Mixing for Markov Chains and Spin Systems DRAFT of August 31 , 2005

Lectures given at the 2005 PIMS Summer School in Probability held at the University of British Columbia from June 6 through June 30. Special thanks is due to Jesse Goodman, Jeffrey Hood, Ben Hough, Sandra Kliem, Lionel Levine, Yun Long, Asaf Nachmias, Alex Skorokhod and Terry Soo for help preparing these notes. These notes have not been subjected to the usual scrutiny reserved for formal publications.

[1]  Prasad Tetali,et al.  Isoperimetric Invariants For Product Markov Chains and Graph Products , 2004, Comb..

[2]  Y. Peres,et al.  Evolving sets, mixing and heat kernel bounds , 2003, math/0305349.

[3]  É. Remy,et al.  Isoperimetry and heat kernel decay on percolation clusters , 2003, math/0301213.

[4]  R. Karman.  Rapid Mixing in Markov Chains , 2002 .

[5]  Ravi Montenegro,et al.  Edge isoperimetry and rapid mixing on matroids and geometric Markov chains , 2001, STOC '01.

[6]  Thierry Coulhon,et al.  A geometric approach to on-diagonal heat kernel lower bounds on groups , 2001 .

[7]  P. Diaconis,et al.  Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques , 2004, math/0401318.

[8]  Elchanan Mossel,et al.  On the mixing time of a simple random walk on the super critical percolation cluster , 2000, math/0011092.

[9]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[10]  László Lovász,et al.  Faster mixing via average conductance , 1999, STOC '99.

[11]  Mu-Fa Chen,et al.  Trilogy of Couplings and General Formulas for Lower Bound of Spectral Gap , 1998 .

[12]  Johan Jonasson,et al.  Rates of convergence for lamplighter processes , 1997 .

[13]  L. Saloff-Coste,et al.  Lectures on finite Markov chains , 1997 .

[14]  Thierry Coulhon,et al.  Ultracontractivity and Nash Type Inequalities , 1996 .

[15]  P. Diaconis,et al.  Nash inequalities for finite Markov chains , 1996 .

[16]  Shing-Tung Yau,et al.  Eigenvalues of Graphs and Sobolev Inequalities , 1995, Combinatorics, Probability and Computing.

[17]  Peter Winkler,et al.  A note on the last new vertex visited by a random walk , 1993, J. Graph Theory.

[18]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[19]  F. Chung.  Laplacians of graphs and Cheeger inequalities , 1993 .

[20]  Jeremy Quastel,et al.  Diffusion of color in the simple exclusion process , 1992 .

[21]  J. A. Fill.  Eigenvalue bounds on convergence to stationarity for nonreversible markov chains , 1991 .

[22]  P. Diaconis,et al.  Strong Stationary Times Via a New Form of Duality , 1990 .

[23]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[24]  Milena Mihail,et al.  Conductance and convergence of Markov chains-a combinatorial treatment of expanders , 1989, 30th Annual Symposium on Foundations of Computer Science.

[25]  A. Sokal,et al.  Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality , 1988 .

[26]  P. Diaconis,et al.  SHUFFLING CARDS AND STOPPING-TIMES , 1986 .

[27]  N. Alon.  Eigenvalues and expanders , 1986, Comb..

[28]  N. Varopoulos.  Isoperimetric inequalities and Markov chains , 1985 .

[29]  N. Alon,et al.  il , , lsoperimetric Inequalities for Graphs , and Superconcentrators , 1985 .