and come with
distinguished vertices on their boundaries that will be discussed later on.

Figure 4: Performing the interval bijection on the labeled map from Figure 3.
Left. Two elementary pieces are represented: one quadrilateral with geodesic
sides in red, and one composite slice in blue. Right. The interval bijection
yields a decomposition into 4 composite slices and 7 quadrilateral with
geodesic sides. Here, only the maximal geodesics and shuttles are depicted. We
let the edges of the original scheme figure on this output map, but these are
neither edges nor chains of edges of this output map (remember that the edges
of the original map are never edges of the output map).

The elementary piece corresponding to the half-edge $e\in\vec{E}(\mathbf{s})$
is encoded by the part of the labeled map $(\mathbf{m},\lambda)$ corresponding
to

  * •

either the interval $I_{e}$ if $e\in\vec{B}(\mathbf{s})$,

  * •

or the union $I_{e}\cup I_{\bar{e}}$ if $e\in\vec{I}(\mathbf{s})$, where
$\bar{e}$ denotes the reverse of $e$.

See Figure 5. Note that, when $e\in\vec{I}(\mathbf{s})$, the elementary pieces
corresponding to $e$ and to its reverse $\bar{e}$ are the same map; only the
distinguished vertices on the boundary will differ (more precisely be given in
a different order). We refer the reader to [Bet16, Section 3.4.1] for more on
this decomposition, keeping in mind that, in the latter reference, the maps
are rooted and the root is encoded in the scheme, which essentially amounts in
seeing the root of the map as an extra external vertex.

Figure 5: The parts of the labeled map from Figure 3 encoding the elementary
pieces. The two parts corresponding to the quadrilateral with geodesic sides
and the composite slice from Figure 4 are extracted. The red and blue colors
match.

##### Finiteness of the number of schemes.

We will elaborate more on elementary pieces in the next two sections and end
this one with a simple combinatorial lemma. We say that a map with holes is a
_scheme_ if it has one internal face, all its external faces have a simple
boundary and do not share a common incident edge, and all its internal
vertices have degree $3$ or more.

######  Lemma 7.

For fixed values of $(g,k)\notin\\{(0,0),(0,1)\\}$, there are finitely many
genus $g$ schemes with $k$ holes and these have at most $3(2g+k-1)$ edges and
$2(2g+k-1)$ vertices.

###### Proof.

As there is a finite number of maps with a given number of edges, the bound on
the number of edges yields the finiteness of the considered set.

Let $v$, $e$, $f$ be the number of vertices, edges and faces of a given scheme
as in the statement, and let $b$ be its number of external faces (so that
$p=k-b$ are external vertices). By construction, we have $f=b+1$ and the
vertices are all of degree at least $3$, except possibly up to $p$ of them,
which have degree at least $1$. The sum of the degrees of the vertices being
twice the number of edges, we obtain $2e\geq 3(v-p)+p$, and we see by the
Euler characteristic formula $v-e+f=2-2g$ that the considered scheme has at
most $6g+3k-p-3\leq 3(2g+k-1)$ edges, and at most $2(2g+k-1)$ vertices, using
that $k=p+b$. ∎

###  2.3 Composite slices

We call _plane forest_ a collection
$\mathbf{f}=(\mathbf{t}^{0},\dots,\mathbf{t}^{l-1},\rho^{l})$, for some $l\geq
1$, of rooted plane trees (the last one being reduced to the vertex-tree),
which we view systematically as a map by taking an embedding of every
$\mathbf{t}^{i}$ in the upper half-plane $\mathbb{R}\times\mathbb{R}_{+}$,
with root $\rho^{i}$ at the point $(i,0)$, and in which $\rho^{i}$ is linked
to $\rho^{i-1}$ by the line segment between $(i,0)$ and $(i-1,0)$, for $1\leq
i\leq l$. The union of these line segments is called the _floor_ of the
forest. The resulting embedded graph, which we still denote by $\mathbf{f}$
(see e.g. the left of Figure 6) is a nonrooted plane map coming with the two
distinguished vertices $\rho=\rho^{0}$ and $\bar{\rho}=\rho^{l}$; it is in
fact a plane tree, but we insist on calling it a forest.

We let $a$ be the total number of edges of $\mathbf{t}^{0}$, …,
$\mathbf{t}^{l-1}$, and $I=\\{c_{0},c_{1},\ldots,c_{2a+l}\\}$ be the interval
of corners of $\mathbf{f}$ that are incident to the upper half-plane (hence
excluding the corners that are “below” the floor), starting from the root
corner of $\mathbf{t}^{0}$ and ending with the only corner incident to
$\rho^{l}$, arranged in the usual contour order. We now equip $\mathbf{f}$
with an integer-valued labeling function $\lambda:V(\mathbf{f})\to\mathbb{Z}$,
again defined up to addition of a constant, that we require to satisfy the
well labeling condition in the interval $I$. In this case, it means that

  * •

$\lambda(u)-\lambda(v)\in\\{-1,0,1\\}$ whenever $u$ and $v$ are neighboring
vertices of the same tree;

  * •

for $1\leq i\leq l$, we have $\lambda(\rho^{i})\geq\lambda(\rho^{i-1})-1$.

Figure 6: The interval Cori–Vauquelin–Schaeffer bijection giving composite
slices. On this example, the forest has $a=7$ edges and $l=3$ trees (remember
that the last vertex-tree does not count as a “real” tree). The boundary of
$\mathbf{q}$ has three parts: the maximal geodesic (in red), the shuttle (in
green) and the base (in burgundy). Its tilt is $4$.

The map $\mathbf{sl}=\operatorname{CVS}(\mathbf{f},\lambda;I)$ is then a
nonrooted plane quadrangulation with one hole having $a$ internal faces.
Setting $\lambda_{*}=\min\\{\lambda(v):v\in V(\mathbf{f})\\}-1$, we see that
the boundary of $\mathbf{sl}$ has length $2(\lambda(\rho^{l})-\lambda_{*}+l)$.
It contains three distinguished vertices: $\rho$, the apex $v_{*}$ (the extra
vertex with label $\lambda_{*}$), and $\bar{\rho}$, as well as three
distinguished paths:

  1. (1)

the maximal geodesic $\gamma$, which has length
$\lambda(\rho^{0})-\lambda_{*}$;

  2. (2)

the shuttle $\xi$, which has length $\lambda(\rho^{l})-\lambda_{*}$;

  3. (3)

the remaining boundary segment, called the _base_ and denoted by $\beta$,
consisting in the arcs connecting the root vertices of the trees. More
precisely, if $c_{j}$ denotes the last corner of the tree $\mathbf{t}^{i}$,
then the part of the boundary of $\mathbf{sl}$ between $\rho^{i}$ and
$\rho^{i+1}$ consists in the arc linking $c_{j}$ to $s(c_{j})$ and the
successive arcs linking $c_{j+1}$, $s(c_{j+1})$, $s(s(c_{j+1}))$, …,
$s(c_{j})$. As a result, this base has length
$\lambda(\rho^{l})-\lambda(\rho^{0})+2l$. Moreover, any vertex of the base is
at distance at most $\max_{1\leq i\leq
l}|\lambda(\rho^{i})-\lambda(\rho^{i-1})|+1$ from some element of the set
$\\{\rho^{0},\ldots,\rho^{l}\\}$.

Note that, as is the case in Figure 6, the base may overlap with the other
distinguished paths. Furthermore, as noted at the end of Section 2.1, the
maximal geodesic and the shuttle are geodesic chains. On the contrary, the
base is not a geodesic in general.

######  Definition 8.

A map obtained by this construction will be called a _discrete composite
slice_ , or simply _slice_ for short: its _area_ is the integer $a$, its
_width_ is the integer $l$ and its _tilt_ is defined as the integer

$\delta=\lambda(\bar{\rho})-\lambda(\rho).$

The terminology of composite slices, width and tilt are borrowed from [Bou19];
however, the reader should mind that our exact definitions differ slightly
from those in that reference555In particular, in [Bou19], the width is the
length of the base, equal to $2l+\delta$ in our notation, and the tilt is the
opposite $-\delta$ of what we call the tilt in this paper.. Note also that, in
the present work, we use the simplified terminology of _slice_ in order to
designate a composite slice. Beware that these have not to be confused with
similar objects existing in the literature, in particular in our previous work
[BM17], called _elementary slices_ or also slices for short; they actually
correspond to composite slices of width $0$, objects that we do not consider
here.

We record the following useful counting result.

######  Proposition 9.

The number of slices with area $a$, width $l$ and tilt $\delta$ is equal to

$3^{a}\,\frac{l}{2a+l}\,\binom{2a+l}{a}\,\binom{2l+\delta-1}{l-1}\,,$

which can also be recast as

$12^{a}\,8^{l}\,2^{\delta}\,Q_{l}(2a+l)\,P_{l}(\delta)\,,$

where $Q_{\ell}(u)$ is the probability that a simple random walk hits $-\ell$
for the first time at time $u$, and
$P_{\ell}(j)=\mathbb{P}(G_{1}+\dots+G_{\ell}=j)$, where $G_{1}$, $G_{2}$, …
are independent random variables with shifted Geometric(1/2) law, i.e., such
that $\mathbb{P}(G_{1}=j)=2^{-j-2}$ for $j\geq-1$.

###### Proof.

The term $\frac{l}{2a+l}\binom{2a+l}{a}$ is the number of forests with $l$
trees and $a$ non-floor edges, the term $\binom{2l+\delta-1}{l-1}$ counts the
number of possible ways to well label the roots, and the term $3^{a}$ counts
the number of ways to well label the other vertices, since it amounts to
choosing a label difference in $\\{-1,0,1\\}$ along each edge.

