Abstract
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.
Itai Benjamini and Ofer Zeitouni
Both authors were supported by their respective Israel Science Foundation grants.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
I. Benjamini, G. Kalai, O. Schramm, First passage percolation has sublinear distance variance. Ann. Prob. 31, 1970–1978 (2003)
S. Bhamidi, R. van der Hofstad, G. Hooghiemstra, First passage percolation on the Erdos-Rényi random graph. Combin. Probab. Comput. 20, 683–707 (2011)
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)
D. Burago, Y. Burago, S. Ivanov, in A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001)
S. Chatterjee, P. Dey, Central limit theorem for first-passage percolation time across thin cylinders. Preprint (2010). http://arxiv.org/abs/0911.5702. To appear, Prob. Th. Rel. Fields (2012)
M. Dekking, B. Host, Limit distributions for minimal displacement of branching random walks. Probab. Theor. Relat. Fields 90, 403–426 (1991)
R. Grigorchuk, I. Pak, Groups of intermediate growth: An introduction for beginners. Enseign. Math. (2) 54, 251–272 (2008). http://arxiv.org/abs/math/0607384
K. Johansson, On some special directed last-passage percolation models. Contemp. Math. 458, 333–346 (2008)
H. Kesten, in Aspects of First Passage Percolation. Lecture Notes in Math., vol. 1180 (Springer, Berlin, 1986), pp. 125–264
R. Lyons, Y. Peres, Probability on Trees and Networks. In preparation, to be published by Cambridge University Press. Current version available at http://mypage.iu.edu/~rdlyons/.
V. Nekrashevych, Self-Similar Groups. A.M.S. Mathematical Surveys and Monographs, vol. 117 (2005), Providence, RI
C. Newman, M. Piza, Divergence of shape fluctuations in two dimensions. Ann. Probab. 23, 977–1005 (1995)
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, 261–264
Acknowledgements
Thanks to Pierre Pansu and Gabor Pete for very useful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Benjamini, I., Zeitouni, O. (2012). Tightness of Fluctuations of First Passage Percolation on Some Large Graphs. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-29849-3_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29848-6
Online ISBN: 978-3-642-29849-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)