Abstract
We determine, up to a log factor, the mixing time of a Markov chain whose state space consists of the successive distances between n labeled “dots” on a circle, in which one dot is selected uniformly at random and moved to a uniformly random point between its two neighbors. The method involves novel use of auxiliary discrete Markov chains to keep track of a vector of quadratic parameters.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Diaconis, P., Shashahani, M.: Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 81 #1, 208–218 (1987)
Dyer, M., Frieze, A., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex sets. J. for the Association for Computing Machinery 38, 1–17 (1991)
Dyer, M., Greenhill, C.: On Markov Chains for Independent Sets. J. Algorithms 35 #1, 17–49 (2000)
Kannan, R.: Markov chains and polynomial time algorithms. In: Proc. 35th FOCS, pp. 656–671 (1994)
Kannan, R., Mahoney, M.W., Montenegro, R.: Rapid mixing of several Markov chains for a hard-core model. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 663–675. Springer, Heidelberg (2003)
Lovász, L., Winkler, P.: Mixing times, Microsurveys in Discrete Probability. In: Aldous, D., Propp, J. (eds.) DIMACS Series in Discrete Math. and Theoretical Computer Science, vol. 41, pp. 85–134 (1998)
Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM Journal on Computing 31, 167–192 (2001)
Luby, M., Vigoda, E.: Fast convergence of the Glauber dynamics for sampling independent sets. Random Structures and Algorithms 15#3-4, 229–241 (1999)
Lee, T.-Y., Yau, H.-T.: Logarithmic Sobolev inequalities for some models of random walks. Ann. Prob. 26 #4, 1855–1873 (1998)
Madras, N., Randall, D.: Markov chain decomposition for convergence rate analysis. Ann. Appl. Prob. 12, 581–606 (2002)
Randall, D.: Mixing. In: Proc. 44th Annual FOCS, pp. 4–15 (2004)
Randall, D., Winkler, P.: Mixing points in an interval. In: Proc. ANALCO 2005, Vancouver BC (2005)
Sinclair, A.: Algorithms for Random Generation & Counting: A Markov Chain Approach. Birkhäuser, Boston (1993)
Wilson, D.B.: Mixing times of lozenge tiling and card shuffling Markov chains. Annals of Appl. Probab. 14, 274–325 (2004)
Wilson, D.B.: Mixing time of the Rudvalis shuffle. Elect. Comm. in Probab. 8, 77–85 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Randall, D., Winkler, P. (2005). Mixing Points on a Circle. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_36
Download citation
DOI: https://doi.org/10.1007/11538462_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28239-6
Online ISBN: 978-3-540-31874-3
eBook Packages: Computer ScienceComputer Science (R0)