The probabilistic form is a simple exercise using the encoding of forests and
geometric walks by simple walks, yielding
$\frac{l}{2a+l}\binom{2a+l}{a}=2^{2a+l}\,Q_{l}(2a+l)$ and
$\binom{2l+\delta-1}{l-1}=2^{2l+\delta}\,P_{l}(\delta)$. See Section 5.4 and
[Bet15, Lemma 6]. ∎

An important feature of the construction is that the labels on
$V(\mathbf{sl})$ inherited from those on $V(\mathbf{f})$ are exactly the
relative distances to $v_{*}$ in $\mathbf{sl}$:

$d_{\mathbf{sl}}(v,v_{*})=\lambda(v)-\lambda_{*}\,,\qquad v\in
V(\mathbf{sl})\,,$

and that the following bound holds:

$d_{\mathbf{sl}}(c_{i},c_{j})\leq\lambda(c_{i})+\lambda(c_{j})-2\min_{i\leq
r\leq j}\lambda(c_{r})+2\,,\qquad i\leq j\,.$ (5)

If $\mathbf{sl}$ is a slice, using a slightly different convention from that
of Section 1.4, we view it as the marked measured metric space in
$\mathbb{M}^{(5,2)}$ given by

$\big{(}V(\mathbf{sl}),d_{\mathbf{sl}},\partial\mathbf{sl},\mu_{\mathbf{sl}},\nu_{\beta}\big{)}\qquad\text{
with
}\qquad\partial\mathbf{sl}=\big{(}\beta,\rho,\gamma,\bar{\rho},\xi\big{)}\,,$
(6)

where each boundary part is identified with the vertices it contains, where
$\mu_{\mathbf{sl}}$ is the counting measure on the vertices of $\mathbf{sl}$
that do not belong to the shuttle, and where $\nu_{\beta}$ is the counting
measure on $\beta\setminus\\{\bar{\rho}\\}$. The measures $\mu_{\mathbf{sl}}$
and $\nu_{\beta}$ are respectively called the _area measure_ and the _base
measure_ of the slice. It might be surprising at this point to include $\rho$
and $\bar{\rho}$ in the marking as these can be found from the other three
marks; they are here to enter the framework of geodesic marks introduced in
Section 3.2. The idea is that the data of $(\rho,\gamma)$ suffice to recover
the maximal geodesic as an _oriented_ path, whereas the data of $\gamma$ (as a
set of vertices) do not give the orientation of the path.

###  2.4 Quadrilaterals with geodesic sides

Consider a _double forest_ , that is, a pair $(\mathbf{f},\bar{\mathbf{f}})$
of plane forests with the same number of trees. Let $h\geq 1$ denote this
common number of trees and recall that this means that $\mathbf{f}$ and
$\bar{\mathbf{f}}$ have $h$ trees plus an additional vertex-tree. Similarly to
the previous section, we represent it by letting

  * •

the floors be both sent to the chain linking the points
$(i,0)\in\mathbb{R}^{2}$, where $0\leq i\leq h$,

  * •

the trees of $\mathbf{f}$ be contained in the upper half-plane
$\mathbb{R}\times\mathbb{R}_{+}$, the $i$-th tree attached to $(i-1,0)$, for
$1\leq i\leq h$,

  * •

and the trees of $\bar{\mathbf{f}}$ be contained in the lower half-plane, the
$i$-th tree attached to $(h-i+1,0)$, for $1\leq i\leq h$.

We obtain a nonrooted plane map, which we denote by
${\mathbf{f}}\cup{\bar{\mathbf{f}}}$, coming with the two distinguished
vertices $\rho=(0,0)$ and $\bar{\rho}=(h,0)$. Here also, it is in fact a plane
tree having two distinguished vertices.

We let $I=\big{\\{}c_{0},c_{1},\ldots,c_{2a+h}\big{\\}}$ be the interval of
corners of ${\mathbf{f}}\cup{\bar{\mathbf{f}}}$, in facial order, that are
incident to the upper half-plane, and
$\bar{I}=\big{\\{}\bar{c}_{0},\bar{c}_{1},\ldots,\bar{c}_{2\bar{a}+h}\big{\\}}$
the interval of those incident to the lower half-plane, where $a$ (resp.
$\bar{a}$) is the number of edges in the trees of the upper (resp. lower)
half-plane. As mentioned during Section 2.1, we use the slightly unusual
convention that $c_{2a+h}\neq\bar{c}_{0}$ (and similarly
$\bar{c}_{2\bar{a}+h}\neq c_{0}$): this means that the first corner incident
to $\rho$ is “split” in two corners, one in the upper half-plane and one in
the lower half-plane.

Finally, assume that, in its unique face, the map
${\mathbf{f}}\cup{\bar{\mathbf{f}}}$ is well labeled by an integer function
$\lambda:V({\mathbf{f}}\cup{\bar{\mathbf{f}}})\to\mathbb{Z}$ defined up to
addition of a constant: this simply means that
$\lambda(u)-\lambda(v)\in\\{-1,0,1\\}$ whenever $u$ and $v$ are neighboring
vertices. See Figure 7 for an example. Note that, equivalently, a well-labeled
double forest $((\mathbf{f},\bar{\mathbf{f}}),\lambda)$ can be seen as a well-
labeled _vertebrate_ , that is a well-labeled tree with two distinct
distinguished vertices $\rho$, $\bar{\rho}$, where the interval $I$
corresponds to the consecutive corners in the contour order from $\rho$ to
$\bar{\rho}$, which contains all the corners incident to $\rho$ and stops at
the first corner incident to $\bar{\rho}$, and $\bar{I}$ is defined similarly
with the roles of $\rho$, $\bar{\rho}$ exchanged.

Figure 7: The interval Cori–Vauquelin–Schaeffer bijection giving
quadrilaterals with geodesic sides. The quadrilateral with geodesic sides has
half-areas $5$ and $4$, width $3$, and tilt $1$.

The map
$\mathbf{qd}=\operatorname{CVS}({\mathbf{f}}\cup{\bar{\mathbf{f}}},\lambda;I,\bar{I})$
is then a nonrooted plane quadrangulation with one hole having $a+\bar{a}+h$
internal faces. Its boundary contains the four distinguished vertices $\rho$,
the apex $v_{*}$ associated with $I$, $\bar{\rho}$, and the apex $\bar{v}_{*}$
associated with $\bar{I}$, as well as the maximal geodesics $\gamma$,
$\bar{\gamma}$ and shuttles $\xi$, $\bar{\xi}$, with obvious notation.

######  Definition 10.

The _quadrilateral with geodesic sides_ , or simply _quadrilateral_ for short,
associated with $((\mathbf{f},\bar{\mathbf{f}}),\lambda)$ is by definition
$\mathbf{qd}=\operatorname{CVS}({\mathbf{f}}\cup{\bar{\mathbf{f}}},\lambda;I,\bar{I})$.
Its _width_ , _half-areas_ , and _tilt_ are respectively the numbers

$h\,,\qquad a\ \text{ and }\
\bar{a}\,,\qquad\lambda(\bar{\rho})-\lambda(\rho).$

Observe that the parameters of a quadrilateral can be recovered from the map
$\mathbf{qd}$ and the distinguished vertices $\rho$, $\bar{\rho}$.
Furthermore, the quadrilateral associated with
$((\bar{\mathbf{f}},\mathbf{f}),\lambda)$ is obtained from the one associated
with $((\mathbf{f},\bar{\mathbf{f}}),\lambda)$ simply by switching the
distinguished elements $\rho$ with $\bar{\rho}$, $\gamma$ with $\bar{\gamma}$,
and $\xi$ with $\bar{\xi}$. It has the same width, its half-areas are switched
and its tilt is reversed.

######  Proposition 11.

The number of quadrilaterals with half-areas $a$, $\bar{a}$, width $h$ and
tilt $\delta$ is equal to

$12^{a+\bar{a}+h}\,Q_{h}(2a+h)\,Q_{h}(2\bar{a}+h)\,M_{h}(\delta)\,,$

where $Q_{\ell}$ has been defined in Proposition 9, and
$M_{\ell}(j)=\mathbb{P}(U_{1}+\ldots+U_{\ell}=j)$, where $U_{1}$, $U_{2}$, …
are independent uniform random variables in $\\{-1,0,1\\}$.

###### Proof.

As in the proof of Proposition 9, the number of forests with $h$ floor edges
and $\alpha$ non-floor edges is $2^{2\alpha+h}Q_{h}(2\alpha+h)$, so the number
of double forests with proper parameters is
$4^{a+\bar{a}+h}Q_{h}(2a+h)Q_{h}(2\bar{a}+h)$ . Then, the number of possible
labelings of the floor vertices is the number of walks with $h$ steps in
$\\{-1,0,1\\}$ going form $0$ to $\delta$, which equals $3^{h}M_{h}(\delta)$.
The final term $3^{a+\bar{a}}$ counts the possible labelings of the non-root
vertices in the double forest. ∎

If $\mathbf{qd}$ is a quadrilateral, we will view it as a marked measured
metric space in $\mathbb{M}^{(6,1)}$ given by

$\big{(}V(\mathbf{qd}),d_{\mathbf{qd}},\partial\mathbf{qd},\mu_{\mathbf{qd}}\big{)}\qquad\text{
with
}\qquad\partial\mathbf{qd}=\big{(}\rho,\gamma,\xi,\bar{\rho},\bar{\gamma},\bar{\xi}\big{)}\,,$
(7)

where each boundary part is identified with the vertices it contains, and
where $\mu_{\mathbf{qd}}$ is the counting measure on the vertices of
$\mathbf{qd}$ that do not belong to the shuttles. We call this measure
$\mu_{\mathbf{qd}}$ the _area measure_ of the quadrilateral.

