$\mathrm{d}^{\star}(f,B_{r}^{\bullet}(\mathfrak{q}_{\infty}^{(1)}))\leqslant\alpha_{0}(s-r)$
(39)

holds for every $0\leqslant r<s\leqslant R$ with $s-r\geqslant\delta R$, and
for every downward triangle $f$ at height $s$. Then, there exists another
positive constant $\tilde{\beta}$ such that for any sufficiently large $R$,

$\mathbb{P}(A_{R}(\delta))\geqslant 1-e^{-\tilde{\beta}R}.$ (40)

###  6.2 Upper bound in finite simple triangulations

We finally turn our attention to finite triangulations. The purpose of this
section is to obtain a result equivalent to [13, Lemma 28] but for uniform
simple triangulations rather than uniform general triangulations and for the
$\mathrm{d}^{\star}$ distance rather than the graph distance or the Eden
model.

For $x\in V(\mathrm{t}_{n})$ and $f\in F(\mathrm{t}_{n})$, write
$x\triangleleft f$ to mean that the vertex $x$ is incident to the face $f$.
Then we have:

######  Proposition 31.

Let $\alpha_{0}$ be as in Lemma 30. Let $\mathfrak{t}_{n}$ be a uniform rooted
simple triangulation with $n+2$ vertices. For $\varepsilon\in(0,1/4)$, define
$E_{n}$ as the event where the bound

$\mathrm{d}^{\star}(f,g)\leqslant\alpha_{0}\mathrm{d}_{\mathrm{gr}}(x,y)+n^{\varepsilon}$
(41)

holds for every $x,y\in V(\mathrm{t}_{n})$ and $f,g\in F(\mathrm{t}_{n})$ such
that $x\triangleleft f$ and $y\triangleleft g$. Then:

$\mathbb{P}(E_{n})\xrightarrow[n\rightarrow\infty]{}1.$ (42)

The same result holds when replacing $\mathfrak{t}_{n}$ by
$\mathfrak{q}_{n}^{(1)}$, where $\mathfrak{q}_{n}^{(1)}$ is a uniform quasi-
simple triangulation of the 1-gon with $n$ inner vertices.

###### Proof.

The main difficulty that arises in our setting and does not exist in [13] is
that we have to derive results for simple triangulations, whereas we
established (in particular in the previous section) results about the local
limit of _quasi-simple_ triangulations. Fortunately, these two models are
similar enough to obtain the results we need.

In this proof, we always consider that the random planar maps are endowed with
random weights on their edges, sampled independently under a law $\nu$ with
support included in $[\eta_{0},\infty)$ for some $\eta_{0}>0$, and with
exponential tails. Let $\mathfrak{q}_{n}^{(1)}$ be a uniform quasi-simple
triangulation of the $1$-gon with $n$ inner vertices. Following the proof of
[13, Lemma 28], we write $\rho_{n}$ for its root vertex, $o_{n}$ for its
pointed vertex and set $d_{n}:=\mathrm{d}_{\mathrm{gr}}(\rho_{n},o_{n})$.
Then, we write $\Theta$ for the set of quasi-simple triangulations
$\mathrm{t}$ that belong to $\mathcal{Z}_{1,r}$ for some $r>1$ and are such
that there exists a face $f$ incident to $\partial^{*}\mathrm{t}$ whose
$\mathrm{d}^{\star}$ from the bottom face is greater than $\alpha r$. By the
exact same chain of arguments as in the proof of [13, Lemma 28] that leads to
their equation (73), we obtain that there exist two constants $c>0$ and $a>0$,
such that:

$\mathbb{P}\left(d_{n}>n^{\epsilon}\,;\,B_{d_{n}-1}^{\bullet}(\mathfrak{q}_{n}^{(1)})\in\Theta\right)\leqslant
c\exp(-an^{\epsilon}).$ (43)

By, again, the same line of arguments as in [13] and given the definition of
$\Theta$, it is then easy to see that on the event
$\\{d_{n}>n^{\epsilon};B_{d_{n}-1}^{\bullet}(\mathfrak{q}_{n}^{(1)})\notin\Theta\\}$,
we have $\mathrm{d}^{\star}(f,g)\leqslant\alpha
d_{n}+3w^{*}\cdot\mathrm{mD}(\mathfrak{q}_{n}^{(1)})$, for any
$f,g\in\mathfrak{q}_{n}^{(1)}$, with $\rho_{n}\triangleleft f$ and
$o_{n}\triangleleft g$ (where we denote by $\mathrm{mD}$ the maximum vertex
degree of a map, and by $w^{*}$ the maximal dual weight of an edge of
$\mathfrak{t}_{n}$).

The original proof continues with a rerooting argument, which cannot apply
here since quasi-simple triangulation are not invariant under rerooting.
However, we can transfer the bound obtained in (43) from quasi-simple
triangulations to simple triangulations (up to changing the value of the
constant $c$) in the following way. The asymptotic enumerative results for
simple triangulations and quasi-simple triangulations of the $1$-gon given
respectively in (24) and (29) imply that there exists $\kappa>0$, such that:

$\mathbb{P}\left(\Psi(\mathfrak{q}_{n}^{(1)})\in\mathcal{T}^{\scalebox{0.4}{$\blacksquare$}}_{n-1}\right)>\kappa,\quad\text{for
any }n\geqslant 1,$ (44)

where we recall that $\mathcal{T}^{\scalebox{0.4}{$\blacksquare$}}_{n}$
denotes the set of rooted simple triangulations with $n+2$ vertices and a
marked inner vertex.

Moreover, conditionally on the fact that
$\Psi(\mathfrak{q}_{n}^{(1)})\in\mathcal{T}^{\scalebox{0.4}{$\blacksquare$}}_{n-1}$,
it is clear from the construction that $\mathfrak{q}_{n}^{(1)}$ has the same
distribution as a uniformly random element of
$\mathcal{T}^{\scalebox{0.4}{$\blacksquare$}}_{n-1}$ (as a random measured
metric space) 777Note that their dual are not exactly isometric due to the
root-edge being doubled, but distances in the dual of both models differ at
most by 1..

Let $\mathfrak{t}_{n}$ be uniformly random in
$\mathcal{T}^{\scalebox{0.4}{$\blacksquare$}}_{n}$. Write $\tilde{\rho}_{n}$
for its root vertex, $\tilde{o}_{n}$ for its pointed vertex and
$\tilde{d}_{n}$ for
$\mathrm{d}_{\mathrm{gr}}(\tilde{\rho}_{n},\tilde{o}_{n})$. It follows from
the discussion above and from the discussion following (43) that there exist
two constants $\tilde{c}>0$ and $\tilde{a}>0$ such that:

$\mathbb{P}\left(\tilde{d}_{n}>n^{\epsilon}\,;\,\mathrm{d}_{\mathrm{gr}}^{\dagger}(f,g)>\alpha_{0}\tilde{d}_{n}+3w^{\star}\cdot\mathrm{mD}(\mathfrak{t}_{n})\text{
whenever }\tilde{\rho}_{n}\triangleleft f\text{ and
}\tilde{o}_{n}\triangleleft
g\right)\leqslant\tilde{c}\exp(-\tilde{a}n^{\epsilon}).$ (45)

Now, by rerooting $\mathfrak{t}_{n}$ at another oriented edge $e_{n}$ chosen
uniformly and independently from $o_{n}$, we obtain a pointed and rooted
simple triangulation with the same distribution. Hence, the bound obtained
above remains valid upon replacing $\tilde{\rho}_{n}$ by $\rho^{\prime}_{n}$,
where $\rho^{\prime}_{n}$ is the initial vertex of $e_{n}$ and $\tilde{d}_{n}$
by $\mathrm{d}_{\mathrm{gr}}(\tilde{o}_{n},\rho^{\prime}_{n})$.

We can then proceed as in the original proof to establish that:

$\mathbb{E}\left[\sum_{u,v\in
V(\mathfrak{t}_{n})}\mathbf{1}_{\\{\mathrm{d}_{\mathrm{gr}}(u,v)>n^{\epsilon}\\}}\mathbf{1}_{\\{\mathrm{d}_{\mathrm{gr}}^{\dagger}(f,g)>\alpha_{0}\mathrm{d}_{\mathrm{gr}}(u,v)+3w^{\star}\cdot\mathrm{mD}(\mathfrak{t}_{n})\text{
whenever }u\triangleleft f\text{ and }v\triangleleft g\\}}\right]\leqslant
6(n+2)^{2}\tilde{c}\exp(-\tilde{a}n^{\epsilon}).$ (46)

Hence, asymptotically almost surely, the following bound holds:

$\mathrm{d}^{\star}(f,g)\leqslant\alpha_{0}\mathrm{d}_{\mathrm{gr}}(u,v)+3w^{\star}\cdot\mathrm{mD}(\mathfrak{t}_{n}),$
(47)

whenever $u,v\in V(\mathfrak{t}_{n})$,
$\mathrm{d}_{\mathrm{gr}}(u,v)>n^{\epsilon}$ and $u\triangleleft f$ and
$v\triangleleft g$.

Since $\nu$ is assumed to have exponential tails, $w^{\star}$ can be bounded
by $n^{\epsilon}$ outside a set of probability tending to 0 as
$n\rightarrow\infty$. Besides, the geometric bound obtained for the degree of
the root vertex in [4, Lemma 4.1] gives the existence of a constant $A>0$ for
which $\mathrm{mD}(\mathfrak{t}_{n})\leqslant A\log n$ with probability
tending to 1 as $n\rightarrow$. This gives the desired result, except for the
requirement that $\mathrm{d}_{\mathrm{gr}}(u,v)>n^{\epsilon}$. This case can
be dealt with by hand using the fact that $\mathrm{d}^{\star}(f,g)\leqslant
w^{\star}\cdot\mathrm{mD}(\mathfrak{t}_{n})(\mathrm{d}_{\mathrm{gr}}(u,v)+1)$,
which concludes the proof for simple triangulations.

To finish the proof for quasi-simple triangulations, we rely on Lemmas 20 and
21. Fix $\epsilon>0$ and recall that we write $T(\mathfrak{q})$ for the
largest simple component of a quasi-simple triangulation $\mathfrak{q}$. By
Lemma 20, there exists $K\in\mathbb{N}$ such that:

$\mathbb{P}\left(|T(\mathfrak{q}_{n}^{(1)})|\geqslant n-K\right)\geqslant
1-\epsilon.$ (48)

We fix $k\in[0,K]$ and $|T(\mathfrak{q}_{n}^{(1)})|=n-k$. Then by Lemma 21, we
can couple $T(\mathfrak{q}_{n}^{(1)})$ with a uniform simple triangulation
$\mathfrak{t}_{n-k}$ of size $n-k$ such that
$\mathbb{P}(T(\mathfrak{q}_{n}^{(1)})\neq\mathfrak{t}_{n-k})<\epsilon$. We
extend this coupling to the model with weights on the edges and make sure that
the weights also coincide when the triangulations are equal.

By applying the result of the proposition for $\mathfrak{t}_{n-k}$, we know
that outside a set of probability smaller than $\epsilon$, the bound (41)
holds. This gives the result when $u,v\in T(\mathfrak{q}_{n}^{(1)})$. For
$w\notin T(\mathfrak{q}_{n}^{(1)})$, we can use the easy fact that there
exists $u\in T(\mathfrak{q}_{n}^{(1)})$ with
$\mathrm{d}_{\mathrm{gr}}(u,w)\leqslant k$, which concludes the proof of the
proposition. ∎

###  6.3 Two-point function

Thanks to the bounds we obtained in the two previous sections, we are now
ready to prove the equivalent of Proposition 21 of [13], stated below as
Proposition 34. Roughly speaking, this proposition establishes that the
difference between the 2-point function for the graph distance and the
$\mathrm{d}^{\star}$ 2-point function is $o(n^{1/4})$.

