First-passage percolation on width-two stretches

In this work we consider two first-passage percolation problems. In the first part we concern ourselves with effectively one-dimensional graphs with vertex set {1 . . . , n}×{0, 1} and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate χ by solving certain recursive distributional equations and invoking results from ergodic theory to identify χ as the expected asymptotic one-step growth of the first-passage time from (0, 0) to (n, 0). In the second part we derive an integral equation to characterize the stationary behavior of a related continuous problem.