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

Abstract

We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set $\{1,\dots,n\}\times\{0,1\}$ and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate $\chi$ by solving certain recursive distributional equations and invoking results from ergodic theory to identify $\chi$ as the expected asymptotic one-step growth of the first-passage time from $(0, 0)$ to $(n, 0)$.