A Central Limit Theorem for First Passage Percolation in the Slab

TL;DR

This paper establishes a Central Limit Theorem for first-passage percolation on a slab graph, extending previous planar results to non-planar structures with edges assigned passage times of 0 or 1.

Contribution

It generalizes existing CLT results for first-passage percolation from planar graphs to slab graphs of fixed width, considering a specific Bernoulli distribution of edge weights.

Findings

Proves CLT for point-to-point passage times in slabs.

Proves CLT for point-to-line passage times in slabs.

Extends prior planar results to non-planar graph structures.

Abstract

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