Evolving sets, mixing and heat kernel bounds

Probability Theory and Related Fields

Volumn 133, Issue 2, 2005, Pages 245-266

  Morris, B ^a   Peres, Yuval ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 24644445142     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-005-0434-7     Document Type: Article

References (25)

1. Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83-96 (1986)
2. 1, Isoperimetric inequalities for graphs and superconcentrators. J. Combinatorial Theory Ser. B 38, 73-88 (1985)
3. Benjamini, I., Mossel, E.: On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Relat. Fields 125, 408-420 (2003)
4. Chung, F.R.K.: Laplacians of graphs and Cheeger's inequalities. In Combinatorics, Paul Erdös is eighty, vol 2, 157-172 J. Bolyai Soc. Math. Stud., Budapest. 1996
5. Chung, F.R.K., Yau, S.T.: Eigenvalues of graphs and Sobolev inequalities. Combinatorics, Probability and Computing 4, 11-26 (1995)
6. Coulhon, T.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141, 510-539 (1996)
7. Coulhon, T., Grigorýan, A., Pittet, C.: A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble) 51, 1763-1827 (2001)
8. Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18, 1483-1522 (1990)
9. Diaconis, P., Saloff-Coste, L.: Nash inequalities for finite Markov chains. J. Theoret. Probab. 9, 459-510(1996)
10. Fill, J.A.: Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1, 62-87 (1991)
11. Häggström, O., Jonasson, J.: Rates of convergence for lamplighter processes. Stochastic Process. Appl. 67, 227-249 (1997)
12. Houdré, C., Tetali, P.: Isoperimetric Invariants for Product Markov Chains and Graph Products. Combinatorica 24, 359-388 (2004)
13. Jerrum, M.R., Sinclair, A.J.: Approximating the permanent. SIAM Journal on Computing 18, 1149-1178 (1989)
14. Kannan, R.: Rapid Mixing in Markov Chains. Proceedings of International Congress of Math, 2002, Vol. III, 673-683
15. 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309, 557-580 (1988)
16. Lovász, L., Kannan, R.: Faster mixing via average conductance. Proceedings of the 27th Annual ACM Symposium on theory of computing, 1999
17. Mathieu, P., Remy, E.: Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32, 100-128 (2004)
18. Mihail, M. : Conductance and convergence of Markov chains - A combinatorial treatment of expanders. Proceedings of the 30th Annual Conference on Foundations of Computer Science, 1989, 526-531
19. Montenegro, R., Son, J.-B.: Edge Isoperimetry and Rapid Mixing on Matroids and Geometric Markov Chains. Proceedings of the 33rd Annual ACM Symposium on theory of computing, 2001
20. Morris, B.: A new, probabilistic approach to heat kernel bounds. Lecture at Sectional AMS meeting, Atlanta, GA, March 2002
21. Pittet, C., Saloff-Coste, L.: A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples. Unpublished manuscript, available at http://www.math.cornell.edu/~lsc/lau.html 2002
22. Saloff-Coste, L.: Lectures on finite Markov chains. Lecture Notes in Math. 1665, 301-413, Springer, Berlin, 1997
23. Sinclair, A.: Algorithms for Random Generation and Counting: A Markov Chain Approach, Birkhäuser, Boston, 1993
24. Varopoulos, Th.N.: Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63, 215-239 (1985)
25. Woess, W.: Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics 138, Cambridge University Press, 2000