Georgia Institute of Technology College of Sciences

In critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large, it is known that there are a.s. no infinite open clusters. In particular, for n large, every path from the origin to the boundary of $[-n, n]^d$ must contain some closed edges. Let $T_n$ be the (random) minimal number of closed edges in such a path. How does $T_n$ grow with $n$? We present results showing that for d larger than the upper critical dimension for Bernoulli percolation ($d > 6$), $T_n$ is typically of the order $\log \log n$. This is in contrast with the $d = 2$ case, where $T_n$ grows logarithmically. Perhaps surprisingly, the model exhibits another major change in behavior depending on whether $d > 8$.