On the Markov Transition Kernels for First Passage Percolation on the Ladder

Abstract

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

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

“…The asymptotic results derived in the present study greatly generalizes earlier results by Schlemm [28], who proved a central limit theorem for first-passage percolation on the (2, 2)tube with exponential weights. A central limit theorem and Donsker theorem have also been obtained independently, and by different methods, by Chatterjee and Dey [7], which we further comment upon below.…”

“…The claim follows from a straightforward inversion argument, which is found in [2]. Theorem 1.2 was also obtained by Schlemm [28] in the particular case of the (2, 2)-tube with exponential passage times. A variant of Theorem 1.2 was proved by other means in an independent work by Chatterjee and Dey [7].…”

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

“…The claim follows from a straightforward inversion argument, which can be found in Ahlberg (2011). Theorem 1.3 was also obtained by Schlemm (2011) in the particular case of the (2, 2)-tube with exponential passage times. A variant of Theorem 1.3 was proved by yet other means in an independent work by Chatterjee and Dey (2012).…”