In this paper, we develop a new “robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let ℙ be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices \(\{{\mathbb{A}}_t\}_{t > 0}\) satisfying \(\pi{\mathbb{A}}_t = \pi\). An AMMC process involves an application of ℙ followed by \({\mathbb{A}}_t\) at time t. The robust mixing time of an irreducible Markov Chain ℙ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains.

Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position \(t {\pmod n}\) with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345+o(1))nlogn and Cnlogn + O(n) (with C ≈4 ×105). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤nlogn + O(n) when the adversarial strategies are limited to those which preserve the symmetry of the underlying Markov Chain.

Reversible liftings: Chen, Lovász and Pak showed that for a reversible ergodic Markov Chain ℙ, any reversible lifting ℚ of ℙ must satisfy \({\mathcal{T}}({\mathbb{P}}) \leq {\mathcal{T}}(\mathbb{Q})\log (1/\pi_*)\) where π * is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that \({\mathcal{T}}(\mathbb{Q}) \geq r({\mathbb{P}})\) where r(ℙ) is the relaxation time of ℙ. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.


Markov Chain Stationary Distribution Convex Combination Transition Probability Matrix Stochastic Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous, D., Diaconis, P.: Shuffling cards and stopping times. The American Mathematical Monthly 93(5), 333–348 (1986)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bezakova, I., Stefankovic, D.: Convex combinations of markov chains and sampling linear orderings (in preperation)Google Scholar
  3. 3.
    Bobkov, S., Tetali, P.: Modified log-sobolev inequalities, mixing and hypercontractivity. In: STOC 2003: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pp. 287–296. ACM Press, New York (2003)CrossRefGoogle Scholar
  4. 4.
    Chen, F., Lovász, L., Pak, I.: Lifting markov chains to speed up mixing. In: Proceedings of the thirty-first annual ACM symposium on Theory of computing, pp. 275–281. ACM Press, New York (1999)CrossRefGoogle Scholar
  5. 5.
    Montenegro, R., Tetali, P., Goel, S.: Mixing time bounds via the spectral profile. Electronic Journal of Probability 11, 1–26 (2006), MathSciNetGoogle Scholar
  6. 6.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. American Statistical Association Journal, 13–30 (1963)Google Scholar
  7. 7.
    Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)MATHGoogle Scholar
  8. 8.
    Miclo, L.: Remarques sur l’hypercontractivité et l’évolution de l’entropie pour des chaînes de markov finies. Séminaire de probabilités de Strasbourg 31, 136–167 (1997)MathSciNetGoogle Scholar
  9. 9.
    Mironov, I. (Not so) random shuffles of RC4. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 304–319. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Montenegro, R.: Duality and evolving set bounds on mixing times (preprint),
  11. 11.
    Mossel, E., Peres, Y., Sinclair, A.: Shuffling by semi-random transpositions (2004)Google Scholar
  12. 12.
    Saloff-Coste, L.: Random walks on finite groups,

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Murali K. Ganapathy
    • 1
  1. 1.University of ChicagoChicagoUSA

Personalised recommendations