###  2.5 Scaling limits of elementary pieces

In this section, we state two important results that will be crucial in the
proof of Theorem 1. These show that, under appropriate hypotheses, random
discrete slices and quadrilaterals converge in distribution in the GHP
topology toward “continuum analogs” of these objects.

We first fix three sequences $(a_{n})\in(\mathbb{Z}_{+})^{\mathbb{N}}$,
$(l_{n})\in\mathbb{N}^{\mathbb{N}}$ and
$(\delta_{n})\in\mathbb{Z}^{\mathbb{N}}$ such that

$\frac{a_{n}}{n}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}A>0\
,\qquad\frac{l_{n}}{\sqrt{2n}}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}L>0\qquad\text{ and
}\qquad\left(\frac{9}{8n}\right)^{1/4}\delta_{n}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}\Delta\in\mathbb{R}.$ (8)

Recall that a slice is seen as an element of $\mathbb{M}^{(5,2)}$ given by (6)
and that $\Omega_{n}$ is the scaling operator defined in (4).

######  Theorem 12.

Let $\mathrm{Sl}_{n}$ be uniformly distributed among composite slices with
area $a_{n}$, width $l_{n}$ and tilt $\delta_{n}$. Then we have the
convergence

$\Omega_{n}(\mathrm{Sl}_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\mathsf{Sl}_{A,L,\Delta}\,,$

in distribution in the space
$\big{(}\mathbb{M}^{(5,2)},\mathrm{d}_{\mathrm{GHP}}^{(5,2)}\big{)}$. The
limit is called a _(continuum composite) slice with area $A$, width $L$ and
tilt $\Delta$_.

This theorem will be proved in Section 5, where a detailed characterization of
the limiting object will be given. For the time being, this theorem should be
taken as a definition of the spaces $\mathsf{Sl}_{A,L,\Delta}$.

The following statement, whose proof is a direct consequence of Theorem 12 and
is left to the reader, deal with the case of vanishing areas and widths, and
will be useful in Section 4.3 below.

######  Corollary 13.

Let the sequences $(a_{n})\in(\mathbb{Z}_{+})^{\mathbb{N}}$ and
$(l_{n})\in(\mathbb{Z}_{+})^{\mathbb{N}}$ satisfy
$l_{n}=\scalebox{0.5}[0.62]{$\mathbfcal{O}$}(\sqrt{n})$ and
$a_{n}+l_{n}=\Theta((l_{n})^{2})$. Let $\mathrm{Sl}_{n}$ be the vertex map
whenever $l_{n}=0$, or be uniformly distributed among slices with area
$a_{n}$, width $l_{n}$ and tilt $0$ otherwise. Then we have the convergence
toward the point space

$\Omega_{n}(\mathrm{Sl}_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\\{\varrho\\}\,,$

in distribution in the space
$\big{(}\mathbb{M}^{(5,2)},\mathrm{d}_{\mathrm{GHP}}^{(5,2)}\big{)}$.

To lift any possible ambiguity, let us stress that the property
“$a_{n}+l_{n}=\Theta((l_{n})^{2})$” means that the sequence
$((a_{n}+l_{n})/(l_{n})^{2})$, restricted to the values of $n$ for which they
are defined, is bounded away from $0$ and $\infty$. This compact way of
writing this property covers in fact the two following situations. If
$(l_{n})$ is a bounded integer sequence, it simply means that $(a_{n})$ is a
bounded integer sequence. If $(l_{n})$ is unbounded, then it means that
$(a_{n}/(l_{n})^{2})$ is bounded away from $0$ and $\infty$. Note that, up to
extracting subsequences, we are always in one of these two situations.

We will derive Theorem 12 from the known convergence of the uniform infinite
half-planar quadrangulation toward the Brownian half-plane. The former
naturally contains a family of slices and the latter contains a continuous
“flow” of continuum slices. These consist in free versions of the objects
considered here so that we will need to finish with a conditioning argument.

We now turn to quadrilaterals, which are seen as elements of
$\mathbb{M}^{(6,1)}$ given by (7). We consider four sequences $(a_{n})$,
$(\bar{a}_{n})\in(\mathbb{Z}_{+})^{\mathbb{N}}$,
$(h_{n})\in\mathbb{N}^{\mathbb{N}}$ and
$(\delta_{n})\in\mathbb{Z}^{\mathbb{N}}$ such that, as $n\to\infty$,

$\frac{a_{n}}{n}\to
A>0\,,\quad\frac{\bar{a}_{n}}{n}\to\bar{A}>0\,,\quad\frac{h_{n}}{\sqrt{2n}}\to
H>0\,,\quad\left(\frac{9}{8n}\right)^{1/4}\delta_{n}\to\Delta\in\mathbb{R}.$
(9)

######  Theorem 14.

Let $\mathrm{Qd}_{n}$ be uniformly distributed among quadrilaterals with half-
areas $a_{n}$ and $\bar{a}_{n}$, width $h_{n}$ and tilt $\delta_{n}$. Then we
have the convergence

$\Omega_{n}(\mathrm{Qd}_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}^{(d)}}\mathsf{Qd}_{A,\bar{A},H,\Delta}\,,$

in distribution in the space
$\big{(}\mathbb{M}^{(6,1)},\mathrm{d}_{\mathrm{GHP}}^{(6,1)}\big{)}$. The
limit is called a _continuum quadrilateral with half-areas $A$ and $\bar{A}$,
width $H$ and tilt $\Delta$_.

As for slices, the proof of this result is postponed, to Section 6, where a
detailed characterization of the limiting object will be given. The idea of
the proof will be similar to that of Theorem 12, using the uniform infinite
planar quadrangulation and Brownian plane as reference spaces instead of the
half-planar versions mentioned above.

##  3 Marking and gluing along geodesics

In our previous work [BM17], we proved Theorem 1 in the case of disks (for the
GH topology) by writing $Q_{n}$ and $\mathsf{S}^{[0]}_{(L^{1})}$ as gluings of
appropriate subspaces along geodesic segments, namely so-called _slices_ in
the discrete setting and their scaling limits in the continuum. The fact that
the number of gluings needed was infinite caused some difficulties (which we
mainly overcame by noticing that any geodesic between two typical points may
be broken down to a finite number of pieces lying in different such
subspaces). In contrast, in this work, we will only need to consider gluings
of a _finite number_ of subspaces along geodesic segments. As this operation
is well behaved in a more general setting, we present it in this section. But
first, we collect a number of useful lemmas on the GHP topology.

We will use the following notation. If $\mu_{\mathcal{X}}$ is a finite
positive measure on a set $\mathcal{X}$, we let
$\bar{\mu}_{\mathcal{X}}=\mu_{\mathcal{X}}/\mu_{\mathcal{X}}(\mathcal{X})$ be
the normalized probability measure. If $\mu_{\mathcal{X}}=0$, we use the
convention $\bar{\mu}_{\mathcal{X}}=0$. If
${\boldsymbol{\mu}}=(\mu^{1},\ldots,\mu^{m})$ is a finite family of
nonnegative measures, we let
$\bar{{\boldsymbol{\mu}}}=(\bar{\mu}^{1},\ldots,\bar{\mu}^{m})$.

###  3.1 Useful facts on the GHP topology and markings

Recall the definitions of
$(\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)})$ and
$(\mathbb{M}^{(\ell)},\mathrm{d}_{\mathrm{GH}}^{(\ell)})$ from Section 1.3. If
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$ is
an element of $\mathbb{M}^{(\ell,m)}$ and $\mathbf{r}\in(\mathbb{Z}_{+})^{m}$
such that $r^{j}=0$ whenever $\mu_{\mathcal{X}}^{j}=0$, we may consider the
random variable
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A}(x^{1}_{1},\dots,x^{1}_{r^{1}})\dots(x^{m}_{1},\dots,x^{m}_{r^{m}}))$
taking values in $\mathbb{M}^{(\ell+\|\mathbf{r}\|)}$, where, for each
$j\in\\{1,\dots,m\\}$, the points $x^{j}_{1}$, …, $x^{j}_{r_{j}}$ are i.i.d.
sampled random variables with law $\bar{\mu}^{j}_{\mathcal{X}}$ (if the latter
measure is $0$, then this still makes sense since $r^{j}=0$); we denote by
$\mathrm{Mark}_{\mathbf{r}}((\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),\cdot)$
the law of this random marked metric space. Some care is actually needed here
since we are considering isometry classes of metric measure spaces. See
[Mie09] for an accurate definition of this notion, which is immediately
generalized to our setting where we incorporate the extra marks given by
$\mathbf{A}$, and several measures. The following lemma states that one can
formulate the GHP convergence entirely in terms of the GH convergence of
randomly marked spaces.

######  Lemma 15.

Let
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$,
$n\geq 1$, and
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$ be
elements of $\mathbb{M}^{(\ell,m)}$. The following statements are equivalent.

  1. (i)

The space
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$
converges to
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$ in
$\big{(}\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}\big{)}$.

  2. (ii)

One has
${\boldsymbol{\mu}}_{\mathcal{X}_{n}}(\mathcal{X}_{n})\to{\boldsymbol{\mu}}_{\mathcal{X}}(\mathcal{X})$
coordinatewise as $n\to\infty$ and, for every
$\mathbf{r}\in(\mathbb{Z}_{+})^{m}$ such that $r^{j}=0$ whenever
$\mu^{j}_{\mathcal{X}}=0$, it holds that

$\mathrm{Mark}_{\mathbf{r}}\big{(}(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}}),\cdot\,\big{)}\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}\mathrm{Mark}_{\mathbf{r}}\big{(}(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),\cdot\,\big{)}$

in the sense of weak convergence of probability measures on
$\big{(}\mathbb{M}^{(\ell+\|\mathbf{r}\|)},\mathrm{d}_{\mathrm{GH}}^{(\ell+\|\mathbf{r}\|)}\big{)}$.

