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Construction of a short path in high-dimensional first passage percolation. (English) Zbl 1231.60109
Summary: For first passage percolation in $$\mathbb Z^{d}$$ 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 D. Dhar [“First passage percolation in many dimensions”, Phys. Lett. A 130, No. 4–5, 308–310 (1988; doi:10.1016/0375-9601(88)90616-0)] shows that the limiting shape in FPP with exponential passage times (and thus that of Eden’s model [M. Eden, Proc. 4th Berkeley Symp. Math. Stat. Probab. 4, 223-239 (1961; Zbl 0104.13801)]) is not the Euclidean ball in dimension larger than 35.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
##### Keywords:
first passage percolation; time constant; limit shape
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