Abstract
In this chapter, we first define coupling and the connection to mixing times of Markov chains via the coupling inequality. We then motivate the method of path coupling with a card shuffling example. In the remaining sections we provide the complete derivation of the path coupling method in full generality and include the definition of the greedy coupling which is the particular coupling used for the statistical mechanical models discussed in this monograph. This will be done in an alternative, and in our opinion, more rigorous way than usually employed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This coupling was constructed as part of the REU project of Jennifer Thompson that was supervised by Yevgeniy Kovchegov in the summer of 2010 at Oregon State University.
- 2.
Here, symbols ∧ and ∨ denote minimum and maximum, respectively, and \((x)_{+} ={ 1 \over 2}\big(\vert x\vert + x\big)\).
- 3.
Here, \(\mathbf{1}_{\{\sigma _{u}\neq \tau _{u}\}} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\text{ if }\sigma _{u}\neq \tau _{u}, \\ 0\quad &\text{ if }\sigma _{u} =\tau _{u}. \end{array} \right.\)
- 4.
Symbol \(\mathop{=}\limits^{ dist}\) means “distributed according to.”
References
D. Aldous, J. Fill, Reversible Markov chains and random walks on graphs, in Unfinished Monograph (2002). Available at https://www.stat.berkeley.edu/~aldous/RWG/book.pdf
N. Bhatnagar, D. Randall, Torpid mixing of simulated tempering on the Potts model, in Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 478–487 (2004)
M. Blume, Theory of the first-order magnetic phase change in UO2. Phys. Rev. 141, 517–524 (1966)
M. Blume, V. J. Emery, R.B. Griffiths, Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A 4, 1071–1077 (1971)
M. Bordewich, M.E. Dyer, Path coupling without contraction. J. Discrete Algorithms 5(2), 280–292 (2007)
P. Brémaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics, vol. 31 (Springer, New York, 1999)
R. Bubley, M.E. Dyer, Path coupling: a technique for proving rapid mixing in Markov chains, in Proceedings of the 38th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 223–231 (1997)
H.W. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 32, 966–988 (1966)
H.W. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting II. Physica 33, 295–331 (1967)
H.W. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting III. Physica 37, 423–441 (1967)
F. Collet, Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach. J. Stat. Phys. 157(6), 1301–1319 (2014)
M. Costeniuc, R.S. Ellis, H. Touchette, Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46, 063301 (2005)
P. Cuff, J. Ding, O. Louidor, E. Lubetzy, Y. Peres, A. Sly, Glauber dynamics for the mean-field Potts model. J. Stat. Phys. 149(3), 432–477 (2012)
A. Dembo, O. Zeitouni, Large deviations techniques and applications, 2nd edn. (Springer, New York, 1998)
F. den Hollander, Probability Theory: The Coupling Method. Lectures Notes-Mathematical Institute (Leiden University, Leiden, 2012)
J. Ding, E. Lubetzky, Y. Peres, The mixing time evolution of Glauber dynamics for the mean-field Ising model. Commun. Math. Phys. 289(2), 725–764 (2009)
J. Ding, E. Lubetzky, Y. Peres, Censored Glauber dynamics for the mean-field Ising model. J. Stat. Phys. 137(1), 161–207 (2009)
W. Doeblin, Exposé de la théorie des chaınes simples constantes de Markova un nombre fini d’états. Mathématique de l’Union Interbalkanique 2(77–105), 78–80 (1938)
M. Dyer, L.A. Goldberg, C. Greenhill, M. Jerrum, M. Mitzenmacher, An extension of path coupling and its application to the Glauber dynamics for graph colorings. SIAM J. Comput. 30(6), 1962–1975 (2001)
M. Ebbers, H. Knöpfel, M. Löwe, F. Vermet, Mixing times for the swapping algorithm on the Blume-Emery-Griffiths model. Random Struct. Algoritm. (2012). https://doi.org/10.1002/rsa.20461
T. Eisele, R.S. Ellis, Multiple phase transitions in the generalized Curie-Weiss model. J. Stat. Phys. 52(1/2), 161–207 (1988).
R.S. Ellis, Entropy, Large Deviations and Statistical Mechanics (Springer, New York, 1985). Reprinted in 2006 in Classics in Mathematics
R.S. Ellis, K. Wang, Limit theorems for the empirical vector of the Curie-Wiess-Potts model. Markov Proc. Their Appl. 35, 59–79 (1990)
R.S. Ellis, K. Haven, B. Turkington, Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101(5/6), 999–1064 (2000)
R.S. Ellis, P.T. Otto, H. Touchette, Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model. Ann. Appl. Probab. 15, 2203–2254 (2005)
R.S. Ellis, J. Machta, P.T. Otto, Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg-Landau polynomials. J. Stat. Phys. 133, 101–129 (2008)
R.S. Ellis, J. Machta, P.T. Otto, Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality. Ann. Appl. Probab. 20, 2118–2161 (2010)
M. Fedele, F. Unguendoli, Rigorous results on the bipartite mean-field model. J. Phys. A: Math. Theor. 45(38), 385001 (2012)
T.P. Hayes, A. Sinclair, A general lower bound for mixing of single-site dynamics on graphs, in 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), pp. 511–520 (2005)
T.P. Hayes, E. Vigoda, Variable length path coupling. Random Struct. Algoritm. 31(3), 251–272 (2007)
J.C. Hernández, Y. Kovchegov, P.T. Otto, The aggregate path coupling method for the Potts model on bipartite graph. J. Math. Phys. 58, 023303 (2017)
G. Jaeger, Ehrenfest classification of phase transitions: introduction and evolution. Arch. Hist. Exact Sci. 53, 51–81 (1998)
B. Jahnel, C. Külske, E. Rudelli, J. Wegener, Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model. Markov Proc. Relat. Fields 20, 601–632 (2014)
Y. Kovchegov, P.T. Otto, Rapid mixing of Glauber dynamics of Gibbs ensembles via aggregate path coupling and large deviations methods. J. Stat. Phys. 161(3), 553–576 (2015)
Y. Kovchegov, P.T. Otto, M. Titus, Mixing times for the mean-field Blume-Capel model via aggregate path coupling. J. Stat. Phys. 144(5), 1009–1027 (2011)
D. Levin, Y. Peres, E. Wilmer, Markov Chains and Mixing Times (American Mathematical Society, Providence, RI, 2009)
D.A. Levin, M. Luczak, Y. Peres, Glauber dynamics of the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146(1), 223–265 (2010)
T.M. Liggett, Interacting Particle Systems (Springer, Berlin, 1985)
T.M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, Berlin, 1999)
T. Lindvall, Lectures on the Coupling Method (Wiley, New York, 1992). Reprint: Dover paperback edition (2002)
M.J. Luczak, Concentration of measure and mixing times of Markov chains, in Proceedings of the 5th Colloquium on Mathematics and Computer Science. Discrete Mathematics and Theoretical Computer Science, pp. 95–120 (2008)
R.V. Solé, Phase Transitions (Princeton University Press, Princeton, NJ, 2011)
S.R.S. Varadhan, Asymptotic properties and differential equations. Commun. Pure Appl. Math. 19, 261–286 (1966)
F.Y. Wu, The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018
About this chapter
Cite this chapter
Kovchegov, Y., Otto, P.T. (2018). Coupling, Path Coupling, and Mixing Times. In: Path Coupling and Aggregate Path Coupling. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-77019-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-77019-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77018-5
Online ISBN: 978-3-319-77019-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)