###### Proof.

The implication $\text{\ref{lemrandmarki}}\implies\text{\ref{lemrandmarkii}}$
is an easy generalization of known results. See [Mie09, Proposition 10] for
the case where the measures are probability measures, and [LG19a, Section 2.2]
for a generalized context with finite measures; our extended context of marked
measured metric spaces adds no difficulty. To show the converse implication,
we argue as follows. By taking the trivial case
$\mathbf{r}={\boldsymbol{0}}^{m}$ of (ii), we obtain that
$\\{(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n}),n\geq 1\\}$ is
relatively compact in
$\big{(}\mathbb{M}^{(\ell)},\mathrm{d}_{\mathrm{GH}}^{(\ell)}\big{)}$.
Together with the fact that the sequences
$(\mu^{j}_{\mathcal{X}_{n}}(\mathcal{X}_{n}),n\geq 1)$ are bounded, this
implies that
$\\{(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}}),n\geq
1\\}$ is relatively compact in
$\big{(}\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}\big{)}$. So
let
$(\mathcal{X}^{\prime},d_{\mathcal{X}^{\prime}},\mathbf{A}^{\prime},{\boldsymbol{\mu}}_{\mathcal{X}^{\prime}})$
be a limit in $\mathbb{M}^{(\ell,m)}$ along some subsequence of
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$.
By using the implication
$\text{\ref{lemrandmarki}}\implies\text{\ref{lemrandmarkii}}$, we obtain that,
for every $\mathbf{r}$ such that $r^{j}=0$ whenever $\mu_{\mathcal{X}}^{j}=0$,

$\mathrm{Mark}_{\mathbf{r}}\big{(}(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),\cdot\,\big{)}=\mathrm{Mark}_{\mathbf{r}}\big{(}(\mathcal{X}^{\prime},d_{\mathcal{X}^{\prime}},\mathbf{A}^{\prime},{\boldsymbol{\mu}}_{\mathcal{X}^{\prime}}),\cdot\,\big{)}\,.$

Now, let $m^{\prime}$ be the number of nonzero elements of
${\boldsymbol{\mu}}_{\mathcal{X}}$, fix $r>0$ and set
$r^{j}=r\mathbf{1}_{\\{\mu_{\mathcal{X}}^{j}\neq 0\\}}$. We let the
$(\ell+rm^{\prime})$-marked metric space
$(\mathcal{X},d_{\mathcal{X}},(A^{1},\ldots,A^{\ell},x^{1}_{1},\dots,x^{1}_{r^{1}},\ldots,x^{m}_{1},\ldots,x^{m}_{r^{m}}))$
have law
$\mathrm{Mark}_{\mathbf{r}}((\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),\cdot)$,
and set $\boldsymbol{\theta}_{r}=(\theta^{j}_{r},1\leq j\leq m)$, where
$\theta^{j}_{r}=r^{-1}\sum_{i=1}^{r}\delta_{x^{j}_{i}}$ if
$\mu_{\mathcal{X}}^{j}\neq 0$ and $\theta^{j}_{r}=0$ if
$\mu_{\mathcal{X}}^{j}=0$. It is a consequence of the law of large numbers
that $(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\boldsymbol{\theta}_{r})$
converges almost surely in $\mathbb{M}^{(\ell,m)}$, as $r\to\infty$, to
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\bar{{\boldsymbol{\mu}}}_{\mathcal{X}})$;
see for instance [LG19a, Lemma 5]. Applying this same result to
$(\mathcal{X}^{\prime},d_{\mathcal{X}^{\prime}},\mathbf{A}^{\prime},{\boldsymbol{\mu}}_{\mathcal{X}^{\prime}})$
shows that
$(\mathcal{X}^{\prime},d_{\mathcal{X}^{\prime}},\mathbf{A}^{\prime},\bar{{\boldsymbol{\mu}}}^{\prime}_{\mathcal{X}^{\prime}})$
is isometry-equivalent to
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},\bar{{\boldsymbol{\mu}}}_{\mathcal{X}})$.
Since
${\boldsymbol{\mu}}_{\mathcal{X}}(\mathcal{X})={\boldsymbol{\mu}}_{\mathcal{X}^{\prime}}(\mathcal{X}^{\prime})$
is the limit of ${\boldsymbol{\mu}}_{\mathcal{X}_{n}}(\mathcal{X}_{n})$, we
conclude. ∎

We also recall that, often, the most useful way to estimate GH distances is
via the notion of _distortion of a correspondence_. A _correspondence_ between
two sets $\mathcal{X}$ and $\mathcal{Y}$ is a subset
$\mathcal{R}\subseteq\mathcal{X}\times\mathcal{Y}$ whose coordinate
projections are $\mathcal{X}$ and $\mathcal{Y}$. We will often write
$x\mathbin{\mathcal{R}}y$ instead of $(x,y)\in\mathcal{R}$. If $\mathcal{X}$
and $\mathcal{Y}$ are endowed with the metrics $d_{\mathcal{X}}$ and
$d_{\mathcal{Y}}$, the _distortion_ of the correspondence $\mathcal{R}$ is the
number

$\operatorname{dis}(\mathcal{R})=\sup\big{\\{}|d_{\mathcal{X}}(x,x^{\prime})-d_{\mathcal{Y}}(y,y^{\prime})|\,:\,x\mathbin{\mathcal{R}}y,\
x^{\prime}\mathbin{\mathcal{R}}y^{\prime}\big{\\}}.$

If $\mathbf{A}=(A^{1},\ldots,A^{\ell})$ and
$\mathbf{B}=(B^{1},\ldots,B^{\ell})$ are markings of $\mathcal{X}$ and of
$\mathcal{Y}$, we say that the correspondence $\mathcal{R}$ between
$\mathcal{X}$ and $\mathcal{Y}$ is _compatible_ with the markings if for every
$1\leq i\leq\ell$, $\mathcal{R}\cap(A^{i}\times B^{i})$ is a correspondence
between $A^{i}$ and $B^{i}$.

######  Lemma 16 ([Mie09, Section 6.4]).

It holds that

$\mathrm{d}_{\mathrm{GH}}^{(\ell)}\big{(}(\mathcal{X},\mathbf{A},d_{\mathcal{X}}),(\mathcal{Y},\mathbf{B},d_{\mathcal{Y}})\big{)}=\frac{1}{2}\inf_{\mathcal{R}}\operatorname{dis}(\mathcal{R}),$

where the infimum is taken over correspondences compatible with the markings.

Correspondences are also useful for estimating GHP distances when used
together with the notion of couplings, which are measures on the product of
the two spaces to be compared. The following is a direct adaptation of [LG19a,
Lemma 4], which treats the case of $\mathbb{M}^{(0,1)}$.

######  Lemma 17.

Let
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$
and
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})$ be
elements of $\mathbb{M}^{(\ell,m)}$ for some $\ell$, $m\geq 0$. Let
$\varepsilon>0$, and $\mathcal{R}$ be a correspondence between $\mathcal{X}$
and $\mathcal{Y}$ compatible with the markings and of distortion bounded above
by $\varepsilon$. For $1\leq j\leq m$, let $\nu^{j}$ be a finite measure on
the product $\mathcal{X}\times\mathcal{Y}$ such that
$\nu^{j}(\mathcal{R}^{c})<\varepsilon$ and, letting $p_{\mathcal{X}}$,
$p_{\mathcal{Y}}$ be the coordinate projections onto $\mathcal{X}$ and
$\mathcal{Y}$,

$d_{\mathcal{X}}^{\mathrm{P}}(\mu^{j}_{\mathcal{X}},(p_{\mathcal{X}})_{*}\nu^{j})\vee
d_{\mathcal{Y}}^{\mathrm{P}}(\mu^{j}_{\mathcal{Y}},(p_{\mathcal{Y}})_{*}\nu^{j})<\varepsilon\,.$

Then
$\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}((\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}}),(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}}))\leq
3\varepsilon$.

Finally, we state an elementary lemma whose proof is straightforward and
omitted.

######  Lemma 18.

The mappings

$\displaystyle(\mathcal{X},d_{\mathcal{X}},(A^{1},\ldots,A^{\ell}),{\boldsymbol{\mu}}_{\mathcal{X}})$
$\displaystyle\longmapsto(\mathcal{X},d_{\mathcal{X}},(A^{1}\cup
A^{2},A^{3},\ldots,A^{\ell}),{\boldsymbol{\mu}}_{\mathcal{X}})\,,$
$\displaystyle(\mathcal{X},d_{\mathcal{X}},(A^{1},\ldots,A^{\ell}),{\boldsymbol{\mu}}_{\mathcal{X}})$
$\displaystyle\longmapsto(\mathcal{X},d_{\mathcal{X}},(A^{1},\ldots,A^{\ell-1}),{\boldsymbol{\mu}}_{\mathcal{X}})$
are $1$-Lipschitz from
$\big{(}\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}\big{)}$ to
$\big{(}\mathbb{M}^{(\ell-1,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell-1,m)}\big{)}$;
the mappings
$\displaystyle(\mathcal{X},d_{\mathcal{X}},\mathbf{A},(\mu^{1}_{\mathcal{X}},\ldots,\mu^{m}_{\mathcal{X}})$
$\displaystyle\longmapsto(\mathcal{X},d_{\mathcal{X}},\mathbf{A},(\mu^{1}_{\mathcal{X}}+\mu^{2}_{\mathcal{X}},\mu^{3}_{\mathcal{X}},\ldots,\mu^{m}_{\mathcal{X}}))$
$\displaystyle(\mathcal{X},d_{\mathcal{X}},\mathbf{A},(\mu^{1}_{\mathcal{X}},\ldots,\mu^{m}_{\mathcal{X}})$
$\displaystyle\longmapsto(\mathcal{X},d_{\mathcal{X}},\mathbf{A},(\mu^{1}_{\mathcal{X}},\ldots,\mu^{m-1}_{\mathcal{X}}))$
are respectively $2$-Lipschitz and $1$-Lipschitz from
$\big{(}\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)}\big{)}$ to
$\big{(}\mathbb{M}^{(\ell,m-1)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m-1)}\big{)}$;
and, for every permutation $\sigma$ of $\\{1,2,\ldots,\ell\\}$ and $\tau$ of
$\\{1,2,\ldots,m\\}$,
$\displaystyle\big{(}\mathcal{X},d_{\mathcal{X}},(A^{1},\ldots,A^{\ell}),(\mu^{1}_{\mathcal{X}},\ldots,\mu^{m}_{\mathcal{X}})\big{)}$
$\displaystyle\longmapsto\big{(}\mathcal{X},d_{\mathcal{X}},\big{(}A^{\sigma(1)},\ldots,A^{\sigma(\ell)}\big{)},\big{(}\mu^{\tau(1)}_{\mathcal{X}},\ldots,\mu^{\tau(m)}_{\mathcal{X}}\big{)}\big{)},$

is an isometry from
$(\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)})$ onto itself.