The proof of Proposition 34 relies on the following similar result obtained
for the _infinite_ quasi-simple triangulation. Given the upper bounds
established in Lemma 30, we can apply verbatim the proof presented in
Proposition 19 (in the primal case) and in Proposition 29 (in the dual case)
of [13] (see the remark at the end of Section 5.6.2) to readily obtain:

######  Proposition 32.

Let $\epsilon>0$ and $\delta>0$. We can find $\eta\in(0,1/2)$ such that, for
every sufficiently large $n$, we have:

$\mathbb{P}\left((1-\epsilon)c_{\nu}\lfloor\eta
n\rfloor\leqslant\mathrm{d}^{\star}\left(f,B_{n-\lfloor\eta
n\rfloor}(\mathfrak{q}_{\infty}^{(1)})\right)\leqslant(1+\epsilon)c_{\nu}\lfloor\eta
n\rfloor,\,\forall
f\in\mathrm{F}_{n}(\mathfrak{q}_{\infty}^{(1)})\right)>1-\delta,$ (49)

where we recall that $\mathrm{F}_{n}(\mathfrak{q}_{\infty}^{(1)})$ is the set
of downward triangles incident to $\partial_{n}(\mathfrak{q}_{\infty}^{(1)})$.
Moreover

$\mathbb{P}\left((c_{\nu}-\epsilon)n\leqslant\mathrm{d}^{\star}\left(f,B_{0}(\mathfrak{q}_{\infty}^{(1)})\right)\leqslant(c_{\nu}+\epsilon)n,\,\forall
f\in\mathrm{F}_{n}(\mathfrak{q}_{\infty}^{(1)})\right)\xrightarrow[n\rightarrow\infty]{}1.$
(50)

To transfer this result to finite simple triangulations, we rely on the
following absolute continuity relation between finite and infinite quasi-
simple triangulations (analogous to [13, Lemma 22]). Recall that
$\mathcal{Z}_{1,r}$ denotes the set of quasi-simple triangulation of the
cylinder of height $r$ and rooted on a loop. For $\Delta\in\mathcal{Z}_{1,r}$,
we denote by $|\Delta|$ the number of inner vertices of $\Delta$. Recall that
$\mathfrak{q}_{n}^{(1)}$ denotes the random quasi-simple triangulation of the
1-gon with $n$ inner vertices. Then, we have:

######  Lemma 33.

There exists a constant $\bar{c}$, such that for every $n\geqslant 1$, for
every $r\geqslant 1$ and every $\Delta\in\mathcal{Z}_{1,r}$ such that
$n>|\Delta|$,

$\mathbb{P}\left(B_{r}^{\bullet}(\mathfrak{q}_{n}^{(1)})=\Delta\right)\leqslant\bar{c}\left(\frac{n}{n-|\Delta|}\right)^{3/2}\mathbb{P}\left(B_{r}^{\bullet}(\mathfrak{q}_{\infty}^{(1)})=\Delta\right).$
(51)

###### Proof.

The result follows directly from the uniform upper bound obtained in Lemma 19,
from a similar lower bound on $|\mathcal{Q}_{n,1}|$ that we deduce from (28),
and from the limit of (34) together with the fact that
$\mathfrak{q}_{\infty}^{(1)}$ is the local limit of $\mathfrak{q}_{n}^{(1)}$
as $n\rightarrow\infty$. ∎

We can now state the main result of this section:

######  Proposition 34.

Denote by $f_{*}$ the bottom face of $\mathfrak{q}_{n}^{(1)}$. Let $o_{n}$ be
a uniformly distributed inner vertex of $\mathfrak{q}_{n}^{(1)}$ and let
$f_{n}$ be a face incident to $o_{n}$ (fixed in some deterministic manner
given $o_{n}$). Then, for any $\epsilon>0$,

$\mathbb{P}\left(|\mathrm{d}^{\star}(f_{n},f_{*})-c_{\nu}\mathrm{d}_{\mathrm{gr}}(o_{n},\rho_{n})|>\epsilon
n^{1/4}\right)\xrightarrow[n\rightarrow\infty]{}0.$ (52)

The same result holds also if we replace $\mathfrak{q}_{n}^{(1)}$ by a random
rooted simple triangulation having $n+2$ vertices and a random marked vertex
$o_{n}$, and $f_{*}$ by its root face.

###### Proof.

Again, we follow closely the proof of [13, Proposition 21]. Only two
modifications must be made. First, if we follow verbatim their proof, then at
the very end, we need to adapt the upper bound they obtain. Indeed, at this
point they use the fact that the first-passage distance on the primal graph
(i.e on the triangulation) is upper-bounded by the graph distance on the
primal. In particular, they fix three close values
$\alpha_{j}<\beta_{j}<\gamma_{j}$ and restrict their attention to the event
$\\{\beta_{j}n^{1/4}\leqslant\mathrm{d}_{\mathrm{gr}}(\rho_{n},o_{n})\leqslant\gamma_{j}n^{1/4}\\}$.
Then, they can give a deterministic upper bound for the first-passage
percolation distance between $o_{n}$ and the hull of radius
$\lfloor\alpha_{j}n^{1/4}\rfloor$. This last step does not apply directly
anymore when working on the dual graph, but can be made valid if we replace
the deterministic bound by the probabilistic bound obtained in Proposition 31.

Secondly, write $\mathfrak{B}_{r}(\mathfrak{q}_{n}^{(1)},o_{n})$ for the ball
of radius $r$ centered at $o_{n}$ in $\mathfrak{q}_{n}^{(1)}$. We have to
establish the following analogous statement of [13, (62)], and prove that for
any $y<1$, there exists $b\in(0,1)$ such that:

$\liminf_{n\rightarrow\infty}\mathbb{P}\left(|\mathfrak{B}_{\lfloor(\beta-\alpha)n^{1/4}\rfloor}(\mathfrak{q}_{n}^{(1)},o_{n})|>bn\right)\geqslant
y.$ (53)

Given the convergence of simple triangulations to the Brownian sphere in the
Gromov–Hausdorff-Prokhorov topology [1], this statement is known to be true
for uniform simple triangulations. It can then be extended to quasi-simple
triangulations, upon extracting their largest simple component and by applying
Lemmas 20 and 21. This concludes the proof for quasi-simple triangulations.

To extend the result to simple triangulations, we rely on the same line of
arguments as in the proof of Proposition 31, and particularly on (44). It
states that a positive proportion of quasi-simple triangulations are in fact
simple triangulations (after applying the mapping $\Psi$, as depicted in
Figure 4) Moreover, when $\Psi(\mathfrak{q}^{(1)}_{n})$ is a simple
triangulation, it yields a direct coupling between a random inner vertex of
$\mathfrak{q}^{(1)}_{n}$ and of $\Psi(\mathfrak{q}^{(1)}_{n})$. Since we have
just shown that (52) holds for quasi-simple triangulations, and since $\Psi$
preserves the underlying metric space, it has to hold for simple
triangulations as well. ∎

###  6.4 Coupling between $\mu_{n}$ and $\mu_{n}^{\dagger}$ and proof of
Theorem 2

######  Theorem 35.

Let $\mathfrak{t}_{n}$ be uniformly random in $\mathcal{T}_{n}$. Consider the
two following weighted metric spaces
$\mathfrak{T}_{n}=(\mathfrak{t}_{n},\mathrm{d}_{\mathrm{gr}},\mu_{n})$ and
$\mathfrak{K}_{n}=(\mathfrak{t}^{\dagger}_{n},\mathrm{d}^{\star},\mu_{n}^{\dagger})$,
where $\mu_{n}$ and $\mu_{n}^{\dagger}$ denote respectively the uniform
measures on $V(\mathfrak{t}_{n})$ and on $V(\mathfrak{t}_{n}^{\dagger})$.

Then, there exists an explicit correspondence $C_{n}$ between
$(\mathfrak{t}_{n},\mathrm{d}_{\mathrm{gr}})$ and
$(\mathfrak{t}^{\dagger}_{n},\mathrm{d}^{\star})$ and an explicit coupling
$\pi_{n}$ between $\mu_{n}$ and $\mu_{n}^{\dagger}$, such that:

$\pi_{n}(C_{n})\geqslant 1-\frac{2}{n+2}\qquad\text{ and
}\qquad\operatorname{dis}(C_{n})\leqslant\epsilon n^{1/4}.$

Before giving the proof of this theorem, notice that in view of the known
convergence of simple triangulations towards the Brownian sphere established
in [1], it concludes the proof of Theorem 2.

###### Proof.

The correspondence $C_{n}$ is defined as the set of pairs:

$C_{n}:=\\{(v,f),\text{ with }v\in V(\mathfrak{t}_{n}),\,f\in
F(\mathfrak{t}_{n})\text{ and }v\triangleleft f\\}.$

Next, the coupling we construct between $\mu_{n}$ and $\mu_{n}^{\dagger}$ is
based on the theory of Schnyder woods for simple triangulations [35], that we
now briefly recall. A simple triangulation $\mathfrak{t}$ can be canonically
endowed by an orientation of its edges, such that for each $v\in
V(\mathfrak{t})$, writing $\mathrm{out}(v)$ for the outdegree of $v$, we have:

$\mathrm{out}(v)=\begin{cases}3&\text{if }v\text{ is not incident to the root
face of }\mathfrak{t}\\\ 1&\text{otherwise}.\end{cases}$ (54)

Note that to ensure that this orientation is canonical, it is enough to
require that directed counterclockwise cycles are not allowed, but this is not
relevant for our purpose. However, this canonical orientation enables to
construct an explicit coupling between $\mu_{n}$ and $\mu_{n}^{\dagger}$.

Letting $o_{n}$ be a random vertex of $\mathfrak{t}_{n}$, we associate to
$o_{n}$ a uniform face of $\mathfrak{t}_{n}$ in the following way. If $o_{n}$
is not incident to the root face of $\mathfrak{t}_{n}$, then let $e_{n}$ be an
edge chosen uniformly among one of the three outgoing edges from $o_{n}$. With
probability 1/2, we pick $f_{n}$ to be the edge on the left of $e_{n}$, and
otherwise $f_{n}$ is the edge on the right of $e_{n}$.

Now, assume $o_{n}$ is incident to the root face of $\mathfrak{t}_{n}$. Then,
with probability 1/3, we let $e_{n}$ to be the unique outgoing edge from
$o_{n}$, and with probability 2/3, we let $e_{n}$ to be a uniform edge of
$\mathfrak{t}_{n}$. In both cases, we proceed as above.

It is clear from the construction, that $e_{n}$ is a uniform edge of
$\mathfrak{t}_{n}$. Since each face has degree $3$, $f_{n}$ is then a uniform
face of $\mathfrak{t}_{n}$. Moreover, apart on an event of probability at most
$2/(n+2)$ (corresponding to the case where $o_{n}$ is incident to the root
face and $e_{n}$ is sampled uniformly in $E(\mathfrak{t}_{n})$), we have
$o_{n}\triangleleft f_{n}$.

