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Mathematische Zeitschrift

, Volume 254, Issue 4, pp 837–870 | Cite as

Large deviation and the tangent cone at infinity of a crystal lattice

  • Motoko KotaniEmail author
  • Toshikazu Sunada
Article

Abstract

We discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as we shall observe, has remarkable combinatorial features, and shows up also in the Gromov-Hausdorff limit of a crystal lattice.

Keywords

Random Walk Crystal Lattice Extreme Point Tangent Cone Large Deviation Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Anantharaman, N.: Counting geodesics which are optimal in homology. Ergod. Th. & Dynam. Sys. 23, 353–388 (2003)Google Scholar
  2. 2.
    Babillot, M.: Théorie du renouvellement pour des chaines semi-markoviennes transients. Ann. Inst. Poincaré Probab. Statist. 24, 507–569 (1988)Google Scholar
  3. 3.
    Babillot, M., Ledrappier, F.: Lalley's theorem on periodic orbits of hyperbolic flows. Ergod. Th. & Dynam. Sys. 18, 17–39 (1998)Google Scholar
  4. 4.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York, 1998Google Scholar
  5. 5.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. I, Commun. Pur Appl. Math. 28, 1–47 (1975)Google Scholar
  6. 6.
    Ellis, R.S.: Large deviations for a general class of random vectors. Ann. Prob. 12, 1–12 (1984)Google Scholar
  7. 7.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, 1999Google Scholar
  8. 8.
    Gromov, M., Lafontaine, J., Pansu, P.: Structures métrique pour les variétés riemaniennes. Cedic / Fernand Nathan, Paris, 1981Google Scholar
  9. 9.
    Kirsch, W., Simon, B.: Comparison theorem for the gap of Schrödeinger operator. J. Funct. Anal. 75, 396–410 (1987)Google Scholar
  10. 10.
    Kotani, M.: An asymptotic of the large deviation for random walks on a crystal lattice. Contemporary Math. 347, 141–151 (2004)Google Scholar
  11. 11.
    Kotani, M., Shirai, S., Sunada, T.: Asymptotic behavior of the transition probability of a random walk on an infinite graph. J. Funct. Anal. 159, 664–689 (1998)Google Scholar
  12. 12.
    Kotani, M., Sunada, T.: Jacobian tori associated with a finite graph and its abelian covering graphs. Advances in Apply. Math. 24, 89–110 (2000)Google Scholar
  13. 13.
    Kotani, M., Sunada, T.: Albanese maps and off diagonal long time asymptotics for the heat kernels. Comm. Math. Phys. 209, 633–670 (2000)Google Scholar
  14. 14.
    Kotani, M., Sunada, T.: Standard realizations of crystal lattices via harmonic maps. Trans. Amer. Math. Soc. 353, 1–20 (2000)Google Scholar
  15. 15.
    Kotani, M., Sunada, T.: Spectral geometry of crystal lattices. Contemporary Math. 338, 271–305 (2003)Google Scholar
  16. 16.
    Lin, V.Ya., Pinchover, Y.: Manifolds with group actions and elliptic operators. Memoirs of Amer. Math. Soc. 112(540), (1994)Google Scholar
  17. 17.
    Manabe, S.: Large deviation for a class of current-valued processes. Osaka J. Math. 29, 89–102 (1992)Google Scholar
  18. 18.
    Pansu, P.: Croissance de boules et des géodésiques fermées dans les nilvariétés. Ergod. Th. & Dynam. Sys. 3, 415–445 (1983)Google Scholar
  19. 19.
    Pansu, P.: Profil isopérimétrique, métiriques périodiques et formes d'équilibre des cristaux. ESAIM Control Optim. Calc. Var. 4, 631–665 (1999)Google Scholar
  20. 20.
    Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structur of hyperbolic dynamics. Asterisque 187–188 (1990)Google Scholar
  21. 21.
    Rockafellar, R.T.: Convex Analysis. Princeton Univ. Press, 1972Google Scholar
  22. 22.
    Shirai, T.: Long time behavior of the transition probability of a random walk with drift on an abelian covering graph. Tohoku Math. J. 55, 255–269 (2003)Google Scholar
  23. 23.
    Woess, W.: Random Walks on Infinite Graphs and Groups, 138. Cambridge Univ. Press, Cambridge 2000Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan
  2. 2.Department of MathematicsMeiji UniversityHigashi-Mita, Tama, KawasakiJapan

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