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Annales Henri Poincaré

, Volume 16, Issue 4, pp 1067–1101 | Cite as

Approximation of the Integrated Density of States on Sofic Groups

  • Christoph SchumacherEmail author
  • Fabian Schwarzenberger
Article

Abstract

In this paper, we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases, we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows one to treat applications from long-range percolation and the Anderson model. Our results apply to operators on \({\mathbb{Z}^d}\) , amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without an ergodic theorem at hand.

Keywords

Matrix Element Cayley Graph Ergodic Theorem Amenable Group Anderson Model 
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.

References

  1. 1.
    Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis: A hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)Google Scholar
  2. 2.
    Adachi T., Sunada T.: Density of states in spectral geometry. Comment. Math. Helv. 68(1), 480–493 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ayadi, S., Schwarzenberger, F., Veselić, I.: Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs. Math. Phys. Anal. Geom (2012) online first. doi: 10.1007/s11040-013-9133-2
  4. 4.
    Aizenman M., Warzel S.: The canopy graph and level statistics for random operators on trees. Math. Phys. Anal. Geom. 9(4), 291–333 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Biggs, N.L.: Girth and residual finiteness. Combinatorica 8(4) 307–312 (1988). MR 981888 (90c:05105)Google Scholar
  6. 6.
    Brooks S., Lindenstrauss E.: Non-localization of eigenfunctions on large regular graphs. Israel J. Math. 193(1), 1–14 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bowen L.: Measure conjugacy invariants for actions of countable sofic groups. J. Am. Math. Soc. 23(1), 217–245 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bowen L.: Sofic entropy and amenable groups. Ergod. Theor. Dyn. Syst. 32, 427–466 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bartholdi, L., Woess, W.: Spectral computations on lamplighter groups and Diestel-Leader graphs. J. Fourier Anal. Appl. 11(2), 175–202 (2005). MR 2131635 (2006e:20052)Google Scholar
  10. 10.
    Cycon H., Froese R., Kirsch W., Simon B.: Schrödinger Operators, 3rd edn. Springer, Berlin (2008)Google Scholar
  11. 11.
    Cornulier Y.: A sofic group away from amenable groups. Math. Ann. 350, 269–275 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    de Branges, L.: The Stone-Weierstrass theorem. Proc. Amer. Math. Soc. 10, 822–824 (1959). MR 0113131 (22 #3970)Google Scholar
  13. 13.
    Dodziuk J., Linnell P., Mathai V., Schick T., Yates S.: Approximating L 2-invariants, and the Atiyah conjecture. Commun. Pur. Appl. Math. 56(7), 839–873 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dodziuk, J., Mathai, V.: Approximating l 2 invariants of amenable covering spaces: A heat kernel approach. In: Contemporary Mathematics, vol. 211, pp. 151–167. AMS (1997)Google Scholar
  15. 15.
    Dodziuk J., Mathai V.: Approximating L 2 invariants of amenable covering spaces: a combinatorial approach. J. Funct. Anal. 154(2), 359–378 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Eckmann B.: Approximating 2-Betti numbers of an amenable covering by ordinary Betti numbers. Comment. Math. Helv. 74(1), 150–155 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Elek G., Szabó E.: On sofic groups. J. Group Theory 9(2), 161–171 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Froese R., Hasler D., Spitzer W.: A geometric approach to absolutely continuous spectrum for discrete Schrödinger operators. Prog Probab 64, 201–226 (2011)MathSciNetGoogle Scholar
  19. 19.
    Gromov M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2), 109–197 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Komarov, N., McNeill, R.T., Webster, J.T.: Normal subgroups of the free group. In: Proceedings of the Oregon State University. REU in Mathematics, pp. 61–80 (2007)Google Scholar
  21. 21.
    Lenz D., Müller P., Veselić I.: Uniform existence of the integrated density of states for models on \({{\mathbb{Z}}^d}\) . Positivity 12(4), 571–589 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lenz D., Peyerimhoff N., Veselić I.: Integrated density of states for random metrics on manifolds. Proc. London Math. Soc. 88(3), 733–752 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lenz D., Peyerimhoff N., Veselić I.: Groupoids, von neumann algebras, and the integrated density of states. Math. Phys. Anal. Geom. 10(1), 1–41 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lück, W., Schick, T.: L 2-torsion of hyperbolic manifolds of finite volume. Geom. Funct. Anal. 9(3), 518–567 (1999). MR 1708444 (2000e:58050)Google Scholar
