Stretch Factor in a Planar Poisson-Delaunay Triangulation with a Large Intensity

Let $X := X_n ∪ \{(0, 0), (1, 0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with $X$ when the intensity of $X_n$ goes to infinity. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to $35/3\pi^2$, giving an upper bound for the expected length of the smallest path.