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

TL;DR

This paper analyzes first-passage percolation on a width-two graph with exponential weights, deriving exact asymptotic growth rates for certain configurations using recursive equations and ergodic theory.

Contribution

It provides exact formulas for the asymptotic percolation rate in specific one-dimensional graph cases with exponential link weights, expanding understanding of such models.

Findings

Exact expressions for the asymptotic percolation rate  in three cases

Use of recursive distributional equations to solve the problem

Application of ergodic theory to identify growth rates

Abstract

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 \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).