To conclude the proof of the theorem, it is then enough to prove that
$\operatorname{dis}(C_{n})\leqslant\epsilon n^{1/4}$, or in other words that:

$\mathbb{P}\left(\sup_{\begin{subarray}{c}x,y\in V(\mathfrak{t}_{n}),f,g\in
F(\mathfrak{t}_{n})\\\ x\triangleleft f,y\triangleleft
g\end{subarray}}|\mathrm{d}^{\star}(f,g)-c_{\nu}\mathrm{d}_{\mathrm{gr}}(x,y)|>\epsilon
n^{1/4}\right)\xrightarrow[n\rightarrow\infty]{}0.$ (55)

To prove this last statement, we adapt only slightly the framework of [13,
Theorem 1]. The general strategy of the proof is the following. We can extend
the result for the 2-point function stated in Proposition 34 to control the
pairwise distance between any finite number of random points. Then, using the
known convergence of simple triangulations to the Brownian sphere, and the
fact that the Brownian sphere is compact, we can approximate the metric
structure of a large simple random triangulation by restricting our attention
to a (large) fixed number of uniform points. We now give the details of the
proof.

Let $\mathfrak{t}_{n}$ be a uniform element of $\mathcal{T}_{n}$. Let
$\Phi_{n}:V(\mathfrak{t}_{n})\rightarrow F(\mathfrak{t}_{n})$ be a
deterministic function (given $\mathfrak{t}_{n}$) which associates to each
vertex $v$ of $\mathfrak{t}_{n}$ a face $f$ with $v\triangleleft\Phi_{n}(v)$.
The conclusion of Proposition 34 can be rewritten as: let $o_{n}$ be a uniform
vertex of $\mathfrak{t}_{n}$, then for any $\epsilon>0$

$\mathbb{P}\left(|\mathrm{d}^{\star}(\Phi_{n}(o_{n}),f_{*})-c_{\nu}\mathrm{d}_{\mathrm{gr}}(o_{n},\rho_{n})|>\epsilon
n^{1/4}\right)\xrightarrow[n\rightarrow\infty]{}0,$ (56)

where $f_{*}$ and $\rho_{n}$ denote respectively the root face and the root
vertex of $\mathfrak{t}_{n}$. Let $C(\mathfrak{t}_{n})$ be the set of corners
of $\mathfrak{t}_{n}$. For $\kappa\in C(\mathfrak{t}_{n})$, the incident
vertex (resp. face) is denoted $\mathrm{v}(\kappa)$ (resp.
$\mathrm{f}(\kappa)$). Let
$\mathfrak{t}_{n}^{\bullet}=(\mathfrak{t}_{n},o_{n})$. Conditionally given
$\mathfrak{t}_{n}^{\bullet}$, let $\kappa_{n}$ be a corner of
$\mathfrak{t}_{n}$ chosen uniformly at random. The law of
$\mathfrak{t}_{n}^{\bullet}$ is clearly invariant by rerooting
$\mathfrak{t}_{n}^{\bullet}$ at $\kappa_{n}$ (while keeping $o_{n}$ marked).
Hence (56) remains valid if we replace $f_{*}$ and $\rho_{n}$ respectively by
$\mathrm{f}(\kappa_{n})$ and $\mathrm{v}(\kappa_{n})$.

Then we can rewrite the probability in (56) as:

$\mathbb{E}\left[\frac{1}{n+2}\frac{1}{6n}\sum_{v\in
V(\mathfrak{t}_{n})}\sum_{\kappa\in
C(\mathfrak{t}_{n})}1_{|\mathrm{d}^{\star}(\Phi_{n}(v),\mathrm{f}(\kappa_{n}))-c_{\nu}\mathrm{d}_{\mathrm{gr}}(v,\mathrm{v}(\kappa_{n}))|}\right]$
(57)

Since any pair $(v,\Phi_{n}(v))$ can be written as
$(\mathrm{v}(\kappa),\mathrm{f}(\kappa))$ for some $\kappa\in
C(\mathfrak{t}_{n})$, the latter quantity is greater than:

$\frac{1}{6(n+2)^{2}}\mathbb{E}\left[\sum_{v\in V(\mathfrak{t}_{n})}\sum_{w\in
V(\mathfrak{t}_{n})}1_{|\mathrm{d}^{\star}(\Phi_{n}(v),\Phi_{n}(w))-c_{\nu}\mathrm{d}_{\mathrm{gr}}(v,w)|}\right].$
(58)

Combined with (56), this implies that, if $o^{\prime\prime}_{n}$ is another
uniform vertex of $\mathfrak{t}_{n}^{\bullet}$, then for any $\epsilon>0$, we
have

$\mathbb{P}\left(|\mathrm{d}^{\star}(\Phi_{n}(o^{\prime}_{n}),\Phi_{n}(o^{\prime\prime}_{n}))-c_{\nu}\mathrm{d}_{\mathrm{gr}}(o^{\prime}_{n},o^{\prime\prime}_{n})|>\epsilon
n^{1/4}\right)\xrightarrow[n\rightarrow\infty]{}0.$ (59)

Now let $(o_{n}^{i})_{i\geqslant 1}$ be a sequence of independent uniform
vertices of $\mathfrak{t}_{n}$. Thanks to the convergence of simple
triangulations for the Gromov–Hausdorff–Prokhorov topology established in [1]
towards the Brownian sphere, we can transfer the compactness of the Brownian
sphere from the continuum world to the discrete world. More precisely, for any
fixed $\delta$, there exist a large enough $N\geqslant 1$, such that for every
$n\in\mathbb{N}$:

$\mathbb{P}\left(\sup_{x\in V(\mathfrak{t}_{n})}\left(\inf_{1\leqslant
j\leqslant N}\mathrm{d}_{\mathrm{gr}}(x,o_{n}^{j})\right)<\epsilon
n^{1/4}\right)>1-\delta.$ (60)

For $N$ fixed, (59) implies that, for $n$ large enough:

$\mathbb{P}\left(\bigcap_{1\leqslant i\leqslant j\leqslant
N}\left\\{|\mathrm{d}^{\star}(\Phi_{n}(o_{n}^{i}),\Phi_{n}(o_{n}^{j}))-c_{\nu}\mathrm{d}_{\mathrm{gr}}(o_{n}^{i},o_{n}^{j})|\right\\}\leqslant\epsilon
n^{1/4}\right)\geqslant 1-\delta.$ (61)

By the triangle inequality, we can then write:

$\sup_{\begin{subarray}{c}x,y\in V(\mathfrak{t}_{n}),f,g\in
F(\mathfrak{t}_{n})\\\ x\triangleleft f,y\triangleleft
g\end{subarray}}|\mathrm{d}^{\star}(f,g)-c_{\nu}\mathrm{d}_{\mathrm{gr}}(x,y)|\\\
\leqslant\quad\sup_{1\leqslant i\leqslant j\leqslant
N}\left\\{\Big{|}\mathrm{d}^{\star}(\Phi_{n}(o_{n}^{i}),\Phi_{n}(o_{n}^{j}))-c_{\nu}\mathrm{d}_{\mathrm{gr}}(o_{n}^{i},o_{n}^{j})\Big{|}\right\\}+2\sup_{x\in
V(\mathfrak{t}_{n})}\left(\inf_{1\leqslant j\leqslant
N}\mathrm{d}_{\mathrm{gr}}(x,o_{n}^{j})\right)\\\
\qquad\qquad\qquad\qquad+2\sup_{f\in
F(\mathfrak{t}_{n})}\left(\inf_{1\leqslant j\leqslant
N}\mathrm{d}^{\star}(f,\Phi_{n}(o_{n}^{j}))\right)\\\ $

We take $N$ large enough so that (60) holds. By Proposition 31, outside a set
of probability at most $\delta$ and provided that $n$ is large enough, the
last term in the upper bound can be bounded by $2\sup_{x\in
V(\mathfrak{t}_{n})}\left(\inf_{1\leqslant j\leqslant
N}\mathrm{d}_{\mathrm{gr}}(x,o_{n}^{j})\right)+n^{1/8}$. In view of (60) and
(61), this concludes the proof. ∎

##  7 Scaling limit of the 3-connected core with distances induced by the
connected graph

Informally, the purpose of this section is to establish a result analogous to
Proposition 34 (and then a result analogous to (55)), but where we relax the
independent hypothesis for the weights on the edges. Recall from Section 3.3
that $\mathcal{C}_{n}^{(q)}$ is the set of cubic connected planar graphs of
size $n$ having a marked $3$-connected component of size $q$, that
$\mathfrak{C}_{n}^{(q)}$ is a uniformly random element of
$\mathcal{C}_{n}^{(q)}$ (for $q>n/2$ it coincides with the random connected
cubic planar graph of size $n$ conditioned to have its $3$-connected core of
size $q$). Similarly, $\mathcal{M}_{n}^{(q)}$ is the set of cubic connected
planar multigraphs of size $n$ having a marked $3$-connected component of size
$q$, and $\mathfrak{M}_{n}^{(q)}$ denotes a uniformly random element of
$\mathcal{M}_{n}^{(q)}$ (for $q>n/2$ it coincides with the random connected
cubic planar multigraph of size $n$ conditioned to have its $3$-connected core
of size $q$). Recall also that for $\mathrm{g}\in\mathcal{C}_{n}$, we write
$K(\mathrm{g})$ for the 3-connected core of $\mathrm{g}$.

We aim at proving the following result (analogous to (55), with a dependence
on the edge-lengths):

######  Theorem 36.

Let $(q(n),n\geqslant 1)$ be such that $q(n)=\alpha n+O(n^{2/3})$ (where
$\alpha\approx 0.85$ is defined in Proposition 7 and with the informal
notation introduced below Lemma 9). Write respectively
$\mathrm{dist}_{\mathfrak{C}_{n}^{(q(n))}}$ and $\mathrm{d}_{\mathrm{gr}}$ for
the distance on $K(\mathfrak{C}_{n}^{(q(n))})$ induced by the graph distance
on $\mathfrak{C}_{n}^{(q(n))}$ and for the graph distance on
$K(\mathfrak{C}_{n}^{(q(n))})$.

Then, there exists a constant $c>0$ such that, for any $\epsilon>0$:

$\mathbb{P}\left(\sup_{x,y\in
V(K(\mathfrak{C}_{n}^{(q(n))}))}|\mathrm{dist}_{K(\mathfrak{C}_{n}^{(q(n))})}(x,y)-c\,\mathrm{d}_{\mathrm{gr}}(x,y)|\geqslant\epsilon\,q^{1/4}\right)\xrightarrow[n\to\infty]{}0$
(62)

A similar result holds for the random multigraph $\mathfrak{M}_{n}^{(q(n))}$,
when $q(n)=\tfrac{199}{316}n+O(n^{2/3})$.

We note that, given Theorem 3, this result ensures that the scaling limit of
$K(\mathfrak{C}_{n}^{(q(n))})$ with distances (rescaled by $c_{1}q^{1/4}$ for
some $c_{1}\in(0,\infty)$) induced by the distances in
$\mathfrak{C}_{n}^{(q(n))}$ is the Brownian sphere. Moreover, combined with
Theorem 2, it gives the joint convergence of the two last coordinates in
Theorem 4.

To prove this theorem, we first establish the following similar result for the
2-point function (analogous to Proposition 34, with a dependence on the edge-
lengths).

######  Proposition 37.

Consider the same setting as in the previous theorem. Let $o,o^{\prime}$ be
two random vertices of $K(\mathfrak{C}_{n}^{(q(n))})$. Then, there exists a
constant $c>0$ (the same as in Theorem 36) such that, for any $\epsilon>0$:

$\mathbb{P}\big{(}|\mathrm{dist}_{\mathfrak{C}_{n}^{(q(n))}}(o,o^{\prime})-c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})|\geqslant\epsilon\,q^{1/4}\big{)}\xrightarrow[q\to\infty]{}0$
(63)

A similar result holds for the random multigraph $\mathfrak{M}_{n}^{(q(n))}$,
when $q(n)=\tfrac{199}{316}n+O(n^{2/3})$.

