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Abstract

Let (Xₜ )ₜ∈𝕫₊ be an irreducible, aperiodic, reversible Markov chain on a finite state space Ω. Let M := 𝑡mix/𝑡L, where 𝑡mix and 𝑡L are the total variation mixing times of the chain and its lazy version, respectively. We show - in a precise quantitative sense - that if M is sufficiently large, then the chain is ”near-bipartite”. That is, there exists a bipartition (A, B) of Ω such that π(A) and π(B) are both close to 1/2, and the Markov chain rarely spends two consecutive time steps within the same set of the bipartition. In particular, we show that for 𝑡 ≫ 𝑡L, the distribution of Xₜ is very close to a mixture of πᴀ and πᴃ.