Construction of a short path in high dimensional First Passage Percolation
Abstract
For First Passage Percolation in Zd with large d, we construct a path connecting the origin to x1 =1, whose passage time has optimal order 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.
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