Sections 7.1 to 7.5 will be devoted to the proof of this proposition. The
proof of Theorem 36 will then follow along the same lines as the part proving
(55) in the proof of Theorem 35 and is given in Section 7.6.

In the rest of Section 7, we always assume that $n$ and $q(n)$ are related as
in Proposition 37. To simplify notations, we drop the index $n$ and simply
write $q$ for $q(n)$. Since $q(n)=\alpha n+O(n^{2/3})$, it makes sense to
write $q\rightarrow\infty$ in place of $n\rightarrow\infty$. Similarly, with a
slight abuse of notation, in all this section, we write $\mathfrak{K}_{q}$ for
$K(\mathfrak{C}_{n,q})$ and
$\widetilde{\mathfrak{K}}_{q}:=(\mathfrak{K}_{q},\widetilde{\mathrm{d}})$ for
the metric space $(K(\mathfrak{C}_{n,q}),\mathrm{dist}_{\mathfrak{C}_{n,q}})$.

On the other hand, we define
$\widehat{\mathfrak{K}}_{q}:=(\mathfrak{K}_{q},\widehat{\mathrm{d}})$ to be
the random metric graph $(K(\mathfrak{C}_{n,q}),(\ell(e))_{e})$, where
$(\ell_{e})$ is a collection of i.i.d random variables sampled from
$\nu_{\star}$. We recall that $\nu_{\star}$ is the law of the pole-distance of
a critical Boltzmann cubic network as defined in Section 4.1. It follows from
Lemma 12 that $\nu_{\star}$ has exponential tails. Since the support of
$\nu_{\star}$ is by definition included in $[1,\infty)$, all the results
obtained in the previous section apply to $\widehat{\mathfrak{K}}_{q}$. In
particular, by applying Proposition 34 twice (once with $\nu$ equals to the
Dirac measure on $\\{1\\}$ and once with $\nu:=\nu_{\star}$), there exists a
constant $c>0$ (the same as in Proposition 37) such that, for $o,o^{\prime}$
two random vertices in $\mathfrak{K}_{q}$ and for every $\epsilon>0$, we have

$\mathbb{P}\big{(}|\widehat{\mathrm{d}}(o,o^{\prime})-c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})|\geqslant\epsilon\,q^{1/4}\big{)}\xrightarrow[q\to\infty]{}0$
(64)

To prove Proposition 37, we are going to construct a coupling between
$\widehat{\mathfrak{K}}_{q}$ and $\widetilde{\mathfrak{K}}_{q}$, so as to
minimize the the number of edges whose edge-lengths differ. We will then show
that, under this coupling, the weighted distance between two random vertices
$o,o^{\prime}$ differs by $o(q^{1/4})$ in probability.

###  7.1 Definition of the coupling between $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$

To define the coupling between $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$, we first introduce a handful of notations.

For $s\in\mathbb{Z}_{\geqslant 0}$, let $\Sigma_{s}$ to be the set of
sequences $(n_{1},n_{2},\ldots)$ of nonnegative integers, such that
$\sum_{i}n_{i}=s$. For $\boldsymbol{\delta}=(\delta_{1},\ldots,\delta_{m})$ a
sequence of positive integers, we write
$N_{i}(\boldsymbol{\delta})=\\#\\{j\in[1..m],\ \delta_{j}=i\\}$ and
$N_{\geqslant i}(\boldsymbol{\delta})=\\#\\{j\in[1..m],\ \delta_{j}\geqslant
i\\}$ for any $i\geqslant 1$. Next, we define
$\mathrm{Mult}(\boldsymbol{\delta})$ as the mapping from $\mathbb{N}_{*}^{m}$
to $\Sigma_{m}$, defined by:

$\mathrm{Mult}(\boldsymbol{\delta}):=(N_{1}(\boldsymbol{\delta}),N_{2}(\boldsymbol{\delta}),\ldots)$

i.e., $\mathrm{Mult}(\boldsymbol{\delta})$ gives at each position $i\geqslant
1$ the multiplicity of $i$ in $\boldsymbol{\delta}$.

Moreover, for $i\geqslant 1$, we denote
$\boldsymbol{\delta}^{(i)}:=\\{j\in[1..m],\ \delta_{j}=i\\}$ the set of
indices at which the sequence is equal to $i$. Fix $q\in\mathbb{Z}_{\geqslant
0}$, and let $\tilde{\mathbf{n}}:=(\tilde{n}_{1},\tilde{n}_{2},\ldots)$ and
$\hat{\mathbf{n}}:=(\hat{n}_{1},\hat{n}_{2},\ldots)$ be two sequences of
$\Sigma_{3q}$. As illustrated in Figure 9, it is easy to build canonically
from $\hat{\mathbf{n}}$ and $\tilde{\mathbf{n}}$, two sequences
$\hat{\boldsymbol{\delta}}=(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})$
and
$\tilde{\boldsymbol{\delta}}=(\widetilde{\delta}_{1},\ldots,\widetilde{\delta}_{3q})$
such that, for each $i\geqslant 1$, we have

$|\hat{\boldsymbol{\delta}}^{(i)}|=\hat{n}_{i},\ \ \
|\tilde{\boldsymbol{\delta}}^{(i)}|=\tilde{n}_{i},\ \ \
|\hat{\boldsymbol{\delta}}^{(i)}\cap\tilde{\boldsymbol{\delta}}^{(i)}|=\mathrm{min}(\hat{n}_{i},\tilde{n}_{i}).$
(65)

Figure 9: The canonical rearrangement of two increasing sequences
$\widehat{\boldsymbol{\delta}},\widetilde{\boldsymbol{\delta}}$, with
$\mathrm{Mult}(\widehat{\boldsymbol{\delta}})=\hat{\mathbf{n}}$ and
$\mathrm{Mult}(\widetilde{\boldsymbol{\delta}})=\tilde{\mathbf{n}}$.

For $q\geqslant 1$, let us now define two probability distributions on
$\Sigma_{3q}$. Let $(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})$ be a
sequence of i.i.d random variables sampled from $\nu_{\star}$. We denote by
$\widehat{\mathcal{V}}_{q}$ the push-forward distribution on $\Sigma_{3q}$ via
the mapping $\mathrm{Mult}(.)$, of the distribution of
$(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})$.

Similarly, let $(\widetilde{\delta}_{1},\ldots,\widetilde{\delta}_{3q})$ be
the random vector giving the edge-lengths in $\widetilde{\mathfrak{K}}_{q}$
inherited from $\mathfrak{C}_{n,q}$. Then, we denote by
$\widetilde{\mathcal{V}}_{q}$ the push-forward distribution on $\Sigma_{3q}$
via the mapping $\mathrm{Mult}(.)$ of the distribution of
$(\widetilde{\delta}_{1},\ldots,\widetilde{\delta}_{3q})$.

We can now describe the coupling between $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$:

  1. 1.

Let $\widehat{\mathbf{N}}=(\widehat{N}_{1},\widehat{N}_{2},\ldots)$ and
$\widetilde{\mathbf{N}}=(\widetilde{N}_{1},\widetilde{N}_{2},\ldots)$ be
independently sampled according to $\widehat{\mathcal{V}}_{q}$ and
$\widetilde{\mathcal{V}}_{q}$ respectively.

  2. 2.

Construct (in the canonical way depicted in Figure 9) two sequences
$\widehat{\boldsymbol{\delta}}=(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})$
and
$\widetilde{\boldsymbol{\delta}}=(\widetilde{\delta}_{1},\ldots,\widetilde{\delta}_{3q})$,
which satisfy (65).

  3. 3.

Draw a uniformly random $3$-connected cubic planar graph $\mathfrak{K}_{q}$
with $2q$ vertices, and sample uniformly at random a permutation of its edges.
List as $(e_{1},\ldots,e_{3q})$ its edges in this random ordering.

  4. 4.

Define $\widehat{\mathfrak{K}}_{q}=(\mathfrak{K}_{q},(\widehat{\ell}(e_{j})))$
and
$\widetilde{\mathfrak{K}}_{q}=(\mathfrak{K}_{q},(\widetilde{\ell}(e_{j})))$ as
the metric graphs built from $\mathfrak{K}_{q}$, where for each $j\in[1..3q]$,
$\widehat{\ell}(e_{j})=\widehat{\delta}_{j}$ and
$\widetilde{\ell}(e_{j})=\widetilde{\delta}_{j}$.

It is clear from the construction that $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$ defined in this manner follow the required
distribution. In the rest of this section, they are always assumed to be
sampled jointly in this manner.

######  Remark 7.

In the following, it will be useful to be able to “choose an edge in
$\mathfrak{K}_{q}$ using some information about $\widehat{\mathfrak{K}}_{q}$
but no information from $\widetilde{\mathfrak{K}}_{q}$”, or symmetrically. To
make rigorous sense of this, we can slightly adapt the sampling procedure as
follows to better differentiate that $\widehat{\mathfrak{K}}_{q}$ is first
drawn, and $\widetilde{\mathfrak{K}}_{q}$ subsequently (we use the notation
$E^{=i}$ for the set of edges of length $i$ in a metric graph with integer
edge-lengths):

  1. 1.

Sample $\mathfrak{K}_{q}$, $\widehat{\mathbf{N}}$ and
$\widetilde{\mathbf{N}}$.

  2. 2.

Consider a random permutation of $E(\mathfrak{K}_{q})$, and list its edges as
$(e_{1},\ldots,e_{3q})$ in this random ordering.

  3. 3.

For each $i\in\mathbb{Z}_{\geqslant 1}$, define
$E^{=i}(\widehat{\mathfrak{K}}_{q})$ as $\\{e_{\widehat{N}_{\geqslant
i-1}+1},\ldots,e_{\widehat{N}_{\geqslant i}}\\}$.

  4. 4.

For each $i\in\mathbb{Z}_{\geqslant 1}$, reorder uniformly at random the edges
of $E^{=i}(\widehat{\mathfrak{K}}_{q})$ and apply the canonical construction
of Figure 9 to construct $\widetilde{\mathfrak{K}}_{q}$ following
$\widetilde{\mathbf{N}}$.

In this way, for every $i\geqslant 1$ we have
$|E^{=i}(\widehat{\mathfrak{K}}_{q})\cap
E^{=i}(\widetilde{\mathfrak{K}}_{q})|=\mathrm{min}(\widehat{N}_{i},\widetilde{N}_{i})$,
and for every $m\in[\widehat{N}_{\geqslant i-1}+1..\widehat{N}_{\geqslant i}]$
we have $\mathbb{P}(e_{m}\in
E^{=i}(\widetilde{\mathfrak{K}}_{q}))=\frac{\mathrm{min}(\widehat{N}_{i},\widetilde{N}_{i})}{\widehat{N}_{i}}$.

Of course, the roles played by $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$ are perfectly symmetric, and we can proceed by
first sampling $\widetilde{\mathfrak{K}}_{q}$ and then
$\widehat{\mathfrak{K}}_{q}$ if needs be.

###  7.2 First properties of the coupling

To quantify the number of edges which do not have the same length in both
models, we start by proving the following estimate:

######  Lemma 38.

Let $\widehat{\mathbf{N}}=(\widehat{N}_{1},\widehat{N}_{2},\ldots)$ and
$\widetilde{\mathbf{N}}=(\widetilde{N}_{1},\widetilde{N}_{2},\ldots)$ be
independently sampled from $\widehat{\mathcal{V}}_{q}$ and from
$\widetilde{\mathcal{V}}_{q}$ respectively. Then there exist positive
constants $a$ and $A$ such that a.a.s. as $q\to\infty$, the following event
$\mathcal{E}_{q}$ holds:

  * •

for every positive integer $i\leqslant a\log(q)$, we have
$\mathrm{min}(\widehat{N}_{i},\widetilde{N}_{i})\geqslant q^{3/4}$ and
$|\widehat{N}_{i}-\widetilde{N}_{i}|\leqslant q^{2/3}$.

