Electronic Communications in Probability

First-passage percolation on width-two stretches with exponential link weights

Eckhard Schlemm

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We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set $\{1,\dots,n\}\times\{0,1\}$ and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate $\chi$ by solving certain recursive distributional equations and invoking results from ergodic theory to identify $\chi$ as the expected asymptotic one-step growth of the first-passage time from $(0, 0)$ to $(n, 0)$.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 41, 424-434.

Accepted: 6 October 2009
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

First-passage percolation percolation rate Markov chains ergodicity

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Schlemm, Eckhard. First-passage percolation on width-two stretches with exponential link weights. Electron. Commun. Probab. 14 (2009), paper no. 41, 424--434. doi:10.1214/ECP.v14-1493. https://projecteuclid.org/euclid.ecp/1465234750

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  • M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Books on Advanced Mathematics, New York: Dover,| c1965, Corrected edition, edited by Abramowitz, Milton; Stegun, Irene A., 1965.
  • Breiman, Leo. Probability. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont. 1968 ix+421 pp.
  • Durrett, Richard. Probability. Theory and examples. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. x+453 pp. ISBN: 0-534-13206-5
  • A. Flaxman, D. Gamarnik, and G.B. Sorkin. First-Passage Percolation on a Width-2 Strip and the Path Cost in a VCG Auction. LECTURE NOTES IN COMPUTER SCIENCE, 4286:99, 2006.
  • R.L. Graham, M. Grötschel, and L. Lovász. Handbook of Combinatorics. MIT Press, 1995.
  • R. Jentzsch. Ãœber Integralgleichungen mit positivem Kern. J. Reine Angew. Math, 141:235–244, 1912.
  • Kesten, Harry. Aspects of first passage percolation. École d'été de probabilités de Saint-Flour, XIV-1984, 125–264, Lecture Notes in Math., 1180, Springer, Berlin, 1986.
  • E. Schlemm. First-passage percolation on width-two stretches. Master's thesis, Freie Universität Berlin, June 2008.