Journal of Applied Probability

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

Eckhard Schlemm

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Appl. Probab., Volume 48, Number 2 (2011), 366-388.

Dates
First available in Project Euclid: 21 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1308662633

Digital Object Identifier
doi:10.1239/jap/1308662633

Mathematical Reviews number (MathSciNet)
MR2840305

Zentralblatt MATH identifier
1223.60054

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J05: Discrete-time Markov processes on general state spaces
Secondary: 33C90: Applications

Keywords
Central limit theorem first passage percolation generating function Markov chain transition kernel

Citation

Schlemm, Eckhard. On the Markov transition kernels for first passage percolation on the ladder. J. Appl. Probab. 48 (2011), no. 2, 366--388. doi:10.1239/jap/1308662633. https://projecteuclid.org/euclid.jap/1308662633


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (eds) (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
  • Ahlberg, D. (2009). Asymptotics of first-passage percolation on 1-dimensional graphs. Preprint. Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University.
  • Aldous, D., Lovász, L. and Winkler, P. (1997). Mixing times for uniformly ergodic Markov chains. Stoch. Process. Appl. 71, 165–185.
  • Altmann, M. (1993). Reinterpreting network measures for models of disease transmission. Social Networks 15, 1–17.
  • Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1965.
  • 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.
  • Chen, X. (1999). Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139, 203 pp.
  • 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.
  • Graham, R. L., Grötschel, M. and Lovász, L. (eds) (1995). Handbook of Combinatorics, Vol. 2. North-Holland, Amsterdam.
  • Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrscheinlichkeitsth. 66, 335–366.
  • 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.
  • Kesten, H. (1987). Percolation theory and first-passage percolation. Ann. Prob. 15, 1231–1271.
  • Nummelin, E. and Tuominen, P. (1982). Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl. 12, 187–202.
  • Renlund, H. (2010). First-passage percolation with exponential times on a ladder. Combinatorics Prob. Comput. 19, 593–601.
  • Schlemm, E. (2009). First-passage percolation rates on width-two stretches with exponential link weights. Electron. Commun. Prob. 140, 424–434.
  • Seppäläinen, T. (1998). Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Prob. 26, 1232–1250.
  • Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice (Lecture Notes Math. 671). Springer, Berlin.
  • Sood, V., Redner, S. and ben Avraham, D. (2005). First-passage properties of the Erdős-Renyi random graph. J. Phys. A 38, 109–123.
  • 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.
  • Wilf, H. S. (2006). Generatingfunctionology, 3rd edn. A K Peters, Wellesley, MA.