  * •

for all $i\geqslant A\log(q)$, we have $\widehat{N}_{i}=\widetilde{N}_{i}=0$.

###### Proof.

Let $p_{i}:=\nu_{\star}(i)$. We can obtain a lower bound for $p_{i}$ by
noticing that any cubic network that is a series of $i$ $L$-networks has pole-
distance $i+1$. Hence:

$p_{i}\geqslant\frac{L(\rho)^{i-1}}{D(\rho)}\geqslant\frac{1}{D(\rho)}e^{-\kappa
i},\text{ where }\kappa=\ln(1/L(\rho))>0,$

and $L(\rho)$ is the generating series of $L$-networks evaluated at its radius
of convergence.

Let $a=\frac{1}{5\kappa}$. Then, for $i\leqslant a\log(q)$, we have
$p_{i}\geqslant\frac{1}{D(\rho)}q^{-1/5}$. On the other hand, by definition,
$\widehat{N}_{i}$ follows a $\mathrm{Binomial}(3q,p_{i})$ distribution, so
that $\mathbb{E}(\widehat{N}_{i})=3q\,p_{i}\geqslant\frac{3}{D(\rho)}q^{4/5}$.
By Hoeffding’s inequality, we have that, for any $x>0$:

$\mathbb{P}(|\widehat{N}_{i}-3q\,p_{i}|\geqslant xq^{1/2})\leqslant
2\exp(-2x^{2}/3),$

so that $\mathbb{P}(|\widehat{N}_{i}-3q\,p_{i}|\geqslant q^{2/3}/2)\leqslant
2\exp(-q^{1/3}/6)$. By Lemma 9, this implies that

$\mathbb{P}(|\widetilde{N}_{i}-3q\,p_{i}|\geqslant
q^{2/3}/2)=O(q^{2/3}\exp(-q^{1/3}/6)).$

Hence,

$\mathbb{P}\left(\forall i\leqslant
a\log(q),|\widehat{N}_{i}-3q\,p_{i}|\leqslant q^{2/3}/2\text{ and
}|\widetilde{N}_{i}-3q\,p_{i}|\leqslant
q^{2/3}/2\right)\xrightarrow[q\rightarrow\infty]{}1,$

which gives the first property (using the lower bound on $3q\,p_{i}$).

Regarding the second property, we proved in Lemma 12 that $\nu_{\star}$ has
exponential tail. Hence, if
$(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})$ is a sequence of i.i.d.
random variables sampled from $\nu_{\star}$, then there exists $b>0$, such
that
$\mathbb{P}(\mathrm{max}(\widehat{\delta}_{1},\ldots,\widehat{\delta}_{3q})\geqslant
i)=O(q\exp(-bi))$. This is equivalent to the fact that
$\mathbb{P}(\widehat{N}_{j}\neq 0\ \mathrm{for\ some}\ j\geqslant
i)=O(q\exp(-bi))$. Using again Lemma 9, we can derive from this property, the
fact that $\mathbb{P}(\widetilde{N}_{j}\neq 0\ \mathrm{for\ some}\ j\geqslant
i)=O(q^{5/2}\exp(-bi))$. Hence, setting $A=3/b$, this gives the desired
result. ∎

######  Corollary 39.

Conditionally on the event $\mathcal{E}_{q}$ defined in the previous lemma,
for any $e\in E(\mathfrak{K}_{q})$ such that $\widehat{\ell}(e)\leqslant
a\log(q)$, and chosen without using any information on
$\widetilde{\mathfrak{K}}_{q}$, we have:

$\mathbb{P}(\widehat{\ell}(e)\neq\widetilde{\ell}(e))\leqslant q^{-1/12}.$

Similarly, for any edge $e\in E(\mathfrak{K}_{q})$, such that
$\widetilde{\ell}(e)\leqslant a\log(q)$, and chosen without using any
information on $\widehat{\mathfrak{K}}_{q}$, we have
$\mathbb{P}(\widehat{\ell}(e)\neq\widetilde{\ell}(e))\leqslant q^{-1/12}$.

###### Proof.

By construction of the coupling (or rather of its variant described in Remark
7), in the first case, we have that:

$\mathbb{P}(\widetilde{\ell}(e)\neq\widehat{\ell}(e)|\widehat{\ell}(e)=i)=\frac{\max(\widetilde{N}_{i}-\widehat{N}_{i},0)}{\widetilde{N}_{i}},\text{
for any }i\in\mathbb{Z}_{>0}.$ (66)

Summing over the values of $i$ in $\\{1,..,\lfloor a\log(q)\rfloor\\}$, and
using the bound for $\widetilde{N}_{i}$ and $\widehat{N}_{i}$, this proves the
result.

The other case is treated in exactly the same manner. ∎

###  7.3 Truncation of edge-lengths

Another important ingredient of our proof is the truncation of the edge-
lengths. Fix $k\geqslant 1$, then for any metric graph $G=(V,E,(\ell(e)_{e\in
E})$, we write $\ell^{(k)}(e):=\min(\ell(e),k)$. For a path
$\gamma=(e_{1},\ldots,e_{m})$ in $G$, the truncated edge-length of $\gamma$ is
defined as $\ell^{(k)}(e_{1})+\cdots+\ell^{(k)}(e_{m})$.

Let us apply this notion to $\widehat{\mathfrak{K}}_{q}$ and
$\widetilde{\mathfrak{K}}_{q}$. For an edge $e$ of $\mathfrak{K}_{q}$, we
denote by $\widehat{\ell}^{(k)}(e)$ (resp. $\widetilde{\ell}^{(k)}(e)$) its
truncated edge-length in $\widehat{\mathfrak{K}}_{q}$ (resp.
$\widetilde{\mathfrak{K}}_{q}$). For $\gamma$ a path in $\mathfrak{K}_{q}$,
its truncated edge-length is denoted by $\widehat{L}^{(k)}(\gamma)$ (resp.
$\widetilde{L}^{(k)}(\gamma)$) in $\widehat{\mathfrak{K}}_{q}$ (resp.
$\widetilde{\mathfrak{K}}_{q}$). Furthermore, for two vertices
$o,o^{\prime}\in\mathfrak{K}_{q}$, we denote by
$\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})$ (resp.
$\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})$) the minimum of
$\widehat{L}^{(k)}(\gamma)$ (resp. of $\widetilde{L}^{(k)}(\gamma)$) over all
paths connecting $o$ and $o^{\prime}$.

Finally, we denote by $\widehat{\mathfrak{K}}_{q}^{(k)}$ (resp.
$\widetilde{\mathfrak{K}}_{q}^{(k)}$) the metric graphs
$(\mathfrak{K}_{q},\widehat{\mathrm{d}}^{(k)})$ (resp.
$(\mathfrak{K}_{q},\widetilde{\mathrm{d}}^{(k)})$). Applying the results of
Section 6 to $\widehat{\mathfrak{K}}_{q}^{(k)}$ instead of
$\widehat{\mathfrak{K}}_{q}$, we obtain that there exists a constant $c^{(k)}$
such that, for every $\epsilon>0$, we have:

$\mathbb{P}\big{(}|\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})-c^{(k)}\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})|\geqslant\epsilon\,q^{1/4}\big{)}\xrightarrow[q\to\infty]{}0,$
(67)

where $o$ and $o^{\prime}$ are two random vertices of $\mathfrak{K}_{q}$
(picked independently from anything else). Note that $c^{(k)}$ is increasing
with $k$ and is bounded by $c$.

The purpose of this section is now to prove the analogous statement on
$\widetilde{\mathfrak{K}}_{q}^{(k)}$:

######  Lemma 40.

Let $k\geqslant 1$. Then, with the same constant $c^{(k)}$ as in (67), we have
for every $\epsilon>0$

$\mathbb{P}\big{(}|\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})-c^{(k)}\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})|\geqslant\epsilon\,q^{1/4}\big{)}\xrightarrow[q\to\infty]{}0,$
(68)

where $o$ and $o^{\prime}$ are two random vertices of $\mathfrak{K}_{q}$
(picked independently from anything else).

###### Proof.

Set $\widehat{N}_{\geqslant k}:=\sum_{i\geqslant k}\widehat{N}_{i}$ and
$\widetilde{N}_{\geqslant k}:=\sum_{i\geqslant k}\widetilde{N}_{i}$, and
define $\mathcal{E}_{q}^{(k)}$ as the event under which:

$\mathrm{min}(\widehat{N}_{i},\widetilde{N}_{i})\geqslant
q^{3/4}\quad\text{and}\quad|\widehat{N}_{i}-\widetilde{N}_{i}|\leqslant
q^{2/3}\text{ for }i<k,\quad\text{and}\quad\mathrm{min}(\widehat{N}_{\geqslant
k},\widetilde{N}_{\geqslant k})\geqslant q^{3/4}.$ (69)

This event holds a.a.s. by Lemma 38.

Let $e\in E(\mathfrak{K}_{q})$, chosen without using any information on
$\widetilde{\mathfrak{K}}_{q}^{(k)}$. Conditionally on
$\mathcal{E}_{q}^{(k)}$, by the same reasoning as in Corollary 39, we have:

$\mathbb{P}(\widetilde{\ell}^{(k)}(e)\neq\widehat{\ell}^{(k)}(e)|\widehat{\ell}^{(k)}(e)<k)\leqslant\max_{1\leqslant
i<k}\frac{\max(\widetilde{N}_{i}-\widehat{N}_{i},0)}{\widetilde{N}_{i}}\leqslant
q^{-1/12}.$ (70)

On the other hand, let $N_{k}^{\prime}$ be the number of edges having length
at least $k$ in $\widehat{\mathfrak{K}}_{q}$ and smaller than $k$ in
$\widetilde{\mathfrak{K}}_{q}$. By construction of the coupling between
$\widehat{\mathfrak{K}}_{q}$ and $\widetilde{\mathfrak{K}}_{q}$,
$N_{k}^{\prime}\leqslant\sum_{i=1}^{k-1}\mathrm{max}(\widetilde{N}_{i}-\widehat{N}_{i},0)$.
Hence, if $\mathcal{E}_{q}^{(k)}$ holds, $N_{k}^{\prime}\leqslant k\,q^{2/3}$.
Therefore, conditionally on $\mathcal{E}_{q}^{(k)}$, we have:

$\mathbb{P}(\widetilde{\ell}^{(k)}(e)\neq\widehat{\ell}^{(k)}(e)|\widehat{\ell}^{(k)}(e)=k)=\frac{N_{k}^{\prime}}{\widehat{N}_{\geqslant
k}}\leqslant kq^{-1/12}.$ (71)

Now, we define $\widehat{\gamma}^{(k)}$ as a canonically chosen path among the
paths $\gamma$ connecting $o$ to $o^{\prime}$ such that
$\widehat{L}^{(k)}(\gamma)=\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})$ (e.g.
$\widehat{\gamma}^{(k)}$ is the lexicographically smallest, for the sequence
of labels of the visited vertices). For any edge of $\widehat{\gamma}^{(k)}$,
we can apply (70) or (71). Moreover, by definition of truncation, for any edge
$e$, we clearly have
$|\widehat{\ell}^{(k)}(e)-\widetilde{\ell}^{(k)}(e)|\leqslant k$. Hence,
conditionally on $\mathcal{E}_{q}^{(k)}$, and in view of (70) and (71), we
have

