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Volume growth and spectrum for general graph Laplacians

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Abstract

We prove estimates relating exponential or sub-exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph Laplacians. The volume growth is computed with respect to a metric adapted to the Laplacian, and use of these metrics produces better results than those obtained from consideration of the graph metric only. Conditions for absence of the essential spectrum are also discussed. These estimates are shown to be optimal or near-optimal in certain settings and apply even if the Laplacian under consideration is an unbounded operator.

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References

  1. Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178, 501–508 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cartier, P.: Fonctions harmoniques sur un arbre. Symposia Mathematica, vol. ix (Convegno di Calcolo delle Probabilitá, INDAM, Rome, 1971), pp. 203–270. Academic Press, London (1972)

  3. Davies, E.B.: Analysis on graphs and noncommutative geometry. J. Funct. Anal. 111, 398–430 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Davies, E.B.: Large deviations for heat kernels on graphs. J. Lond. Math. Soc. 47, 65–72 (1993)

    Article  MATH  Google Scholar 

  5. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: “Geometry of random motion” Contemporary Mathematics. Am. Math. Soc. 73, 25–40 (1988)

    Google Scholar 

  6. Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks on graphs. Elec. J. Prob. 62, 1693–1722 (2011)

    MathSciNet  Google Scholar 

  7. Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. (To appear)

  8. Frank, R.L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. (To appear)

  9. Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku. Math. J. 48, 293–302 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fujiwara, K.: The Laplacian on rapidly branching trees. Duke Math. J. 83, 191–202 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  12. Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)

    Google Scholar 

  13. Huang, X.: On uniqueness class for a heat equation on graphs. J. Math. Anal. Appl. 393, 377–388 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Haeseler, S., Keller, M., Wojciechowski, R.: Volume growth and bounds for the essential spectrum for Dirichlet forms. J. London. Math. Soc. arXiv: 1205.4985v1

  15. Keller, M.: The essential spectrum of the Laplacian on rapidly branching tesselations. Math. Ann. 346, 51–66 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)

    Google Scholar 

  17. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5, 198–224 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Keller, M., Lenz, D., Wojciechowski, R.: Volume growth, spectrum, and stochastic completeness of infinite graphs. Math. Z. (To appear)

  19. Keller, M., Lenz, D., Vogt, H., Wojciechowski, R.: Note on basic features of large time behaviour of heat kernels. Preprint

  20. Sturm, K.-T.: Analysis on local Dirichlet spaces I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine. Angew. Math. 456, 173–196 (1994)

    MATH  MathSciNet  Google Scholar 

  21. Sunada, T.: Fundamental Groups and Laplacians, Lecture Notes in Mathematics 1339. Springer, New York (1988)

    Google Scholar 

  22. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370, 146–158 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was completed at the Statistical Laboratory at the University of Cambridge during a visit from the author. The author also thanks Matthias Keller for some helpful comments concerning the relation between positivity of the bottom of the spectrum, discreteness of the spectrum, and stochastic completeness.

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Authors and Affiliations

  1. Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Matthew Folz

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  1. Matthew Folz
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Correspondence to Matthew Folz.

Additional information

Research supported by a NSERC Alexander Graham Bell Canada Graduate Scholarship and a NSERC Michael Smith Foreign Study Supplement.

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Folz, M. Volume growth and spectrum for general graph Laplacians. Math. Z. 276, 115–131 (2014). https://doi.org/10.1007/s00209-013-1189-y

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  • DOI: https://doi.org/10.1007/s00209-013-1189-y

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