A Central Limit Theorem for First Passage Percolation in the Slab

We consider first-passage percolation on the edges of $\mathbb{Z}^2 \times \{1, \cdots, k\},$ namely the slab $\mathbb{S}_k$ of width $k$. Each edge is assigned independently a passage time of either 0 (with probability $p_c(\mathbb{S}_k)$) or 1 ( with probability $1-p_c(\mathbb{S}_k)$) where $p_c$ is the critical probability for Bernouilli percolation. We prove central limit theorems for point-to-point and point-to-line passage times. These generalize the results of Kesten and Zhang \cite{kestenzhang} to non-planar graphs.