$\mathbb{E}\left(|\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})-\widetilde{L}^{(k)}(\widehat{\gamma}^{(k)})|\,\Big{|}\,|\widehat{\gamma}^{(k)}|=s\right)\leqslant
k^{2}q^{-1/12}s.$

Therefore, conditionally on $\mathcal{E}_{q}^{(k)}$, and by Markov’s
inequality,

$\mathbb{P}\left(|\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})-\widetilde{L}^{(k)}(\widehat{\gamma}^{(k)})|\geqslant\epsilon
q^{1/4}\,\Big{|}\,|\widehat{\gamma}^{(k)}|\leqslant
s\right)\leqslant\frac{k^{2}}{\epsilon}q^{-1/3}s.$

It follows directly from Theorem 3 that a.a.s.
$\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\leqslant q^{1/4}\log(q)$. Since
$k\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\geqslant\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})\geqslant|\widehat{\gamma}^{(k)}|$,
we also have a.a.s. $|\widehat{\gamma}^{(k)}|\leqslant k\,q^{1/4}\log(q)$, so
that:

$\mathbb{P}\left(|\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})-\widetilde{L}^{(k)}(\widehat{\gamma}^{(k)})|\geqslant\epsilon
q^{1/4}\right)\xrightarrow[q\rightarrow\infty]{}0.$

For any $\epsilon>0$, the event
$\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})\leqslant
c^{(k)}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})+\epsilon q^{1/4}$ holds a.a.s.
according to (67). Since
$\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})=\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})$
and
$\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})\leqslant\widetilde{L}^{(k)}(\widehat{\gamma}^{(k)})$,
we conclude that, for any $\epsilon>0$,

$\mathbb{P}\left(\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})\leqslant
c^{(k)}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})+\epsilon
q^{1/4}\right)\xrightarrow[q\rightarrow\infty]{}1.$ (72)

Symmetrically, applying the argument to $\widetilde{\gamma}^{(k)}$, we obtain
that, for every $\epsilon>0$, we have a.a.s.
$|\widehat{L}^{(k)}(\widetilde{\gamma}^{(k)})-\widetilde{L}^{(k)}(\widetilde{\gamma}^{(k)})|\leqslant\epsilon
q^{1/4}$. For any $\epsilon>0$, the event
$\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})\geqslant
c^{(k)}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})-\epsilon q^{1/4}$ holds a.a.s.
according to (67). Since
$\widetilde{L}^{(k)}(\widetilde{\gamma}^{(k)})=\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})$
and
$\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})\leqslant\widehat{L}^{(k)}(\widetilde{\gamma}^{(k)})$,
we get that, for any $\epsilon>0$,

$\mathbb{P}\left(\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})\geqslant
c^{(k)}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})-\epsilon
q^{1/4}\right)\xrightarrow[q\rightarrow\infty]{}1,$ (73)

which concludes the proof. ∎

###  7.4 Lengths of truncated geodesic paths in both models

We now state and prove additional estimates in view of completing the proof of
Proposition 37.

######  Lemma 41.

Let $k\geqslant 1$. Then, for every $\epsilon>0$, we have

$\mathbb{P}\big{(}|\widetilde{L}(\widehat{\gamma}^{(k)})-\widehat{L}(\widehat{\gamma}^{(k)})|\geqslant\epsilon\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\big{)}\xrightarrow[q\to\infty]{}0,$

where $o$ and $o^{\prime}$ are two random vertices for $\mathfrak{K}_{q}$
(picked independently from anything else), and where we recall that
$\widehat{\gamma}^{(k)}$ is a canonical geodesic path in
$\widehat{\mathfrak{K}}_{q}^{(k)}$ between them.

###### Proof.

In all this proof, we work conditionally on the a.a.s. event $\mathcal{E}_{q}$
of Lemma 38, and consider $q$ large enough such that $a\log(q)>k$. For $e$ an
edge chosen using solely information given by
$\widehat{\mathfrak{K}}_{q}^{(k)}$ and for $i\geqslant k$, we have:

$\mathbb{P}(\widehat{\ell}(e)\geqslant
i\,|\,\widehat{\ell}^{(k)}(e)=k)=\nu_{\star}([i,\infty))/\nu_{\star}([k,\infty)).$

Hence, by Lemma 12, letting $\kappa=a_{\star}/\nu_{\star}([k,\infty))$, we
have $\mathbb{P}(\widehat{\ell}(e)\geqslant
a\log(q))\leqslant\kappa\,q^{-ab_{\star}}$. Since

$\mathbb{E}\big{(}|\widehat{\ell}(e)-\widetilde{\ell}(e)|\big{)}\leqslant
A\,\log(q)\left(\mathbb{P}\Big{(}\widetilde{\ell}(e)\neq\widehat{\ell}(e)|\widehat{\ell}(e)<a\log
q\Big{)}+\mathbb{P}\Big{(}\widehat{\ell}(e)\geqslant a\log q\Big{)}\right),$

by Corollary 39, we get that:

$\mathbb{E}\big{(}|\widehat{\ell}(e)-\widetilde{\ell}(e)|\big{)}\leqslant(q^{-1/12}+\kappa\,q^{-ab_{\star}})A\log(q).$

We further work conditionally on the a.a.s. event $\mathcal{E}_{q}^{\prime}$
under which
$q^{1/4}/\log(q)\leqslant\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\leqslant
q^{1/4}\log(q)$. Since $|\widehat{\gamma}^{(k)}|\leqslant
k\cdot\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})$, we have

$\mathbb{E}\big{(}|\widetilde{L}(\widehat{\gamma}^{(k)})-\widehat{L}(\widehat{\gamma}^{(k)})|\big{)}\leqslant
k(q^{-1/12}+\kappa\,q^{-ab_{\star}})A\,q^{1/4}\log(q)^{2}=O(q^{1/4-\xi}),$

where $\xi=\mathrm{min}(1/12,ab_{\star})/2$. Hence, by Markov’s inequality we
have

$\displaystyle\mathbb{P}\Big{(}|\widetilde{L}(\widehat{\gamma}^{(k)})-\widehat{L}(\widehat{\gamma}^{(k)})|\geqslant\epsilon\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}$
$\displaystyle\leqslant\mathbb{P}\Big{(}|\widetilde{L}(\widehat{\gamma}^{(k)})-\widehat{L}(\widehat{\gamma}^{(k)})|\geqslant\epsilon\,q^{1/4}/\log(q)\Big{)}+\mathbb{P}({}^{c}(\mathcal{E}_{q}\cap\mathcal{E}_{q}^{\prime}))$
$\displaystyle=O\big{(}\epsilon^{-1}\log(q)\,q^{-\xi}\big{)}+o(1)=o(1).$

∎

######  Lemma 42.

Let $X$ be a random variable drawn under $\nu_{\star}$. There exist positive
constants $a^{\prime},b^{\prime}$, and an infinite subset
$\mathbb{N}^{\prime}\subset\mathbb{N}$ such that, for every
$k\in\mathbb{N}^{\prime}$ and every $i\geqslant 0$, we have

$\mathbb{P}(X\geqslant k+i\ |\ X\geqslant k)\leqslant
a^{\prime}\,\exp(-b^{\prime}\,i).$

###### Proof.

Let $p_{i}=\mathbb{P}(X=i)$. We have seen in the proof of Lemma 38 that
$p_{i}>0$ for any $i\geqslant 0$ and that $p_{i}=O(\exp(-bi))$ for some $b>0$.
Set $p_{i}^{\prime}:=p_{i}\exp(bi/2)$. Then $p_{i}^{\prime}\to 0$ as
$i\to\infty$. Hence, we can define a subset $\mathbb{N}^{\prime}$ of integers
such that, for every $k\in\mathbb{N}^{\prime}$,
$p_{k}^{\prime}=\mathrm{max}(p_{i}^{\prime},\ i\geqslant k)$ holds. Hence, for
$k\in\mathbb{N}^{\prime}$ and $i\geqslant k$, we have $p_{i}\leqslant
p_{k}\exp(-b(i-k)/2)$. This gives

$\mathbb{P}(X\geqslant k+i\ |\ X\geqslant k)\leqslant\frac{\sum_{j\geqslant
i}p_{k}\,e^{-bj/2}}{\mathbb{P}(\nu_{\star}\geqslant
k)}\leqslant\sum_{j\geqslant i}e^{-bj/2}=\frac{1}{1-e^{-b/2}}e^{-bi/2}.$

∎

######  Lemma 43.

There exists a positive constant $\hat{a}$ such that, for every
$k\in\mathbb{N}^{\prime}$,

$\mathbb{P}\Big{(}\,\widehat{L}(\widehat{\gamma}^{(k)})\leqslant\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})+\frac{\hat{a}}{k}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}\xrightarrow[q\to\infty]{}1,$

where $o$ and $o^{\prime}$ are two random vertices for $\mathfrak{K}_{q}$
(picked independently from anything else), and where we recall that
$\widehat{\gamma}^{(k)}$ is a canonical geodesic path in
$\widehat{\mathfrak{K}}_{q}^{(k)}$ between them.

Moreover, we have $\lim_{k\to\infty}c^{(k)}=c$.

###### Proof.

Let
$\Delta=\widehat{L}(\widehat{\gamma}^{(k)})-\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})$.
Then $\Delta=\sum_{e\in E_{k}}(\ell(e)-k)$, where $E_{k}$ is the set of edges
$e\in\widehat{\gamma}^{(k)}$ with $\ell^{(k)}(e)=k$. Since
$\widehat{\gamma}^{(k)}$ is selected based solely on
$\widehat{\mathfrak{K}}_{q}^{(k)}$, $\Delta$ is the sum of $|E_{k}|$ i.i.d.
copies of the random variable $X^{(k)}$, whose distribution is given by:

$\mathbb{P}(X^{(k)}\geqslant
i)=\frac{\nu_{\star}([k+i,\infty))}{\nu_{\star}([k,\infty))}.$

Lemma 42 gives
$\mathbb{E}(X^{(k)})\leqslant\frac{a^{\prime}}{1-e^{-b^{\prime}}}$. For
$m\geqslant 1$, let $S_{m}$ be the sum of $m$ i.i.d. copies of $X^{(k)}$.
Given the universal exponential bound on $X^{(k)}$ in Lemma 42, Chernoff’s
bound ensures that there exist positive constants
$a^{\prime\prime},b^{\prime\prime}$ (not depending on $k$) such that

$\mathbb{P}\Big{(}S_{m}\geqslant\frac{2a^{\prime}}{1-e^{-b^{\prime}}}m\Big{)}\leqslant
a^{\prime\prime}\,e^{-b^{\prime\prime}m},$

and thus
$\mathbb{P}\Big{(}S_{m^{\prime}}\geqslant\frac{2a^{\prime}}{1-e^{-b^{\prime}}}m\Big{)}\leqslant
a^{\prime\prime}\,e^{-b^{\prime\prime}m}$ whenever $m^{\prime}\leqslant m$.

Next, let us define the event $\mathcal{H}_{q}$ as:

$\mathcal{H}_{q}:=\left\\{\widehat{L}^{(k)}(\widehat{\gamma}^{(k)})\leqslant
2c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\right\\}\,\cap\,\left\\{\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\geqslant\frac{q^{1/6}k}{2c}\right\\}.$

By (67) and Theorem 3, we know that $\mathbb{P}(\mathcal{H}_{q})\to 1$ as
$q\to\infty$. If $\mathcal{H}_{q}$ holds, we have
$|E_{k}|\leqslant\frac{2c}{k}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})$, so that:

$\mathbb{P}\left(\Big{\\{}\Delta\geqslant\frac{2a^{\prime}}{1-e^{-b^{\prime}}}\frac{2c}{k}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{\\}}\cap\mathcal{H}_{q}\right)\leqslant
a^{\prime\prime}\,e^{-b^{\prime\prime}\,q^{1/6}}=o(1),$

so that the first statement is proved (with
$\hat{a}=\frac{2a^{\prime}}{1-e^{-b^{\prime}}}2c$).

It ensures that we have a.a.s.
$\widehat{\mathrm{d}}(o,o^{\prime})\leqslant\widehat{\mathrm{d}}^{(k)}(o,o^{\prime})+\frac{\hat{a}}{k}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})$.
Hence, using (67), for any $\epsilon>0$ we have a.a.s.
$\widehat{\mathrm{d}}(o,o^{\prime})\leqslant(c^{(k)}+\frac{\hat{a}}{k}+\epsilon)\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})$.
This implies that $c^{(k)}+\frac{\hat{a}}{k}\geqslant c$. Since
$c^{(k)}\leqslant c$, this gives $\lim_{k\to\infty}c^{(k)}=c$. ∎

###  7.5 Proof of Proposition 37

We start by establishing the lower bound for
$\widetilde{\mathrm{d}}(o,o^{\prime})$ given in (63) of Proposition 37. Fix
$\epsilon>0$ and $\eta>0$. By Theorem 3, we know that there exists $b>0$ such
that for $q$ large enough:

$\mathbb{P}(\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\leqslant
b\,q^{1/4})\geqslant 1-\eta/2.$

On the other hand, by Lemma 40, we have for $q$ large enough:

$\mathbb{P}\big{(}\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})\geqslant
c^{(k)}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})-\frac{\epsilon}{2}q^{1/4}\big{)}\geqslant
1-\eta/2.$

Then, Lemma 43 grants the existence of $k$ such that $c^{(k)}\geqslant
c-\epsilon/(2b)$. Combined with the two preceding inequalities and since
$\widetilde{\mathrm{d}}^{(k)}(o,o^{\prime})\leqslant\widetilde{\mathrm{d}}(o,o^{\prime}))$
it yields for $q$ large enough:

$\mathbb{P}\big{(}\widetilde{\mathrm{d}}(o,o^{\prime})\geqslant
c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})-\epsilon q^{1/4}\big{)}\geqslant
1-\eta.$

Regarding the upper bound, let $k\in\mathbb{N}^{\prime}$ be large enough such
that $\frac{\hat{a}}{k}\leqslant\frac{\epsilon}{4b}$. By Lemma 43, there
exists $q_{1}$ such that, for $q\geqslant q_{1}$, we have

$\mathbb{P}\Big{(}\,\widehat{L}(\widehat{\gamma}^{(k)})\leqslant\widehat{\mathrm{d}}(o,o^{\prime})+\frac{\epsilon}{4b}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}\geqslant
1-\eta/8.$

By Lemma 41, there exists $q_{2}$ such that, for $q\geqslant q_{2}$,

$\mathbb{P}\Big{(}\,\widetilde{L}(\widehat{\gamma}^{(k)})\leqslant\widehat{L}(\widehat{\gamma}^{(k)})+\frac{\epsilon}{4b}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}\geqslant
1-\eta/8.$

Hence, for $q\geqslant\mathrm{max}(q_{1},q_{2})$, we have

$\mathbb{P}\Big{(}\widetilde{L}(\widehat{\gamma}^{(k)})\leqslant\widehat{\mathrm{d}}(o,o^{\prime})+\frac{\epsilon}{2b}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}\geqslant
1-\eta/4.$

Since
$\widetilde{\mathrm{d}}(o,o^{\prime})\leqslant\widetilde{L}(\widehat{\gamma}^{(k)})$,
we have
$\mathbb{P}\Big{(}\widetilde{\mathrm{d}}(o,o^{\prime})\leqslant\widehat{\mathrm{d}}(o,o^{\prime})+\frac{\epsilon}{2b}\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})\Big{)}\geqslant
1-\eta/4$. Hence,
$\mathbb{P}\Big{(}\widetilde{\mathrm{d}}(o,o^{\prime})\leqslant\widehat{\mathrm{d}}(o,o^{\prime})+\frac{\epsilon}{2}q^{1/4}\Big{)}\geqslant
1-3\eta/4$.

Finally, by (64), there exists $q_{3}$ such that, for $q\geqslant q_{3}$, we
have

$\mathbb{P}\Big{(}\widehat{\mathrm{d}}(o,o^{\prime})\leqslant
c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})+\frac{\epsilon}{2}q^{1/4}\Big{)}\geqslant
1-\eta/4.$

Hence, for $q\geqslant\mathrm{max}(q_{1},q_{2},q_{3})$, we have
$\mathbb{P}\big{(}\widetilde{\mathrm{d}}(o,o^{\prime})\leqslant
c\,\mathrm{d}_{\mathrm{gr}}(o,o^{\prime})+\epsilon q^{1/4}\big{)}\geqslant
1-\eta$. This concludes the proof.

###  7.6 Proof of Theorem 36

Given Proposition 37, the proof of Theorem 36 follows the exact same lines as
in the proof of (55). The only missing point is a result analogous to
Proposition 31, which we now state and prove:

######  Lemma 44.

Let $K$ be any constant larger than $\mathbb{E}(\nu_{\star})$. For
$\epsilon\in(0,1/4)$, define $\tilde{\mathcal{E}}_{q}$ as the event where the
bound

$\widetilde{\mathrm{d}}(x,y)\leqslant
K\,\mathrm{d}_{\mathrm{gr}}(x,y)+q^{\epsilon}$

holds for every pair of vertices $x,y$ in $\mathfrak{K}_{q}$. Then
$\mathbb{P}(\tilde{\mathcal{E}}_{q})\to 1$ as $q\to\infty$.

###### Proof.

We start by proving a similar result but for $\widehat{\mathfrak{K}}_{q}$
rather than $\widetilde{\mathfrak{K}}_{q}$. Let $\hat{\mathcal{E}}_{q}$ be the
event where the bound

$\widehat{\mathrm{d}}(x,y)\leqslant
K\,\mathrm{d}_{\mathrm{gr}}(x,y)+q^{\epsilon}$

holds for every pair of vertices $x,y$ in $\mathfrak{K}_{q}$. Let $x,y$ be two
fixed vertices in $\mathfrak{K}_{q}$. Since $\nu_{\star}$ has exponential
tail, the Chernoff bound ensures that there exist positive constants
$a^{\prime},b^{\prime}$ (not depending on $x,y$) such that

$\mathbb{P}(\widehat{\mathrm{d}}(x,y)>K\,\mathrm{d}_{\mathrm{gr}}(x,y))\leqslant
a^{\prime}\exp(-b^{\prime}\,\mathrm{d}_{\mathrm{gr}}(x,y)).$

In particular, if $\mathrm{d}_{\mathrm{gr}}(x,y)>q^{\epsilon/2}$, this implies
that
$\mathbb{P}\Big{(}\widehat{\mathrm{d}}(x,y)>K\,\mathrm{d}_{\mathrm{gr}}(x,y)\Big{)}\leqslant
a^{\prime}\exp(-b^{\prime}\,q^{\epsilon/2})$.

We now deal with the case where $\mathrm{d}_{\mathrm{gr}}(x,y)\leqslant
q^{\epsilon/2}$. Let $w^{\star}$ be the maximal weight over the edges in
$\widehat{\mathfrak{K}}_{q}$. The union bound and the fact that $\nu_{\star}$
has exponential tail ensure that (with $a_{\star},b_{\star}$ the constants in
Lemma 12)

$\mathbb{P}(w^{\star}>q^{\epsilon/2})\leqslant 3q\cdot
a_{\star}\cdot\exp(-b_{\star}\,q^{\epsilon/2}).$

Hence, if $\mathrm{d}_{\mathrm{gr}}(x,y)\leqslant q^{\epsilon/2}$, we have

$\mathbb{P}(\widehat{\mathrm{d}}(x,y)>q^{\epsilon})\leqslant 3q\cdot
a_{\star}\cdot\exp(-b_{\star}\,q^{\epsilon/2})$

Letting $b^{\prime\prime}$ be a positive constant smaller than $b_{\star}$ and
$b^{\prime}$, this guarantees the existence of a constant $a^{\prime\prime}$
such that, for every given pair $x,y$ of vertices in $\mathfrak{K}_{q}$,

$\mathbb{P}(\widehat{\mathrm{d}}(x,y)>K\,\mathrm{d}_{\mathrm{gr}}(x,y)+q^{\epsilon})\leqslant
a^{\prime\prime}\exp(-b^{\prime\prime}\,q^{\epsilon/2}).$

By the union bound, this gives
$\mathbb{P}(\neg{\hat{\mathcal{E}}_{q}})\leqslant
q^{2}\,a^{\prime\prime}\exp(-b^{\prime\prime}\,q^{\epsilon/2})$. Finally, by
Lemma 9, we obtain that:
$\mathbb{P}(\neg{\tilde{\mathcal{E}}_{q}})=O(q^{2/3}\mathbb{P}(\neg{\hat{\mathcal{E}}_{q}}))=O(q^{8/3}\,a^{\prime\prime}\exp(-b^{\prime\prime}\,q^{\epsilon/2}))=o(1)$.
∎

Given the preceding discussion, this concludes the proof of Theorem 36.

##  8 Gromov–Hausdorff–Prokhorov distance between a random connected cubic
planar graph and its 3-connected core

The aim of this section is to establish the following result:

######  Proposition 45.

Let $(q(n),n\geqslant 1)$ be such that $q(n)=\alpha n+O(n^{2/3})$ (where
$\alpha\approx 0.85$ is defined in Proposition 7). Let
$\mathfrak{C}_{n}^{(q(n))}$ be a uniformly random element of
$\mathcal{C}_{n}^{(q(n))}$. Write respectively
$\mathrm{dist}_{\mathfrak{C}_{n}^{(q(n))}}$ and $\mathrm{d}_{\mathrm{gr}}$ for
the distance on $K(\mathfrak{C}_{n}^{(q(n))})$ induced by the graph distance
on $\mathfrak{C}_{n}^{(q(n))}$ and for the graph distance on
$K(\mathfrak{C}_{n}^{(q(n))})$. Then

$d_{\mathrm{GHP}}\left(\Big{(}\mathfrak{C}_{n}^{(q(n))},\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{\mathfrak{C}_{n}^{(q(n))}}\Big{)},\Big{(}K(\mathfrak{C}_{n}^{(q(n))}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q(n))}},\mu_{K(\mathfrak{C}_{n}^{(q(n))})}\Big{)}\right)\xrightarrow[n\to\infty]{(p)}0,$

where, as usual, $\mu_{\mathfrak{C}_{n}^{(q(n))}}$ and
$\mu_{K(\mathfrak{C}_{n}^{(q(n))})}$ denote respectively the uniform
distribution on $V(\mathfrak{C}_{n}^{(q(n))})$ and on
$V(K(\mathfrak{C}_{n}^{(q(n))}))$.

A similar results holds for $\mathfrak{M}_{n}^{(q(n))}$, with
$q(n)=\tfrac{199}{316}n+O(n^{2/3})$.

By Theorem 8 of Section 3.2, the size of the $3$-connected core of a random
cubic planar graph is $\alpha n+O(n^{2/3})$ with high probability. We
established in Section 6 (see the discussion below Theorem 3) that
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{q^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}},\mu_{K(\mathfrak{C}_{n}^{(q)})})$
converges to the Brownian sphere. Hence Proposition 45 directly implies the
joint convergence of
$(K(\mathfrak{C}_{n}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}},\mu_{K(\mathfrak{C}_{n})})$
and of
$(\mathfrak{C}_{n},\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{\mathfrak{C}_{n}})$,
which corresponds to the joint convergence of the first and second coordinates
stated in Theorem 4.