###  3.2 Geodesics in metric spaces

We now discuss the important notion of geodesics in metric spaces, as well as
its relations with GHP limits.

In a metric space $(\mathcal{X},d_{\mathcal{X}})$, compact or not, a
_geodesic_ is a mapping $\chi:[0,\mathpzc{l}]\to\mathcal{X}$ defined on some
compact interval666We allow $\mathpzc{l}=0$ in this definition.
$[0,\mathpzc{l}]$ and that is isometric, i.e., satisfies

$d_{\mathcal{X}}(\chi(s),\chi(t))=|t-s|\,,\qquad 0\leq s,t\leq\mathpzc{l}\,.$
(10)

The points $\chi(0)$, $\chi(\mathpzc{l})$ are called the _extremities_ of
$\chi$, and the quantity
$\mathpzc{l}=d_{\mathcal{X}}(\chi(0),\chi(\mathpzc{l}))$ is the _length_ of
the geodesic, denoted by $\operatorname{length}_{d_{\mathcal{X}}}(\chi)$, or
simply $\operatorname{length}(\chi)$ when there is little risk of ambiguity.
The space $(\mathcal{X},d_{\mathcal{X}})$ is called a _geodesic space_ if, for
every pair of points $x$, $y\in\mathcal{X}$, there exists a geodesic with
extremities $x$, $y$.

The range $\chi([0,\operatorname{length}(\chi)])$ of a geodesic path is called
a _geodesic segment_. An _oriented geodesic segment_ is a pair
$(\chi(0),\chi([0,\operatorname{length}(\chi)]))$ made of a geodesic segment
and a distinguished extremity, called its _origin_. Note that an oriented
geodesic segment uniquely determines the geodesic $\chi$, since $\chi(t)$ is
the unique point at distance $t$ away from the origin. For this reason, we
will systematically identify geodesics with oriented geodesic segments and use
the same piece of notation for both of them.

In a marked measured metric space
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$,
some pairs $(A^{i},A^{j})$ of marks might be oriented geodesic segments; such
pairs are called _geodesic marks_.

##### Geodesic marks and GHP limits.

The following proposition states that geodesic marks nicely pass to the limit
in the GHP topology.

######  Proposition 19.

Let
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$,
$n\geq 1$, be a sequence of marked measured compact metric spaces that
converges to some limit
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$ in
the GHP topology. Suppose that $i$, $j$ are fixed and that, for every $n$, the
pair of marks $(A_{n}^{i},A_{n}^{j})=\gamma_{n}$ is a geodesic mark. Then the
pair of marks $(A^{i},A^{j})=\gamma$ of $\mathbf{A}$ is also a geodesic mark.
Moreover, it holds that

$\operatorname{length}(\gamma)=\lim_{n\to\infty}\operatorname{length}(\gamma_{n})\,.$

###### Proof.

The wanted property deals only with the marks and not with the measures, so it
suffices to establish the proposition in the space $\mathbb{M}^{(\ell)}$ of
marked, nonmeasured spaces. Without loss of generality (by Lemma 18), we may
and will assume that $i=1$ and $j=2$.

By Lemma 16, we may find a sequence of correspondences $\mathcal{R}_{n}$
between $\mathcal{X}_{n}$ and $\mathcal{X}$ that is compatible with the
markings $\mathbf{A}_{n}$ and $\mathbf{A}$, and whose distortion
$\varepsilon_{n}:=\mathrm{dis}(\mathcal{R}_{n})$ goes to zero. From now on, we
will never need to refer to marks other than the first two.

Let $y$, $z\in A^{1}$. Since $A_{n}^{1}$ contains a single point, which we
denote by $x_{n}=\gamma_{n}(0)$, we have $x_{n}\mathbin{\mathcal{R}_{n}}y$ and
$x_{n}\mathbin{\mathcal{R}_{n}}z$, so that
$d_{\mathcal{X}}(y,z)\leq\varepsilon_{n}$ for every $n\geq 1$, entailing
$y=z$. So $A^{1}$ is a singleton, which we denote by $A^{1}=\\{x\\}$.

Next, let $a\in A^{2}$ and $a_{n}\in A_{n}^{2}$ be such that
$a_{n}\mathbin{\mathcal{R}_{n}}a$. Then
$|d_{\mathcal{X}}(x,a)-d_{\mathcal{X}_{n}}(x_{n},a_{n})|\leq\varepsilon_{n}$,
which implies that $d_{\mathcal{X}_{n}}(x_{n},a_{n})\to d_{\mathcal{X}}(x,a)$
as $n\to\infty$, and in particular,
$d_{\mathcal{X}}(x,a)\leq\liminf_{n\to\infty}\operatorname{length}(\gamma_{n})$,
and therefore

$\max_{a\in
A^{2}}d_{\mathcal{X}}(x,a)\leq\liminf_{n\to\infty}\operatorname{length}(\gamma_{n})\,.$

In the other direction, let
$t\leq\limsup_{n\to\infty}\operatorname{length}(\gamma_{n})$. We claim that
there exists at least a point $c_{t}\in A^{2}$ such that
$d_{\mathcal{X}}(x,c_{t})=t$. This will entail that
$\operatorname{length}(\gamma_{n})$ converges to $\mathpzc{l}=\max_{a\in
A^{2}}d_{\mathcal{X}}(x,a)$. To see the claim, observe that, at least along a
suitable extraction, there exists a sequence $t_{n}\to t$ such that
$\operatorname{length}(\gamma_{n})\geq t_{n}$. Along this extraction, let
$\gamma_{n}(t_{n})$ be the unique point of $\gamma_{n}$ such that
$t_{n}=d_{\mathcal{X}_{n}}(x_{n},\gamma_{n}(t_{n}))$, and let $g_{n}$ be an
element of $A^{2}$ such that
$\gamma_{n}(t_{n})\mathbin{\mathcal{R}_{n}}g_{n}$. Then
$|d_{\mathcal{X}}(x,g_{n})-t_{n}|\leq\varepsilon_{n}$, so that, possibly by
further extracting, $(g_{n})$ converges to a limit $c_{t}\in A^{2}$. It then
holds that $d_{\mathcal{X}}(x,c_{t})=t$, as claimed.

Now fix $s$, $t\in[0,\mathpzc{l}]$ with $s\leq t$, and let $a$, $b\in A^{2}$
be such that $d_{\mathcal{X}}(x,a)=s$ and $d_{\mathcal{X}}(x,b)=t$. By the
triangle inequality, we have $d_{\mathcal{X}}(a,b)\geq t-s$, and, on the other
hand, if $a_{n}\mathbin{\mathcal{R}_{n}}a$ and
$b_{n}\mathbin{\mathcal{R}_{n}}b$ with $a_{n}$, $b_{n}\in A_{n}^{2}$, then

$\displaystyle d_{\mathcal{X}}(a,b)\leq
d_{\mathcal{X}_{n}}(a_{n},b_{n})+\varepsilon_{n}$
$\displaystyle=|d_{\mathcal{X}_{n}}(x_{n},b_{n})-d_{\mathcal{X}_{n}}(x_{n},a_{n})|+\varepsilon_{n}$
$\displaystyle\leq|d_{\mathcal{X}}(x,b)-d_{\mathcal{X}}(x,a)|+3\varepsilon_{n}$
$\displaystyle=t-s+3\varepsilon_{n}$

where in the second line, we have used the fact that $a_{n}$, $b_{n}$ lie on a
geodesic having $x_{n}$ as one of its extremities. Letting $n\to\infty$, this
shows that $d_{\mathcal{X}}(a,b)=t-s$, and in particular, taking $t=s$ shows
that the point $c_{t}$ of the preceding paragraph is the unique point of
$A^{2}$ at distance $t$ from $x$. We conclude that $\gamma$ is an oriented
geodesic segment with length $\mathpzc{l}$ and origin $x$. ∎

##### Maps as compact geodesic metric spaces.

So far, we have been seeing maps as finite metric spaces. We may also
interpret a map $\mathbf{m}$ as a compact geodesic metric space, by viewing
each edge as isometric to a real segment of length $1$ (this is called the
metric graph [BBI01] associated with $\mathbf{m}$). Note that the restriction
of the metric to the subset corresponding to the vertex set of $\mathbf{m}$ is
the graph metric, so that the two metric spaces corresponding to $\mathbf{m}$
are at $\mathrm{d}_{\mathrm{GH}}$-distance less than $1/2$. In the scaling
limit, this bears no effects.

