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

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Eckhard Schlemm
Eckhard Schlemm*
Affiliation:
Technische Universität München
*
∗ Postal address: TUM Institute for Advanced Study & Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany. Email address: schlemm@ma.tum.de
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Abstract

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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 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 → ∞. 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.

Keywords

Central limit theoremfirst passage percolationgenerating functionMarkov chaintransition kernel

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J05: Discrete-time Markov processes on general state spaces 33C90: Applications

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 366 - 388
Copyright
Copyright © Applied Probability Trust 2011 

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