Electronic Journal of Probability

Total variation and separation cutoffs are not equivalent and neither one implies the other

Jonathan Hermon, Hubert Lacoin, and Yuval Peres

Full-text: Open access

Abstract

The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple random walk on a sequence of uniformly bounded degree expander graphs. These examples give a negative answer to a question of Ding, Lubetzky and Peres

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 44, 36 pp.

Dates
Received: 18 December 2015
Accepted: 23 May 2016
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1469557137

Digital Object Identifier
doi:10.1214/16-EJP4687

Mathematical Reviews number (MathSciNet)
MR3530321

Zentralblatt MATH identifier
1345.60077

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Markov chains mixing time cutoff counter example

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hermon, Jonathan; Lacoin, Hubert; Peres, Yuval. Total variation and separation cutoffs are not equivalent and neither one implies the other. Electron. J. Probab. 21 (2016), paper no. 44, 36 pp. doi:10.1214/16-EJP4687. https://projecteuclid.org/euclid.ejp/1469557137


Export citation

References

  • [1] M. Ajtai Recursive construction for 3-regular expanders. Combinatorica 14.4 (1994): 379–416.
  • [2] D. Aldous, J. Fill. Reversible Markov chains and random walks on graphs. In preparation, http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [3] N. Alon, V. Milman. $\lambda _1$ isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B 38.1 (1985): 73–88.
  • [4] N. Alon. $\lambda _1$ Eigenvalues and expanders. Combinatorica 6.2 (1986): 83–96.
  • [5] R. Basu, J. Hermon, Y. Peres. Characterization of cutoff for reversible Markov chains. preprint: http://arxiv.org/abs/1409.3250.
  • [6] G. Y. Chen, L. Saloff-Coste. The cutoff phenomenon for ergodic Markov processes. Electronic Journal of Probability 13.3 (2008): 26–78.
  • [7] G. Y. Chen, L. Saloff-Coste. Comparison of cutoffs between lazy walks and Markovian semigroups. Journal of Applied Probability 50.4 (2013): 943–959.
  • [8] A. Dembo, O. Zeitouni. Large Deviation Techniques and Application, 2nd Edition. Stochastic Modelling and Applied Probability 38, Springer.
  • [9] P. Diaconis and Laurent Saloff-Coste. Separation cutoff for birth and death chains, The Annals of Applied Probability 16.4 (2006): 2098–2122.
  • [10] J. Ding, E. Lubetzky, Y. Peres. Total variation cutoff in birth-and-death chains. Probability theory and related fields 146.1–2 (2010): 61–85.
  • [11] J. A. Fill. The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof. Journal of Theoretical Probability 22.3 (2009): 543–557.
  • [12] J. Hermon, A technical report on hitting times, mixing and cutoff arXiv:1501.01869 [math.PR].
  • [13] S. Karlin and J. McGregor. Coincidence properties of birth and death processes. Pacific J. Math. 9 (1959): 1109–1140.
  • [14] J. Keilson. Markov chain models, rarity and exponentiality. Applied Mathematical Sciences, vol. 28, Springer-Verlag, New York, 1979.
  • [15] H. Lacoin, Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion, (preprint) arXiv:1309.3873.
  • [16] D. Levin, Y. Peres, E. Wilmer, Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, (2009).
  • [17] E. Lubetzky, A. Sly. Cutoff phenomena for random walks on random regular graphs. Duke Mathematical Journal 153.3 (2010): 475–510.
  • [18] E. Lubetzky, A. Sly. Explicit expanders with cutoff phenomena. Electronic Journal Probability 16.15 (2011): 419–435.
  • [19] E. Lubetzky, A. Sly. Cutoff for the Ising model on the lattice, Inventiones Mathematicae 191 (2013), 719–755.
  • [20] O. Reingold, S. Vadhan, A. Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of mathematics (2002): 157–187.
  • [21] A. Sinclair, M. Jerrum. Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computation 82.1 (1989): 93–133.