First-passage percolation with exponential times on a ladder

TL;DR

This paper analyzes first-passage percolation on a ladder graph with exponential edge times, deriving an explicit formula for the time constant and residual time, revealing the process's long-term average speed.

Contribution

It introduces a Markov chain approach to explicitly compute the time constant and residual time in first-passage percolation on a ladder with exponential times.

Findings

Time constant approximately 0.6827

Explicit expression for the time constant

Calculated average residual time

Abstract

We consider first-passage percolation on a ladder, i.e. the graph {0,1,...}*{0,1} where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is approximately 0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time.