First-passage percolation on width-two stretches with exponential link weights

We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set {1,...,n}\times{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).

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