Back to the home page of A.Grigor'yan

Seminar "Analysis and Probability on Graphs" 241772

WS 2007/8, 15.10.2007 - 8.02.2008

Wednesday 16-18, room  V3-204

Topics and talks

  1. The proof of the Polya theorem using electric networks - [5], Ch 5,6 (Oyar, 6.02.2008)
  2. The proof of the Polya theorem by probabilistic method - [5], Ch 7. (Wiens, 9.1.2008)
  3. The type problem on arbitrary graphs - [5] , Ch 8. (Feldmann, 14.11.2007)
  4. Markov chains from analytic point of view (including the Perron - Frobenius theorem) - [8], Ch 1. (Nehring, 23.01.2008)
  5. Estimates of the eigenvalues of a finite graph - [2], Ch 1, and [4], Section 3.3. (Rolletschke, 28.11.2007)
  6. Isoperimetric inequalities on graphs - [2], Ch 2. (Schoppe, 9.01.2008)
  7. Isoperimetric inequalities on graphs - [1], Sections 5,10,11. (Bachmann, 16.01.2008)
  8. Eigenvalues and diameters of graphs - [2], Ch 3 (except Section 3.4); also [3], Section 2. (Brangewitz, 21.11.2008)
  9. Construction of expander graphs - [2], Ch 6. (Salle, 5.12.2007)
  10. Convergence to equilibrium and log-Sobolev inequality - [2], Ch 12. (more analytic)
  11. Convergence to equilibrium and log-Sobolev inequality - [8], Sections 2.1-2.2
  12. Recurrence and transience of infinite networks and application to random walks - [9], Sections 2,3 (except sub-section 3C).
  13. Isoperimetric and Sobolev inequalities - [9], Section 4. (Sommer, 16.01.2008)
  14. Isoperimetric inequalities and the relations to the Poincaré, Nash, Sobolev inequalities - [8], Ch. 3.
  15. The asymptotic behavior of transition probabilities - [9], Section 14
  16. Upper bounds of transition probabilities - [6], Section 5 (Sepanta, 30.01.2008)
  17. Random walks on Sierpinski graphs - [9], Section 15 (requires some material from previous Chapters).
  18. The spectrum on the infinite Sierpinski graph - [7], Section 4.

References

[1]   Chung F., Discrete isoperimetric inequalities,  in: “Eigenvalues of Laplacians and other geometric operators”,  Surveys in Differential Geometry IX, (2004)  pages 53-82. ISBN 1-57146-115-9. Catalogue no. in Uni Bielefeld QA680 S9D5G[9.

[2]   Chung F.R.K., “Spectral Graph Theory”, CBMS Regional Conference Series in Mathematics 92,  AMS publications, 1996. ISBN 0-8218-0315-8.  Catalogue no. in Uni Bielefeld QA052%Y94 C559.

[3]   Chung F.R.K., Grigor’yan A., Yau S.-T., Upper bounds for eigenvalues of the discrete and continuous Laplace operators,  Advances in Math., vol. 117  (1996)  pages 165-178. Available at http://www.math.uni-bielefeld.de/~grigor/eigen.pdf

[4]   Coulhon T., Grigor’yan A., Random walks on graphs with regular volume growth,  Geom. Funct. Anal., vol. 8  (1998)  pages 656-701. Available at http://www.math.uni-bielefeld.de/~grigor/grpheps.pdf

[5]   Doyle P.G., Snell J.L., “Random walks and electric networks”, Carus Mathematical Monographs 22,  Mathematical Association of America, Washington, DC,  1984.ISBN 0-88385-024-9.  Catalogue no. in Uni Bielefeld QC540 D754. Available at http://math.dartmouth.edu/ doyle/docs/walks/walks.pdf

[6]   Grigor’yan A., Telcs A., Sub-Gaussian estimates of heat kernels on infinite graphs,  Duke Math. J., vol. 109  (2001) no.3,  pages 452-510. Available at http://www.math.uni-bielefeld.de/~grigor/traeps.pdf

[7]   Higuchi Y., Shirai T., Some spectral and geometric properties for infinite graphs,  in: “Discrete geometric analysis”,  Contemporary mathematics 347, (2004)  pages 29-56. ISBN 0-8218-3351-0.  Catalogue no. in Uni Bielefeld QA052%Z02 D6G3A.

[8]   Saloff-Coste L., Lectures on finite Markov chains,  in: “Lectures on probability theory and statistics”,  Lecture Notes Math. 1665, Springer, 1997. pages 301-413. ISBN 3-540-63190-9.  Catalogue no. in Uni Bielefeld QA052%Y96 E1E8P. Available at http://www.springerlink.com/content/27114435w5149665/

[9]   Woess W., “Random walks on infinite graphs and groups”, Cambridge Tracts in Mathematics 138,  Cambridge Univ. Press., 2000. ISBN 0-521-55292-3.  Catalogue no. in Uni Bielefeld QB120 W843.