With this point of view on maps, note that, in the notation of Sections 2.3
and 2.4,

  * •

$(\rho,\gamma)$ and $(\bar{\rho},\xi)$ are geodesic marks of $\mathbf{sl}$;

  * •

$(\rho,\gamma)$, $(\bar{\rho},\xi)$, $(\bar{\rho},\bar{\gamma})$, and
$(\rho,\bar{\xi})$ are geodesic marks of $\mathbf{qd}$.

###  3.3 Gluing along geodesics

##### Quotient pseudometrics.

Let $(\mathcal{X},d)$ be a pseudometric space, that is, a set equipped with a
symmetric function $d:\mathcal{X}^{2}\to\mathbb{R}_{+}\sqcup\\{\infty\\}$ that
vanishes on the diagonal and satisfies the triangle inequality. Then
$\\{d=0\\}$ is an equivalence relation on $\mathcal{X}$, and the quotient set
$\mathcal{X}/\\{d=0\\}$ equipped with the function induced by $d$ (still
denoted by $d$ for simplicity), is a true metric space, meaning that $d$ is
also separated.

Let $R$ be an equivalence relation on $\mathcal{X}$. Let $d/R$ be the largest
pseudometric on $\mathcal{X}$ such that $d/R\leq d$ and that satisfies
$d/R(x,y)=0$ as soon as $x\mathbin{R}y$. By [BBI01, Theorem 3.1.27], it is
given by the formula

$d/R(x,y)=\inf\left\\{\sum_{i=1}^{m}d(x_{i},y_{i}):\\!\begin{array}[]{l}m\geq
1,x_{1},\ldots,x_{m},y_{1},\ldots,y_{m}\in\mathcal{X},\\\
x_{1}=x,y_{m}=y,y_{i}\mathbin{R}x_{i+1}\mbox{ for
}i\in\\{1,\ldots,m-1\\}\end{array}\right\\}\\!.$ (11)

In this setting, the set $\\{d/R=0\\}$ is another equivalence relation on
$\mathcal{X}$ that contains $R$, possibly strictly. We let
$(\mathcal{X},d)/R=(\mathcal{X}/\\{d/R=0\\},d/R)$ and call it the gluing of
$(\mathcal{X},d)$ along $R$.

A simple observation is that if $R_{1}$, $R_{2}$ are two equivalence relations
on $\mathcal{X}$, then we have the equality of pseudometrics on $\mathcal{X}$

$(d/R_{1})/R_{2}=(d/R_{2})/R_{1}=d/R\,,$ (12)

where $R$ is the coarsest equivalence relation containing $R_{1}\cup R_{2}$.
This expression is indeed a direct consequence of (11) and the fact that
$x\mathbin{R}y$ if and only if there exists some integer $m$ and points
$x_{0}=x$, $x_{1}$, …, $x_{m}=y$ such that $(x_{i-1},x_{i})\in R_{1}\cup
R_{2}$ for every $i\in\\{1,2,\ldots,m\\}$.

##### Gluing two spaces along geodesics.

Let $(\mathcal{X},d_{\mathcal{X}})$, $(\mathcal{Y},d_{\mathcal{Y}})$ be two
pseudometric spaces and $\gamma$, $\xi$ be two geodesics in $\mathcal{X}$ and
$\mathcal{Y}$, respectively, where the definition (10) of a geodesic is
naturally extended to pseudometric spaces. The pseudometric of the disjoint
union $\mathcal{X}\sqcup\mathcal{Y}$ is defined by

$d_{\mathcal{X}\sqcup\mathcal{Y}}(x,y)=\begin{cases}d_{\mathcal{X}}(x,y)&\text{
if }x,y\in\mathcal{X}\\\ d_{\mathcal{Y}}(x,y)&\text{ if }x,y\in\mathcal{Y}\\\
\infty&\text{ otherwise }\end{cases}.$

We define the metric gluing of $\mathcal{X}$ and $\mathcal{Y}$ along $\gamma$
and $\xi$ by letting
$\mathpzc{l}=\operatorname{length}(\gamma)\wedge\operatorname{length}(\xi)$
and by setting

$G(\mathcal{X},\mathcal{Y};\gamma,\xi)=(\mathcal{X}\sqcup\mathcal{Y},d_{\mathcal{X}\sqcup\mathcal{Y}})/R\,,$
(13)

where $R$ is the coarsest equivalence relation satisfying
$\gamma(t)\mathbin{R}\xi(t)$ for every $t\in[0,\mathpzc{l}]$.

In this particular case, the fact that $\gamma$ and $\xi$ are geodesics
greatly simplifies (11). Indeed, $y_{i}\mathbin{R}x_{i+1}$ and
$y_{i+1}\mathbin{R}x_{i+2}$ imply that
$d_{\mathcal{X}\sqcup\mathcal{Y}}(y_{i},x_{i+2})=d_{\mathcal{X}\sqcup\mathcal{Y}}(x_{i+1},y_{i+1})$.
In other words, using twice the relation $R$ does not create shortcuts. As a
result, the pseudometric of the gluing is the function $d_{G}$ whose
restrictions to $\mathcal{X}\times\mathcal{X}$ and
$\mathcal{Y}\times\mathcal{Y}$ are $d_{\mathcal{X}}$ and $d_{\mathcal{Y}}$
respectively, and

$d_{G}(x,y)=d_{G}(y,x)=\inf_{t\in[0,\mathpzc{l}]}\big{\\{}d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{Y}}(\xi(t),y)\big{\\}}\qquad\text{
if }x\in\mathcal{X},\,y\in\mathcal{Y}\,.$ (14)

######  Remark 20.

In fact, Equation (14) holds in the more general setting of gluing along
isometric subspaces [BH99, Chapter I.5], the underlying isometry in our
context being $\gamma(t)\mapsto\xi(t)$. We will not need this level of
generality here.

If
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})\in\mathbb{M}^{(\ell,m)}$
and
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})\in\mathbb{M}^{(\ell^{\prime},m^{\prime})}$
are marked measured metric spaces, we may view
$G(\mathcal{X},\mathcal{Y};\gamma,\xi)$ as an element of
$\mathbb{M}^{(\ell+\ell^{\prime},m+m^{\prime})}$ by assigning marks and
measures

$(\mathbf{p}(A^{1}),\ldots,\mathbf{p}(A^{\ell}),\mathbf{p}(B^{1}),\ldots,\mathbf{p}(B^{\ell^{\prime}}))\quad\text{
and
}\quad(\mathbf{p}_{*}\mu_{\mathcal{X}}^{1},\ldots,\mathbf{p}_{*}\mu_{\mathcal{X}}^{m},\mathbf{p}_{*}\mu_{\mathcal{Y}}^{1},\ldots,\mathbf{p}_{*}\mu_{\mathcal{Y}}^{m^{\prime}})\,,$

where $\mathbf{p}:\mathcal{X}\sqcup\mathcal{Y}\to
G(\mathcal{X},\mathcal{Y};\gamma,\xi)$ is the canonical projection. With a
slight abuse of notation, we will keep denoting these by
$\mathbf{A}\mathbf{B}$ and
${\boldsymbol{\mu}}_{\mathcal{X}}{\boldsymbol{\mu}}_{\mathcal{Y}}$. Observe
that $\gamma$ and $\xi$ may themselves be part of the marking, in which case
they induce the same marks $\mathbf{p}(\gamma)=\mathbf{p}(\xi)$ in the glued
space. Observe also that geodesic marks in $\mathbf{A}$ or in $\mathbf{B}$
remain geodesic marks in $\mathbf{A}\mathbf{B}$, due to the fact that, by
definition, $(\mathcal{X},d_{\mathcal{X}})$ and
$(\mathcal{Y},d_{\mathcal{Y}})$ are isometrically embedded in
$G(\mathcal{X},\mathcal{Y};\gamma,\xi)$.

Finally observe that the gluing of the point space as an element of
$\mathbb{M}^{(\ell,m)}$ with
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})$
along $\xi$ only has the effect of prepending $\ell$ times $\xi(0)$ to
$\mathbf{B}$ and $m$ times the zero measure to
${\boldsymbol{\mu}}_{\mathcal{Y}}$.

##### Gluing two geodesics in the same space.

A similar gluing procedure777In fact, since we are allowing points at infinite
distance, the gluing $G(\mathcal{X},\mathcal{Y};\gamma,\xi)$ could be seen as
a particular case of gluing of a single space along two geodesics, but we
refrain to do so as we will mostly be interested in gluing true metric spaces.
can be defined for two geodesics $\gamma$, $\xi$ in the same pseudometric
space $(\mathcal{X},d_{\mathcal{X}})$. We again set
$\mathpzc{l}=\operatorname{length}(\gamma)\wedge\operatorname{length}(\xi)$
and then define

$G(\mathcal{X};\gamma,\xi)=(\mathcal{X},d_{\mathcal{X}})/R\,,$ (15)

where $R$ is the coarsest equivalence relation satisfying
$\gamma(t)\mathbin{R}\xi(t)$ for every $t\in[0,\mathpzc{l}]$. The quotient
pseudometric $d_{G}(x,y)$ may be condensed into

$d_{\mathcal{X}}(x,y)\wedge\inf_{t\in[0,\mathpzc{l}]}\big{\\{}d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{X}}(\xi(t),y)\big{\\}}\wedge\inf_{t\in[0,\mathpzc{l}]}\big{\\{}d_{\mathcal{X}}(x,\xi(t))+d_{\mathcal{X}}(\gamma(t),y)\big{\\}}\,.$
(16)

Similarly to the above, the space $G(\mathcal{X};\gamma,\xi)$ naturally
inherits the marking $\mathbf{A}$ and measures
${\boldsymbol{\mu}}_{\mathcal{X}}$ that $\mathcal{X}$ may be endowed with,
simply by pushing those forward by the canonical projection $\mathcal{X}\to
G(\mathcal{X};\gamma,\xi)$; by a slight abuse of notation, we keep the piece
of notation $\mathbf{A}$, ${\boldsymbol{\mu}}_{\mathcal{X}}$ for these
inherited objects. Note however that it is _not true in general_ that geodesic
marks in $(\mathcal{X},d_{\mathcal{X}})$ remain geodesic marks in
$G(\mathcal{X};\gamma,\xi)$. Let us state a useful comparison result between
$d_{\mathcal{X}}$ and $d_{G}$.

