Construction of a short path in high dimensional First Passage Percolation

TL;DR

This paper constructs a short path in high-dimensional First Passage Percolation with optimal passage time order and shows the limiting shape differs from a Euclidean ball in dimensions over 35.

Contribution

It introduces a new construction of short paths in high-dimensional FPP and provides bounds on the shape of the growth cluster.

Findings

Constructed a path with passage time order log d/d in high dimensions.

Proved the limiting shape is not Euclidean ball in dimensions > 35.

Provided improved lower bounds for the cluster speed.

Abstract

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.