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Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson pointprocess of intensity $n$. We provide a first non-trivial lower bound for thedistance 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. Experimental values indicate that the correct value isabout 1.04. We also prove that the expected number of Delaunay edges crossed bythe line segment $[(0,0),(1,0)]$ is equivalent to $2.16\sqrt{n}$ and that theexpected length of a particular path converges to 1.18 giving an upper bound onthe stretch factor.

Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation