Construction of a short path in high-dimensional first passage percolation

Olivier Couronné, Nathanaël Enriquez, Lucas Gerin
2011 Electronic Communications in Probability  
For first passage percolation in 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.
doi:10.1214/ecp.v16-1595 fatcat:a3r3s5pmnvhdjhafau6lkxpdaa