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

Construction of a short path in high dimensional First Passage Percolation