Tightness of Fluctuations of First Passage Percolation on Some Large Graphs

  • Itai BenjaminiEmail author
  • Ofer Zeitouni
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)


The theorem of Dekking and Host [Probab. Theor. Relat. Fields 90, 403–426 (1991)] regarding tightness around the mean of first passage percolation on the binary tree, from the root to a boundary of a ball, is generalized to a class of graphs which includes all lattices in hyperbolic spaces and the lamplighter graph over \(\mathbb{N}\). This class of graphs is closed under product with any bounded degree graph. Few open problems and conjectures are gathered at the end.


First-passage Percolation (FPP) Lamp Light Bounded Degree Graphs Cayley Graph Subadditive Ergodic Theorem 
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Thanks to Pierre Pansu and Gabor Pete for very useful discussions.


  1. 1.
    I. Benjamini, G. Kalai, O. Schramm, First passage percolation has sublinear distance variance. Ann. Prob. 31, 1970–1978 (2003)Google Scholar
  2. 2.
    S. Bhamidi, R. van der Hofstad, G. Hooghiemstra, First passage percolation on the Erdos-Rényi random graph. Combin. Probab. Comput. 20, 683–707 (2011)Google Scholar
  3. 3.
    E. Bolthausen, J.-D. Deuschel, O. Zeitouni, Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field. Elect. Comm. Prob. 16, 114–119 (2011)Google Scholar
  4. 4.
    D. Burago, Y. Burago, S. Ivanov, in A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001)Google Scholar
  5. 5.
    S. Chatterjee, P. Dey, Central limit theorem for first-passage percolation time across thin cylinders. Preprint (2010). To appear, Prob. Th. Rel. Fields (2012)
  6. 6.
    M. Dekking, B. Host, Limit distributions for minimal displacement of branching random walks. Probab. Theor. Relat. Fields 90, 403–426 (1991)Google Scholar
  7. 7.
    R. Grigorchuk, I. Pak, Groups of intermediate growth: An introduction for beginners. Enseign. Math. (2) 54, 251–272 (2008).
  8. 8.
    K. Johansson, On some special directed last-passage percolation models. Contemp. Math. 458, 333–346 (2008)Google Scholar
  9. 9.
    H. Kesten, in Aspects of First Passage Percolation. Lecture Notes in Math., vol. 1180 (Springer, Berlin, 1986), pp. 125–264Google Scholar
  10. 10.
    R. Lyons, Y. Peres, Probability on Trees and Networks. In preparation, to be published by Cambridge University Press. Current version available at
  11. 11.
    V. Nekrashevych, Self-Similar Groups. A.M.S. Mathematical Surveys and Monographs, vol. 117 (2005), Providence, RIGoogle Scholar
  12. 12.
    C. Newman, M. Piza, Divergence of shape fluctuations in two dimensions. Ann. Probab. 23, 977–1005 (1995)Google Scholar
  13. 13.
    R. Pemantle, Y. Peres, Planar first-passage percolation times are not tight, in Probability and Phase Transition (Cambridge, 1993). Nato Adv. Sci. Inst. Ser. C., Math Phys. Sci., Volume 420, Kluwer Academic Pub., Dordrecht, 261–264Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

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