Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are some progress on the analysis of the vertex-wise discrepancy (i.e., L_∞ discrepancy), little is known about the total variation discrepancy (i.e., L₁ discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of the L₁ discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound O(mt*) of the L₁ discrepancy for any ergodic Markov chains, where m is the number of edges of the transition diagram and t* is the mixing time of the Markov chain. Then, we give a better upper bound for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.