Figure 14: Some leftmost mirror geodesics (depicted in blue) in a portion of an Eulerian cylinder triangulation.
6 The Lower Half-Plane Eulerian Triangulation
We now construct a triangulation of the lower half-plane $\mathbb{R} \times \mathbb{R}_{-}$ that will be crucial to prove Theorem 1.2, and that also is an object of interest in itself. Note that this construction is very similar that of the LHPT in [13, Section 3.2].
We start with a doubly infinite sequence $(\mathcal{T}i){i \in \mathbb{Z}}$ of independent Galton-Watson trees with offspring distribution $\theta$. They are embedded in the lower half-plane so that, for every $i \in \mathbb{Z}$, the root of $\mathcal{T}i$ is $(\frac{1}{2} + i, 0)$, and such that the collection of all vertices of all the $\mathcal{T}i$ is $(\frac{1}{2} + \mathbb{Z}) \times \mathbb{Z}{\le 0}$, with vertices at height $k$ being of the form $(\frac{1}{2} + i, -k)$. We also assume that the embedding is such that the collection of vertices of the $\mathcal{T}i$, for $i \ge 0$, is $(\frac{1}{2} + \mathbb{Z}{\ge 0}) \times \mathbb{Z}{\le 0}$ (see Figure 15).
We can now build the triangulation itself. We start with the “distinguished” modules, which will play the role of skeleton modules for our infinite triangulation. They are naturally associated with the vertices of the infinite collection of trees in the following way. To each vertex $(\frac{1}{2} + i, j)$ in one of the trees, we associate a module whose type $n+1$ vertices are $(i,j)$ and $(i+1,j)$. The type $n$ vertex is $(k, j-1)$, where $k$ is the minimal integer such that $(\frac{1}{2} + k, j-1)$ is the child of $(\frac{1}{2} + i', j)$, for some $i' > i$. The last vertex, of type $n+2$, is set to be $(\frac{1}{2} + i, j + \varepsilon)$, for an arbitrary $0 < \varepsilon < 1$. As for the (outer) edges of these skeleton modules, we draw them such that they are all distinct, and do not cross. Having completely determined the configuration of the skeleton edges from the infinite collection of trees, we fill in the slots bounded by these modules, with independent Boltzmann Eulerian triangulation of appropriate perimeters. (Note that each point of the form $(i,j)$, with $j \ge 1$, is at the top of a slot of perimeter $2(c_{i,j} + 1)$, where $c_{i,j}$ is the number of children of $(\frac{1}{2} + i, j)$ in the infinite collection of trees.)
We obtain an Eulerian triangulation of the lower-half plane, which we will note $\mathcal{L}$ and call the Lower Half-Plane Eulerian Triangulation (LHPET). It is rooted at the edge from $(0,0)$ to $(\frac{1}{2}, \varepsilon)$.
We will denote by $\mathcal{L}{[0,r]}$ the infinite rooted planar map obtained by keeping only the first $r$ layers of $\mathcal{L}$ (having the skeleton modules at level $r$ as ghost modules), and denote by $\mathcal{L}r$ the lower boundary of $\mathcal{L}{[0,r]}$. For integers $0 \le m < n$, we also define $\mathcal{L}{[m,n]}$, to be