7 Distances along the half-plane boundary
To fulfill our goal of showing the asymptotic equivalence between the oriented and non-oriented distances in uniform Eulerian triangulations, we need as a technical ingredient some estimates on the (oriented) distances along the boundary of $\mathcal{L}$.
Note that the vertices on $\partial\mathcal{L}$ are of two types, those of coordinates $(i, 0)$ for some $i \in \mathbb{Z}$, and those of coordinates $(i + 1/2, \varepsilon)$, for some $i \in \mathbb{Z}$. To simplify notation, the results in this section only deal with the distances between vertices of the first type, since we are interested in asymptotic estimates, and including the vertices of the second type only adds 1 or 2 to the considered distances. We will lay the stress on this generalization whenever it arises later in the paper.
In the sequel, we will use leftmost mirror geodesics, that were defined in Section 5 for finite cylinder triangulations, and that we generalize now to $\mathcal{L}$. For any $i \in \mathbb{Z}$, the leftmost mirror geodesic from $(i, 0)$ in $\mathcal{L}$ is an infinite path $\omega$ in $\mathcal{L}$, whose reverse is an oriented geodesic, and that visits a vertex $\omega(n)$ in $\mathcal{L}n$ at every step $n \ge 0$. It starts at $(i, 0)$, and is obtained by choosing at step $n+1$ the leftmost edge between $\omega(n)$ and $\mathcal{L}{n+1}$. As before, for $i < j$, the leftmost mirror geodesics from $(i, 0)$ and $(j, 0)$ will coalesce before hitting $\mathcal{L}r$, if and only if all the trees $\mathcal{T}i, \mathcal{T}{i+1}, \dots, \mathcal{T}{j-1}$ all have height strictly smaller than $r$.
7.1 Block decomposition and lower bounds
We first want to obtain lower bounds on the distances along the boundary of $\mathcal{L}$. For that purpose, we adapt the block decomposition of causal triangulations [14, Section 2.1], to $\mathcal{L}$.
Figure 16: The block of height 3 between $\mathcal{T}_0$ and $\mathcal{T}_1$ in the triangulation of Figure 15. As before, ghost modules are shown in pale grey.
For $r \ge 1$, we define the random map $G_r$ to be the planar map obtained from $\mathcal{L}_{[0,r]}$ by keeping only the faces and edges that are between $\mathcal{T}0$ and $\mathcal{T}{i_r}$, where $i_r$ is the smallest integer $i > 0$ such that $\mathcal{T}_i$ has height at least $r$. More precisely, we only keep the skeleton modules that are at height smaller than or equal to $r$, belonging to trees $\mathcal{T}i$, with $0 \le i \le i_r$, and the slots that are to the left of all these skeleton modules (see Figure 16 for an example). Thus, $G_r$ has one boundary that is naturally divided into four parts: the upper and lower parts that it shares with $\mathcal{L}{[0,r]}$, and the left and right parts.
Note that $\mathcal{L}$ contains a lot of submaps that have the same law as $G_r$: if $\mathcal{T}_i, \mathcal{T}_j$ are two