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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: [email protected]
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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 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.

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

References

[1] Abramowitz, M. and Stegun, I. A. (eds) (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
[2] Ahlberg, D. (2009). Asymptotics of first-passage percolation on 1-dimensional graphs. Preprint. Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University.Google Scholar
[3] Aldous, D., Lovász, L. and Winkler, P. (1997). Mixing times for uniformly ergodic Markov chains. Stoch. Process. Appl. 71, 165–185.CrossRefGoogle Scholar
[4] Altmann, M. (1993). Reinterpreting network measures for models of disease transmission. Social Networks 15, 1–17.CrossRefGoogle ScholarPubMed
[5] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). {First passage percolation on random graphs with finite mean degrees}. Ann. Appl. Prob. 20, 1907–1907.Google Scholar
[6] Chatterjee, S. and Dey, P. S. (2009). Central limit theorem for first-passage percolation time across thin cylinders. Preprint. Available at http://arxiv.org/abs/0911.5702v2.Google Scholar
[7] Chen, X. (1999). Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139, 203 pp.Google Scholar
[8] Flaxman, A., Gamarnik, D. and Sorkin, G. (2006). First-passage percolation on a width-2 strip and the path cost in a VCG auction. In Internet and Network Economics (Lecture Notes Comput. Sci. 4286), Springer, Berlin, pp. 99–111.CrossRefGoogle Scholar
[9] Graham, R. L., Grötschel, M. and Lovász, L. (eds) (1995). Handbook of Combinatorics, Vol. 2. North-Holland, Amsterdam.Google Scholar
[10] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrscheinlichkeitsth. 66, 335–366.CrossRefGoogle Scholar
[11] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernouli, Bayes, Laplace. Anniversary Volume, Springer, New York, pp. 61–110.Google Scholar
[12] Kesten, H. (1987). Percolation theory and first-passage percolation. Ann. Prob. 15, 1231–1271.Google Scholar
[13] Nummelin, E. and Tuominen, P. (1982). Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl. 12, 187–202.Google Scholar
[14] Renlund, H. (2010). First-passage percolation with exponential times on a ladder. Combinatorics Prob. Comput. 19, 593–601.Google Scholar
[15] Schlemm, E. (2009). {First-passage percolation rates on width-two stretches with exponential link weights}. Electron. Commun. Prob. 140, 424–434.Google Scholar
[16] Seppäläinen, T. (1998). Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Prob. 26, 1232–1250.Google Scholar
[17] Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice (Lecture Notes Math. 671). Springer, Berlin.Google Scholar
[18] Sood, V., Redner, S. and ben Avraham, D. (2005). First-passage properties of the Erdős-Renyi random graph. J. Phys. A 38, 109–123.CrossRefGoogle Scholar
[19] Van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2001). First-passage percolation on the random graph. Prob. Eng. Inf. Sci. 15, 225–237.Google Scholar
[20] Wilf, H. S. (2006). Generatingfunctionology, 3rd edn. A K Peters, Wellesley, MA.Google Scholar