  25. 25.
    Lenz D., Stollmann P.: An ergodic theorem for Delone dynamical systems and existence of the density of states. J. Anal. Math. 97(1), 1–23 (2005)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Lenz, D., Schwarzenberger, F., Veselić, I.: A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states. Geom. Dedicata 150, 1–34 (2011). MR 2753695 (2012c:22010).Google Scholar
  27. 27.
    Lück W.: Approximating l 2-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4(4), 455–481 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    McDiarmid, C.: Concentration. In: Probabilistic Methods for Algorithmic Discrete Mathematics, pp. 1–46 (1998)Google Scholar
  29. 29.
    McKay B.D.: The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40, 203–216 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Mathai V., Schick T., Yates S.: Approximating spectral invariants of Harper operators on graphs II. Proc. Amer. Math. Soc. 131(6), 1917–1923 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Mathai V., Yates S.: Approximating spectral invariants of Harper operators on graphs. J. Funct. Anal. 188(1), 111–136 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Newman M.: Free subgroups and normal subgroups of the modular group. Am. J. Math. 86(1), 262–265 (1964)CrossRefGoogle Scholar
  33. 33.
    Pastur L.A.: Selfaverageability of the number of states of the Schrödinger equation with a random potential. Mat. Fiz. i Funkcional. Anal. 238(2), 111–116 (1971)MathSciNetGoogle Scholar
  34. 34.
    Pestov V.: Hyperliner and sofic groups: a brief guide. B. Symb. Log. 14(4), 449–480 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. In: Grundlehren der mathematischen Wissenschaften, vol. 297. Springer, Berlin (1992)Google Scholar
  36. 36.
    Pogorzelski, F., Schwarzenberger, F.: A banach space-valued ergodic theorem for amenable groups and applications. J. Anal. Math. 55 (2014, to appear)Google Scholar
  37. 37.
    Peyerimhoff N., Veselić I.: Integrated density of states for ergodic random Schrödinger operators on manifolds. Geometriae Dedicata 91(1), 117–135 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Reed M., Simon B.: Functional Analysis, vol. 1. Academic Press Inc, New York (1930)Google Scholar
  39. 39.
    Rudin W.: Real and Complex Analysis, 3 edn. McGraw-Hill Book Co, Singapore (1987)zbMATHGoogle Scholar
  40. 40.
    Schwarzenberger F.: Uniform approximation of the integrated density of states for long-range percolation Hamiltonians. J. Stat. Phys. 146(6), 1156–1183 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  41. 41.
    Schwarzenberger, F.: The integrated density of states for operators on groups. Ph.D. thesis, Chemnitz University of Technology (2013). http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-123241
  42. 42.
    Shubin M.A.: Spectral theory and the index of elliptic operators with almost-periodic coefficients. Russ. Math. Surv. 34(2), 109–157 (1979)CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    Szabó E.: Soficgroups and direct finiteness. J. Algebra 280(2), 426–434 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Sznitman A.-S.: Lifschitz tail and Wiener sausage on hyperbolic space. Commun. Pure Appl. Math. 42(8), 1033–1065 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Sznitman A.-S.: Lifschitz tail on hyperbolic space: Neumann conditions. Commun. Pure Appl. Math. 43(1), 1–30 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Thom A.: Sofic groups and diophantine approximation. Commun. Pure Appl. Math. 511, 1155–1171 (2008)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Veselić I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331(4), 841–865 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Weiss, B.: Sofic groups and dynamical systems. Sankhy\({\bar{a}}\) Ser. A 62(3), 350–359 (2000). Ergodic theory and harmonic analysis (Mumbai, 1999). MR 1803462 (2001j:37022)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikTU ChemnitzChemnitzGermany

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