Robust Mixing

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 {At}t>0 satisfying πAt = π. An AMMC process involves an application of P followed by At at time t. The robust mixing time of an ergodic 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 Lt := t (mod n) with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of (0.0345 + o(1))n log n for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of Lt is adversarial, and proved an upper bound of Cn logn+O(n) (with C ≈ 4× 10) on the mixing time. We reduce the constant to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤ n logn+O(n) when the adversarial strategies are limited to holomorphic strategies, i.e. those strategies which preserve the symmetry of the underlying Markov Chain. We also show a O(n log n) bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies. Reversible liftings: Chen, Lovász and Pak showed that for a reversible ergodic Markov Chain P, any reversible lifting Q of P must satisfy T (P) ≤ T (Q) log(1/π∗) where π∗ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that T (Q) ≥ r(P) where r(P) is the relaxation time of P. This gives an alternate proof of This work was done when the author was a student at the Univ. of Chicago