Figure 12: The geodesic oriented distances in an Eulerian triangulation $A$ with a (distinguished) alternating boundary $\partial_0 A$ (left) are given by the distances from the root in the triangulation $A'$, obtained from $A$ by gluing into pairs the edges of $\partial_0 A$ (right). The triangles adjacent to the boundary are depicted in black and white, the rest of triangulation is sketched in gray, and the boundary is in red.
Now that we have a satisfactory notion of distance, as before, we will be interested in the union of faces of $A$ incident to vertices at (oriented) distance less than $n$ from the boundary. We will denote this union $B_{n+1}(A)$. As was the case for usual balls, the faces of $B_{n+1}(A)$ adjacent to its boundary parts, other than the original boundary $\partial A$, will correspond to modules of type $n+1$, for the oriented distance from $\partial_0 A$. Once again, we will have the convention that these boundary parts are simple, and we will glue them to semi-simple boundaries. If $A$ is pointed at a vertex $v$ at oriented distance at least $n+2$ from the boundary, we can also define a notion of hull for $B_{n+1}(A)$, which will be an Eulerian triangulation with two boundaries of specific types. In this section, we will first develop the description of such triangulations, before dealing with random Eulerian triangulations with one boundary, and their hulls.
Note that the chain of arguments and notation of this section follow closely those of [13, Section 5]: in order to be both concise and precise, we detail in the proofs of this section only the additional subtleties and difficulties arising in our case compared to the equivalent results of [13].
5.1 Cylinder triangulations
Definition 5.1. We call Eulerian cylinder triangulation of height $r \ge 1$, an Eulerian triangulation with two boundaries, one (the bottom of the cylinder) being alternating and semi-simple, the other one (the top) being a succession of modules (see Figure 13), and such that any module adjacent to the top boundary is of distance type r with respect to the bottom.
We denote by $\partial\Delta$ its bottom boundary, and by $\partial^*\Delta$ its top boundary. The root is an edge on $\partial\Delta$ oriented such that the bottom face sits on its right.
Let $\Delta$ be an Eulerian cylinder triangulation of height $r$. Let $2p$ be the bottom boundary length, and $2q$ the top boundary length. For $1 \le j \le r$, the ball $B_j(\Delta)$ is defined as the union of all edges and faces of $\Delta$ incident to at least a vertex at distance $<j$ from the bottom boundary, and the hull $B_j^\bullet(\Delta)$ is obtained from $B_j(\Delta)$ by adding all the connected components of its complement except the one containing the top boundary. Therefore $B_j^\bullet(\Delta)$ is a cylinder triangulation of height $j$, and we denote by $\partial_j\Delta$ the set of modules adjacent to its top boundary. Let $\mathcal{M}(\Delta)$ be the set of modules of $\Delta$ belonging to some