Jérémie BettinelliÉcole polytechniqueLaboratoire d'informatique (LIX) 91128 Palaiseau Cedex FRANCE E-mail: firstname « . » lastname « at » normalesup « . » org |
Geodesics |
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As stated before, we often look at random maps as random metric spaces. Geodesics (or shortest paths) in these spaces then play a crucial part (see for example Geodesics in Brownian surfaces (Brownian maps) for a study of some geodesics in random maps and their scaling limits). Here are some simulations of geodesic cliques in various random maps. Notice that the geodesics from different vertices toward the same vertex have a tendency to merge before reaching their goal. In fact, Le Gall showed that this property always happens in a large plane quadrangulation (and we later generalized this result to quadrangulations of any orientable surface in the previously mentionned reference).
Geodesic triangle 1k [jpg1 – jpg2 – 3D pdf – u3d&tex] Geodesic 5-clique 20k [jpg1 – jpg2 – 3D pdf – u3d&tex] Geodesic 6-clique 30k [jpg1 – jpg2] Geodesic 6-clique 30k_250 [3D pdf – u3d&tex] |
Geodesic cliques between uniform vertices in uniform random maps |