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with a boundary.

We will say a forest $\mathcal{F}$ with a distinguished vertex is a $(p, q, r)$-admissible forest if it consists of an ordered sequence $(\tau_1, \tau_2, \dots, \tau_q)$ of $q$ rooted plane trees of maximal height $r$, with $p$ vertices at height $r$, with the distinguished vertex at height $r$ in $\tau_1$.

If $\mathcal{F}$ is a $(p, q, r)$-admissible forest, we write $\mathcal{F}^*$ for the set of all vertices of $\mathcal{F}$ at height strictly smaller than $r$.

From the preceding decomposition, we obtain the following result:

Proposition 5.2. The Eulerian triangulations of the cylinder $\Delta$ of height $r$ with a bottom boundary length $2p$ and a top boundary length $2q$, are in bijection with pairs consisting of a $(p, q, r)$-admissible forest $\mathcal{F}$ and a collection $(M_v)_{v \in \mathcal{F}^*}$ such that, for every $v \in \mathcal{F}^*$, $M_v$ is an Eulerian triangulation of the $2(c_v + 1)$-gon with a semi-simple alternating boundary, with $c_v$ being the number of children of $v$ in $\mathcal{F}$.

Note that the bijection of Proposition 5.2 is an adaptation of similar constructions that have been made for usual triangulations and quadrangulations, starting with Krikun's works [18, 17], with more recent versions by Curien and Le Gall for usual triangulations [13], and by Le Gall and Lehéricy for quadrangulations [21]. Following the vocabulary used in these works, we call this bijection the skeleton decomposition, and say that $\mathcal{F}$ is the skeleton of the triangulation $\Delta$. We will also call skeleton modules the modules of $\mathcal{M}(\Delta)$.

5.2 Skeleton decomposition of random triangulations

We will now use the bijection derived in Section 5.1 to obtain the asymptotic behavior of the laws of the hulls of random uniform Eulerian triangulations with a boundary.

We first need a bit of additional notation.

Consider an Eulerian triangulation with a boundary $\Delta$, pointed at $v$. We can define the hull $B_r^\bullet(\Delta)$ of $\Delta$ like for cylinder triangulations, if $\bar{d}(\partial\Delta, v) > r+1$. If $\bar{d}(\partial\Delta, v) \le r+1$, we can set $B_r^\bullet(\Delta) = \Delta$.

Let $\mathcal{T}_n^{(p)}$ be a uniform random triangulation over the set of Eulerian triangulations with a semi-simple alternating boundary of length $2p$ and with $n$ black triangles. We denote by $\overline{\mathcal{T}}_n^{(p)}$ the pointed triangulation obtained by choosing a uniform random inner vertex of $\mathcal{T}n^{(p)}$. Let $\Delta$ be a cylinder triangulation of height $r$, of respective bottom and top boundary lengths $2p$ and $2q$, with $N$ black triangles, with $n \ge N$. Using the skeleton decomposition, we associate to $\Delta$ a $(p, q, r)$-admissible forest $\mathcal{F}$, together with triangulations $(M_v){v \in \mathcal{F}}$ filling in the "slots" between the modules of $\mathcal{M}(\Delta)$. We write $N(M_v)$ for the number of black triangles of $M_v$, for every $v \in \mathcal{F}^*$.

Lemma 5.3. We have

$$\underset{n\to \mathrm{\infty }}{\lim }\mathbb{P}(B_r^{\bullet }\left({\bar{\mathcal{T}}}_n^{(p)}\right)=\mathrm{\Delta })=\frac{4^{-q}C(q)}{4^{-p}C(p)}\prod _{v\in {\mathcal{F}}^{\ast }}\theta (c_v)\frac{8^{-N(M_v)}}{Z(c_v+1)},(5.1)\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash mathbb\{P\}\; \backslash left(\; B\_r^\backslash bullet\; \backslash left(\; \backslash overline\{\backslash mathcal\{T\}\}\_n^\{(p)\}\; \backslash right)\; =\; \backslash Delta\; \backslash right)\; =\; \backslash frac\{4^\{-q\}\; C(q)\}\{4^\{-p\}\; C(p)\}\; \backslash prod\_\{v\; \backslash in\; \backslash mathcal\{F\}^*\}\; \backslash theta(c\_v)\; \backslash frac\{8^\{-N(M\_v)\}\}\{Z(c\_v\; +\; 1)\},\; \backslash quad\; (5.1)$$

where

$$\theta (k)=\frac{1}{8}{4}^{-k+1}Z(k+1),(5.2)\backslash theta(k)\; =\; \backslash frac\{1\}\{8\}\; 4^\{-k+1\}\; Z(k+1),\; \backslash quad\; (5.2)$$

with $Z(k)$ defined as in (4.6).

Proof. First note that this result is the equivalent of [13, Lemma 2]. It is obtained very similarly, though in our case we start from a slightly less explicit expression, as shown in (5.3): this stems from the fact that our triangulations do not necessarily have simple boundaries.