######  Lemma 21.

Let $(\mathcal{X},d_{\mathcal{X}})$ be a pseudometric space with two
distinguished geodesics $\gamma$, $\xi$. Denote by $d_{G}$ the pseudometric on
$G(\mathcal{X};\gamma,\xi)$ as in (16).

  1. (i)

For every $x$, $y\in\mathcal{X}$,

$d_{G}(x,y)\leq d_{\mathcal{X}}(x,y)\leq d_{G}(x,y)+R(\gamma,\xi)\,,$

where $R(\gamma,\xi)$ is the Hausdorff distance in the space
$(\mathcal{X},d_{\mathcal{X}})$ between the initial segments of $\gamma$,
$\xi$ that are glued together, i.e., of length $\mathpzc{l}$.

  2. (ii)

For every $\varepsilon>0$ and $x$, $y\in\mathcal{X}$, if
$d_{\mathcal{X}}(x,y)<\varepsilon$ and $d_{\mathcal{X}}(x,\gamma)\wedge
d_{\mathcal{X}}(y,\gamma)>\varepsilon$ (or if
$d_{\mathcal{X}}(x,\xi)\wedge\mathrm{d}_{\mathcal{X}}(y,\xi)>\varepsilon$), it
holds that

$d_{G}(x,y)=d_{\mathcal{X}}(x,y)\,.$

###### Proof.

Let us first prove (i). The first inequality is a direct consequence of (16).
To prove the other bound, simply observe that for every $t\in[0,\mathpzc{l}]$
we have

$\displaystyle d_{\mathcal{X}}(x,y)$ $\displaystyle\leq
d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{X}}(\gamma(t),\xi(t))+d_{\mathcal{X}}(\xi(t),y)$
$\displaystyle\leq
d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{X}}(\xi(t),y)+R(\gamma,\xi)\,.$

Taking the infimum over $t$, and then applying the same reasoning with the
roles of $\gamma$, $\xi$ interchanged, we obtain the result by (16).

The proof of (ii) is even more straightforward. Under our assumptions, it
holds that both $d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{X}}(y,\xi(t))$ and
$d_{\mathcal{X}}(x,\xi(t))+d_{\mathcal{X}}(y,\gamma(t))$ are greater than
$\varepsilon>d_{\mathcal{X}}(x,y)$, for every choice of $t$, so that
$d_{G}(x,y)$ must be equal to $d_{\mathcal{X}}(x,y)$. ∎

##### Gluing and GHP limits.

From now on, we mostly focus on compact geodesic spaces. The gluing of one or
two geodesic spaces along geodesics is again a geodesic space by general
results presented in [BBI01]. The gluing operation also preserves the
compactness of the spaces that are glued together. Furthermore, if a compact
metric space is the Gromov–Hausdorff limit of a sequence of compact geodesic
spaces, then it is also a compact geodesic space [BBI01, Theorem 7.5.1].

The next result shows that the gluing operations behave well with respect to
the GHP metric. For simplicity, we state it with the first marks of the
markings but it obviously holds up to index permutations, using Lemma 18 for
instance.

######  Proposition 22.

Let
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$,
$(\mathcal{Y}_{n},d_{\mathcal{Y}_{n}},\mathbf{B}_{n},{\boldsymbol{\mu}}_{\mathcal{Y}_{n}})$,
$n\geq 0$, be geodesic marked measured metric spaces that converge in the
marked GHP topology to
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$,
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B},{\boldsymbol{\mu}}_{\mathcal{Y}})$.
Assume that the first pairs of marks $(A_{n}^{1},A_{n}^{2})=\gamma_{n}$ and
$(B_{n}^{1},B_{n}^{2})=\xi_{n}$ of $\mathcal{X}_{n}$ and of $\mathcal{Y}_{n}$
are geodesic marks for every $n\geq 0$. Then the first two marks of
$\mathbf{A}$ and of $\mathbf{B}$ are geodesic marks $\gamma$, $\xi$, and

$G(\mathcal{X}_{n},\mathcal{Y}_{n};\gamma_{n},\xi_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}G(\mathcal{X},\mathcal{Y};\gamma,\xi)$

in the GHP topology.

Similarly, if we now assume that the first four marks are such that
$(A_{n}^{1},A_{n}^{2})=\gamma_{n}$ and $(A_{n}^{3},A_{n}^{4})=\xi_{n}$ are
geodesic marks, then the same holds for the first marks $\gamma$, $\xi$ of
$\mathbf{A}$, and

$G(\mathcal{X}_{n};\gamma_{n},\xi_{n})\mathrel{\mathop{\kern
0.0pt\longrightarrow}\limits_{n\to\infty}}G(\mathcal{X};\gamma,\xi)$

in the GHP topology.

In order to prove this proposition, we are first going to state and prove a
useful lemma that allows one to deal only with the situations where the
geodesics along which the spaces of interest are glued have the same lengths.
While Lemma 18 showed that the operation of merging two marks is continuous on
$(\mathbb{M}^{(\ell,m)},\mathrm{d}_{\mathrm{GHP}}^{(\ell,m)})$, this lemma
states that in the case of geodesic marks, the natural splitting operation is
continuous. If $\gamma$ is a geodesic mark and
$r\in[0,\operatorname{length}(\gamma)]$, the _splitting_ of $\gamma$ at level
$r$ is the two geodesic marks $(\gamma(0),\\{\gamma(t):0\leq t\leq r\\})$,
$(\gamma(r),\\{\gamma(t),r\leq t\leq\operatorname{length}(\gamma)\\})$.

######  Lemma 23.

Let
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n},{\boldsymbol{\mu}}_{\mathcal{X}_{n}})$,
$n\geq 0$, be a sequence of geodesic marked measured metric spaces converging
in the GHP topology toward a geodesic marked measured metric space
$(\mathcal{X},d_{\mathcal{X}},\mathbf{A},{\boldsymbol{\mu}}_{\mathcal{X}})$,
and assume that, say, the first pairs of marks, are geodesic marks
$\gamma_{n}$ and $\gamma$. Denote by
$\mathpzc{l}_{n}=\operatorname{length}(\gamma_{n})$ and
$\mathpzc{l}=\operatorname{length}(\gamma)$ their lengths. Let
$r_{n}\in(0,\mathpzc{l}_{n})$ be real numbers such that $r_{n}\to
r\in(0,\mathpzc{l})$. Then the convergence $\mathcal{X}_{n}\to\mathcal{X}$
still holds in the GHP topology after replacing the marks $\gamma_{n}$ and
$\gamma$ in $\mathbf{A}_{n}$ and $\mathbf{A}$, with their splittings
$\gamma_{n}^{\prime}$, $\gamma_{n}^{\prime\prime}$ and $\gamma^{\prime}$,
$\gamma^{\prime\prime}$, respectively at levels $r_{n}$ and $r$.

###### Proof.

Since the desired property does not involve the measures, it suffices to
establish it in the space of marked, nonmeasured spaces. Let $\mathcal{R}$ be
a correspondence between $\mathcal{X}_{n}$ and $\mathcal{X}$ compatible with
the markings. We fix $\varepsilon>\operatorname{dis}(\mathcal{R})$ and
consider the enlarged correspondence

$\mathcal{R}^{\varepsilon}=\big{\\{}(x,y)\in\mathcal{X}_{n}\times\mathcal{X}:\exists(x^{\prime},y^{\prime})\in\mathcal{R},\,d_{\mathcal{X}_{n}}(x,x^{\prime})\vee\mathrm{d}_{\mathcal{X}}(y,y^{\prime})<\varepsilon\big{\\}}\,.$

By the triangle inequality, the distortion of $\mathcal{R}^{\varepsilon}$ is
at most $\operatorname{dis}(\mathcal{R})+4\varepsilon$. Moreover, we claim
that for $s\in[0,\mathpzc{l}_{n}]$ and $t\in[0,\mathpzc{l}]$ such that
$|t-s|<\varepsilon-\operatorname{dis}(\mathcal{R})$, it holds that
$\gamma_{n}(s)\mathbin{\mathcal{R}}^{\varepsilon}\gamma(t)$. Indeed, since
$\mathcal{R}$ is compatible with the markings, for every $s$, $t$ as above,
there exists $u$ such that $\gamma_{n}(s)\mathbin{\mathcal{R}}\gamma(u)$, so

$|s-u|=\big{|}d_{\mathcal{X}_{n}}(\gamma_{n}(s),\gamma_{n}(0))-d_{\mathcal{X}}(\gamma(u),\gamma(0))\big{|}\leq\operatorname{dis}\mathcal{R}\,,$

and therefore
$|d_{\mathcal{X}}(\gamma(t),\gamma(u))|=|t-u|\leq|t-s|+|s-u|<\varepsilon$, as
wanted. From this, we conclude that $\mathcal{R}^{\varepsilon}$ is compatible
with the geodesic marks $\gamma_{n}^{\prime}$, $\gamma^{\prime}$ and
$\gamma_{n}^{\prime\prime}$, $\gamma^{\prime\prime}$, as soon as
$\varepsilon>|r_{n}-r|\vee|\mathpzc{l}_{n}-\mathpzc{l}|$. Choosing a sequence
of correspondences $\mathcal{R}_{n}$ with vanishing distortion and taking
$\varepsilon_{n}=(\operatorname{dis}(\mathcal{R}_{n})\vee|r_{n}-r|\vee|\mathpzc{l}_{n}-\mathpzc{l}|)+1/n$
yields the result. ∎

###### Proof of Proposition 22.

With the help of Lemma 23, we may and will assume that $\gamma_{n}$, $\xi_{n}$
have same length for every $n$. The fact that the limiting marks are geodesics
marks with same length comes from Proposition 19. We will first establish the
result for marked, nonmeasured spaces, from which we will deduce the full
result thanks to Lemma 15.

Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be two correspondences
respectively between $\mathcal{X}_{n}$ and $\mathcal{X}$ and between
$\mathcal{Y}_{n}$ and $\mathcal{Y}$, compatible with the considered markings.
Identifying $\mathcal{X}_{n}$ and $\mathcal{Y}_{n}$ (resp. $\mathcal{X}$ and
$\mathcal{Y}$) with their canonical embeddings into
$G(\mathcal{X}_{n},\mathcal{Y}_{n};\gamma_{n},\xi_{n})$ (resp. into
$G(\mathcal{X},\mathcal{Y};\gamma,\xi)$), we see
$\mathcal{R}^{\prime\prime}=\mathcal{R}\cup\mathcal{R}^{\prime}$ as a
correspondence between $G(\mathcal{X}_{n},\mathcal{Y}_{n};\gamma_{n},\xi_{n})$
and $G(\mathcal{X},\mathcal{Y};\gamma,\xi)$, obviously compatible with the
other marks. In order to bound its distortion, let us take
$x_{n}\mathbin{\mathcal{R}}x$ and $y_{n}\mathbin{\mathcal{R}}^{\prime}y$.

We let
$\mathpzc{l}_{n}=\operatorname{length}(\gamma_{n})=\operatorname{length}(\xi_{n})$
and $\mathpzc{l}=\operatorname{length}(\gamma)=\operatorname{length}(\xi)$ be
the lengths of the considered geodesics and we denote by $d_{n}$ and $d$ the
metrics in the previous gluings. From (14) and by compactness, there exists
$t\in[0,\mathpzc{l}]$ such that

$d(x,y)=d_{\mathcal{X}}(x,\gamma(t))+d_{\mathcal{Y}}(\xi(t),y).$

Then there exist $t_{n}$, $t^{\prime}_{n}\in[0,\mathpzc{l}_{n}]$ such that
$\gamma_{n}(t_{n})\mathbin{\mathcal{R}}\gamma(t)$ and
$\xi_{n}(t^{\prime}_{n})\mathbin{\mathcal{R}}^{\prime}\xi(t)$. As a result,

$\displaystyle d_{n}(x_{n},y_{n})$ $\displaystyle\leq
d_{\mathcal{X}_{n}}\big{(}x_{n},\gamma_{n}(t_{n})\big{)}+d_{n}\big{(}\gamma_{n}(t_{n}),\xi_{n}(t^{\prime}_{n})\big{)}+d_{\mathcal{Y}_{n}}\big{(}\xi_{n}(t^{\prime}_{n}),y_{n}\big{)}$
$\displaystyle\leq
d_{\mathcal{X}}(x,\gamma(t))+\operatorname{dis}(\mathcal{R})+d_{n}\big{(}\gamma_{n}(t_{n}),\xi_{n}(t^{\prime}_{n})\big{)}+d_{\mathcal{Y}}(\xi(t),y)+\operatorname{dis}(\mathcal{R}^{\prime})$
$\displaystyle\leq
d(x,y)+\operatorname{dis}(\mathcal{R})+\operatorname{dis}(\mathcal{R}^{\prime})+d_{n}\big{(}\gamma_{n}(t_{n}),\xi_{n}(t^{\prime}_{n})\big{)}\,.$

Using the facts that $\gamma_{n}$, $\xi_{n}$, $\gamma$, $\xi$ are geodesics
and $\gamma_{n}(0)\mathbin{\mathcal{R}}\gamma(0)$,
$\xi_{n}(0)\mathbin{\mathcal{R}}^{\prime}\xi(0)$, we easily obtain

$\displaystyle d_{n}\big{(}\gamma_{n}(t_{n}),\xi_{n}(t^{\prime}_{n})\big{)}$
$\displaystyle=\big{|}d_{\mathcal{X}_{n}}\big{(}\gamma_{n}(0),\gamma_{n}(t_{n})\big{)}-d_{\mathcal{Y}_{n}}\big{(}\xi_{n}(0),\xi_{n}(t^{\prime}_{n})\big{)}\big{|}$
$\displaystyle\leq|d_{\mathcal{X}}(\gamma(0),\gamma(t))-d_{\mathcal{Y}}(\xi(0),\xi(t))|+\operatorname{dis}(\mathcal{R})+\operatorname{dis}(\mathcal{R}^{\prime})$
$\displaystyle=\operatorname{dis}(\mathcal{R})+\operatorname{dis}(\mathcal{R}^{\prime}).$

Using a symmetric argument, we see that $|d_{n}(x_{n},y_{n})-d(x,y)|\leq
2\,(\operatorname{dis}(\mathcal{R})+\operatorname{dis}(\mathcal{R}^{\prime}))$.
Adding to this the simpler cases where the pairs of points we compare belong
both to $\mathcal{R}$ or both to $\mathcal{R}^{\prime}$, we obtain

$\operatorname{dis}(\mathcal{R}^{\prime\prime})\leq
2\,(\operatorname{dis}(\mathcal{R})+\operatorname{dis}(\mathcal{R}^{\prime}))$

and the first statement easily follows for the GH topology (without the
measures).

Let us show that the result still holds when considering the measures. We
assume for simplicity that the terms of ${\boldsymbol{\mu}}_{\mathcal{X}}$,
${\boldsymbol{\mu}}_{\mathcal{Y}}$ are all nonzero, since the case of a
vanishing measure, say $\mu^{j}_{\mathcal{X}}$, is equivalent to the fact that
$\mu^{j}_{\mathcal{X}_{n}}(\mathcal{X}_{n})\to 0$. Denote by $m$, $m^{\prime}$
the numbers of coordinates of ${\boldsymbol{\mu}}_{\mathcal{X}}$,
${\boldsymbol{\mu}}_{\mathcal{Y}}$ and sample $r(m+m^{\prime})$ independent
points ${\boldsymbol{x}}=(x^{j}_{i},1\leq i\leq r,1\leq j\leq m)$ and
${\boldsymbol{y}}=(y_{i}^{j},1\leq i\leq r,1\leq j\leq m^{\prime})$ where
$x_{i}^{j}$ has law $\bar{\mu}^{j}_{\mathcal{X}}$ and $y_{i}^{j}$ has law
$\bar{\mu}^{j}_{\mathcal{Y}}$. We identify these points with their images in
the glued space by the canonical projection $\mathcal{X}\sqcup\mathcal{Y}\to
G(\mathcal{X},\mathcal{Y};\gamma,\xi)$. We assume that
$\mu_{\mathcal{X}_{n}}(\mathcal{X}_{n})>0$ and
$\mu_{\mathcal{Y}_{n}}(\mathcal{Y}_{n})>0$, which hold for $n$ sufficiently
large since, as $n\to\infty$,
$\mu_{\mathcal{X}_{n}}(\mathcal{X}_{n})\to\mu_{\mathcal{X}}(\mathcal{X})>0$
and
$\mu_{\mathcal{Y}_{n}}(\mathcal{Y}_{n})\to\mu_{\mathcal{Y}}(\mathcal{Y})>0$.
We proceed similarly to sample $r(m+m^{\prime})$ random points
${\boldsymbol{x}}_{n}=(x_{n,i}^{j})$, ${\boldsymbol{y}}_{n}=(y_{n,i}^{j})$ in
$G(\mathcal{X}_{n},\mathcal{Y}_{n};\gamma_{n},\xi_{n})$ with laws
$\bar{\mu}^{j}_{\mathcal{X}_{n}}$ and $\bar{\mu}^{j}_{\mathcal{Y}_{n}}$ as
appropriate. Lemma 15 guarantees that the marked spaces
$(\mathcal{X}_{n},d_{\mathcal{X}_{n}},\mathbf{A}_{n}{\boldsymbol{x}}_{n})$ and
$(\mathcal{Y}_{n},d_{\mathcal{Y}_{n}},\mathbf{B}_{n}{\boldsymbol{y}}_{n})$
converge to $(\mathcal{X},d_{\mathcal{X}},\mathbf{A}{\boldsymbol{x}})$ and
$(\mathcal{Y},d_{\mathcal{Y}},\mathbf{B}{\boldsymbol{y}})$ in distribution in
the GH topology. Applying the result of Proposition 22 proved above in the
case without measures, we obtain the convergence in distribution of the glued
space $G(\mathcal{X}_{n},\mathcal{Y}_{n};\gamma_{n},\xi_{n})$ with markings
$\mathbf{A}_{n}\mathbf{B}_{n}{\boldsymbol{x}}_{n}{\boldsymbol{y}}_{n}$ to
$G(\mathcal{X},\mathcal{Y};\gamma,\xi)$ with the marking
$\mathbf{A}\mathbf{B}{\boldsymbol{x}}{\boldsymbol{y}}$. Since
${\boldsymbol{x}}_{n}$, ${\boldsymbol{y}}_{n}$ and ${\boldsymbol{x}}$,
${\boldsymbol{y}}$ are also independent samples from the renormalized measures
$\bar{\boldsymbol{\mu}}_{\mathcal{X}_{n}}$,
$\bar{\boldsymbol{\mu}}_{\mathcal{Y}_{n}}$ and
$\bar{\boldsymbol{\mu}}_{\mathcal{X}}$, $\bar{\boldsymbol{\mu}}_{\mathcal{Y}}$
viewed as measures on the glued spaces, an application of the converse
implication of Lemma 15 implies the result.

The second part of the statement dealing with metric spaces that are glued