Construction of a short path in high dimensional First Passage Percolation

Olivier Couronné (1) , Nathanaël Enriquez (1, 2) , Lucas Gerin (1)

1 MODAL'X - Modélisation aléatoire de Paris X
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

For First Passage Percolation in Z^d with large d, we construct a path connecting the origin to {x_1 =1}, whose passage time has optimal order \log d/d. Besides, an improved lower bound for the "diagonal" speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35.

Mots clés: First passage percolation; time constant; limit shape

Domaines: Probabilités [math.PR]

Olivier Couronné, Nathanaël Enriquez, Lucas Gerin. Construction of a short path in high dimensional First Passage Percolation. Electronic Communications in Probability, 2011, 16, pp.22--28. ⟨hal-00512399v2⟩