On the Markov transition kernels for first-passage percolation on the ladder

TL;DR

This paper studies first-passage percolation on a ladder graph with exponential weights, proving a central limit theorem for passage times and explicitly calculating the asymptotic variance using Markov chain transition kernels.

Contribution

It extends previous results by establishing a CLT for passage times and provides explicit formulas for transition kernels and asymptotic variance.

Findings

Proves a central limit theorem for first-passage times on the ladder graph.

Provides explicit computation of transition kernels for a related Markov chain.

Calculates the asymptotic variance in the CLT explicitly.

Abstract

We consider the first-passage percolation problem on the random graph with vertex set N\times{0,1}, edges joining vertices at Euclidean distance equal to unity and independent exponential edge weights. We provide a central limit theorem for the first-passage times l_n between the vertices (0,0) and (n,0), thus extending earlier results about the almost sure convergence of l_n/n as n goes to infinity. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to calculate explicitly the asymptotic variance in the central limit theorem.