As in the previous section, in all this section $q,n$ are related by $q=\alpha
n+O(n^{2/3})$. To lighten notation, we drop the explicit dependency of $q$ in
$n$ and write $q$ instead of $q(n)$.

Roughly speaking, this result relies on two main steps. First, we have to
prove that the Gromov–Hausdorff distance between these two measured metric
spaces is small, or in other terms, that any vertex of
$\mathfrak{C}_{n}^{(q)}$ is at distance $o(n^{1/4})$ from a vertex of
$K(\mathfrak{C}_{n}^{(q)})$. Then, we need to prove that the Levy–Prokhorov
distance between both spaces is small. Roughly speaking, it means that the
uniform measure on $K(\mathfrak{C}_{n}^{(q)})$ and the projection of the
masses of the uniform measure on $\mathfrak{C}_{n}^{(q)}$ into its 3-connected
core are close for the $\mathrm{d}_{\mathrm{LP}}$ distance.

Our approach follows closely the strategy developed in an article by
Addario–Berry and Wen [2], dealing with the joint convergence of a
quadrangulation and its simple core. We give the arguments only for graphs,
but the exact same approach can be developed for multigraphs.

Following [2], we first define a measure on $V(K(\mathrm{C}_{n}^{(q)}))$ that
reflects a “projection of masses”. Let $v_{1},\ldots,v_{2q}$ be the vertices
of $K(\mathrm{C}_{n}^{(q)})$, ordered by increasing label. For $i\in[1..2q]$,
let $s_{i}$ be the total number of vertices of (non-pole) vertices in the
attached networks at the $3$ edges incident to $v_{i}$, and let
$n_{i}=1+s_{i}/2$. Then the measure $\mu_{K(\mathrm{C}_{n}^{(q)})}^{\prime}$
on $V(\mathrm{C}_{n}^{(q)})$ is defined by:

$\mu_{K(\mathrm{C}_{n}^{(q)})}^{\prime}(v_{i})=\frac{n_{i}}{2n},\text{ for any
}i\in[1..2q].$

For a vector $\mathbf{x}=(x_{1},\ldots,x_{2q})\in[0,\infty)^{2q}$, we write
$|\mathbf{x}|_{p}=\big{(}\sum_{i=1}^{2q}x_{i}^{p}\big{)}^{1/p}$, for
$p\geqslant 1$. Note that, with the notations from above and setting
$\mathbf{n}=(n_{1},\ldots,n_{2q})$, we have $|\mathbf{n}|_{1}=2n$.

######  Lemma 46.

Let $\mathfrak{C}_{n}^{(q)}$ be as in Proposition 45. Write
$\mathrm{d}_{\mathrm{LP}}$ for the Lévy-Prokhorov distance on the metric space
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}})$. Then
we have

$\mathrm{d}_{\mathrm{LP}}\left(\mu_{K(\mathfrak{C}_{n}^{(q)})},\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}\right)\xrightarrow[q\to\infty]{(p)}0.$

###### Proof.

Let $\mathbf{n}=(n_{1},\ldots,n_{2q})$ be defined from
$\mathfrak{C}_{n}^{(q)}$ as above. Lemma 6.3 in [2]888In [2], it is assumed
that $|\mathbf{n}|_{2}$ is deterministic, which is not the case here. However,
we can apply their lemma to the normalized vector
$\tilde{\mathbf{n}}=\mathbf{n}/|\mathbf{n}|_{2}$, which gives the desired
result. implies that for any vertex-subset $W\subset
V(K(\mathfrak{C}_{n}^{(q)}))$, and for any $t>0$, we have

$\mathbb{P}\left(|\mu_{K(\mathfrak{C}_{n}^{(q)})}(W)-\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}(W)|>\frac{2t|\mathbf{n}|_{2}}{|\mathbf{n}|_{1}}+\frac{1}{2q+1}\right)\leqslant
4\exp(-2t^{2}).$ (74)

Here, $|\mathbf{n}|_{1}=2n$, and by Lemma 10 we know that
$\mathrm{max}(n_{1},\ldots,n_{2q})\leqslant n^{3/4}$ a.a.s., which implies
$|\mathbf{n}|_{2}\leqslant 2\,n^{7/8}$ (since
$\sum_{i=1}^{2q}n_{i}^{2}\leqslant\mathrm{max}(n_{1},\ldots,n_{2q})\cdot|\mathbf{n}|_{1}$).

Taking $t=n^{1/16}$ and letting $\epsilon_{q}=2n^{-1/16}+\frac{1}{2q+1}=o(1)$,
we thus have a.a.s.
$|\mu_{K(\mathfrak{C}_{n}^{(q)})}(W)-\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}(W)|\leqslant\epsilon_{q}$
for all $W\subseteq V(K(\mathfrak{C}_{n}^{(q)}))$.

We can then proceed with the exact same arguments as in Corollary 7.2 in [2]
(compactness of the Brownian sphere $\mathrm{m}_{\infty}$, and the fact that
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{q^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{K(\mathfrak{C}_{n}^{(q)})})$
converges toward $\mathrm{m}_{\infty}$ for the GHP topology, as we have
already established in Theorem 3). ∎

######  Lemma 47.

Write $\tilde{\mathrm{d}}_{\mathrm{LP}}$ for the Lévy-Prokhorov distance on
the metric space
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}})$.
Then, we have:

$\tilde{\mathrm{d}}_{\mathrm{LP}}(\mu_{K(\mathfrak{C}_{n}^{(q)})},\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime})\xrightarrow[q\to\infty]{(p)}0.$

###### Proof.

We use the characterization of the Lévy-Prokhorov distance in terms of
couplings. Let $V_{q}=V(K(\mathfrak{C}_{n}^{(q)}))$. Let $\epsilon>0$ and
$\eta>0$. Our aim is to show that, for $q$ large enough, with probability at
least $1-\eta$ there exists a coupling $\nu$ between
$\mu_{K(\mathfrak{C}_{n}^{(q)})}$ and
$\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}$ such that, for the distance
$\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}}$, we have, with
the notation of Section 2.2,

$\nu\Big{(}(V_{q}\times V_{q})_{\epsilon}\Big{)}\geqslant 1-\epsilon.$

Let $\beta=2\mathbb{E}(\nu_{\star})$, and let
$\bar{\epsilon}=\epsilon/(\beta+1)$. By Lemma 46 there exists $q_{1}$ such
that, for $q\geqslant q_{1}$, with probability at least $1-\eta/2$, there
holds the event $\mathcal{F}_{q}$ that there exists a coupling $\nu$ between
$\mu_{K(\mathfrak{C}_{n}^{(q)})}$ and
$\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}$ such that, for the distance
$\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}}$, we have

$\nu\Big{(}(V_{q}\times V_{q})_{\bar{\epsilon}}\Big{)}\geqslant
1-\bar{\epsilon}.$

Let $\mathcal{E}_{q}$ be the event that
$\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}}(x,y)\leqslant\beta\mathrm{d}_{\mathrm{gr}}(x,y)+q^{1/6}$
for $x,y\in V_{q}$. Then Lemma 44 guarantees that there exists $q_{0}$ such
that $\mathbb{P}(\mathcal{E}_{q})\geqslant 1-\eta/2$ for $q\geqslant q_{0}$.
Let $q_{2}$ be large enough so that
$n^{-1/4}q^{1/6}\leqslant\frac{\epsilon}{\beta+1}$ for $q\geqslant q_{2}$. If
$q\geqslant q_{2}$ and $\mathcal{E}_{q}$ holds, then for every $x,y\in V_{q}$,
we have that $n^{-1/4}\mathrm{d}_{\mathrm{gr}}(x,y)\leqslant\bar{\epsilon}$
implies
$n^{-1/4}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}}(x,y)\leqslant\epsilon$.
Hence, if $q\geqslant q_{2}$ and if $\mathcal{E}_{q}$ and $\mathcal{F}_{q}$
hold, then with the same coupling $\nu$ as above, but now for the distance
$\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}}$, we have

$\nu\Big{(}(V_{q}\times V_{q})_{\epsilon}\Big{)}\geqslant 1-\epsilon.$

This concludes the proof, since the event $\mathcal{E}_{q}\cap\mathcal{F}_{q}$
occurs with probability at least $1-\eta$ for
$q\geqslant\mathrm{max}(q_{0},q_{1},q_{2})$. ∎

######  Lemma 48.

The GHP distance between
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}},\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime})$
and
$(\mathfrak{C}_{n}^{(q)},\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{\mathfrak{C}_{n}^{(q)}})$
is $o(1)$ in probability.

###### Proof.

We follow the argument of projection of masses given in [2]. Let $\mathcal{R}$
be the correspondence on
$V(K(\mathfrak{C}_{n}^{(q)})),V(\mathfrak{C}_{n}^{(q)})$ where $u\mathcal{R}v$
if $u=v$ or if $v$ is a non-pole vertex in one of the $3$ networks substituted
at the edges incident to $u$. Let $\nu$ be the distribution on pairs in
$\mathcal{R}$ corresponding to drawing a vertex $v\in\mathfrak{C}_{n}^{(q)}$
uniformly, then letting $u=v$ if $v\in K(\mathfrak{C}_{n}^{(q)})$, and
otherwise letting $u$ be a random extremity of the edge $e\in
K(\mathfrak{C}_{n}^{(q)})$ in which the network containing $v$ is substituted.
Clearly, $\nu$ gives a coupling between $\mu_{\mathfrak{C}_{n}^{(q)}}$ and
$\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime}$.

For any $\epsilon>0$, by Lemma 10 and Proposition 13, and the union bound, we
have a.a.s. the property that all networks attached to the edges of
$K(\mathfrak{C}_{n}^{(q)})$ have diameter at most
$n^{(2/3+\epsilon)(1/4+\epsilon)}$. Taking $\epsilon$ small enough (e.g.
$\epsilon=1/30$) so that $(2/3+\epsilon)(1/4+\epsilon)\leqslant 1/5$, we thus
have the a.a.s. property $\mathcal{E}_{q}$ that all networks attached to the
edges of $K(\mathfrak{C}_{n}^{(q)})$ have diameter at most $n^{1/5}$. When
$\mathcal{E}_{q}$ holds, the distorsion $\mathrm{dis}(\mathcal{R})$ with
respect to $\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}}$ thus satisfies
$\mathrm{dis}(\mathcal{R})\leqslant 2\,n^{1/5}n^{-1/4}=2\,n^{-1/20}$. Since
the coupling gives zero probability to pairs that are not in $\mathcal{R}$, we
conclude that, when $\mathcal{E}_{q}$ holds, the GHP distance between
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}},\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime})$
and
$(\mathfrak{C}_{n}^{(q)},\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{\mathfrak{C}_{n}^{(q)}})$
is at most $2\,n^{-1/20}=o(1)$. Since $\mathcal{E}_{q}$ holds a.a.s., the GHP
distance between
$(K(\mathfrak{C}_{n}^{(q)}),\frac{1}{n^{1/4}}\operatorname{dist}_{\mathfrak{C}_{n}^{(q)}},\mu_{K(\mathfrak{C}_{n}^{(q)})}^{\prime})$
and
$(\mathfrak{C}_{n}^{(q)},\frac{1}{n^{1/4}}\mathrm{d}_{\mathrm{gr}},\mu_{\mathfrak{C}_{n}^{(q)}})$
is $o(1)$ in probability. ∎

Proposition 45 then directly follows from Lemma 47 and Lemma 48 (and the fact
that the GHP distance satisfies the triangle inequality).

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