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Bipartitness in reversible Markov chains Langer, Benjamin
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.
Item Metadata
| Title |
Bipartitness in reversible Markov chains
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| Creator | |
| Supervisor | |
| Publisher |
University of British Columbia
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| Date Issued |
2022
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| Description |
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.
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| Genre | |
| Type | |
| Language |
eng
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| Date Available |
2022-08-19
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0417455
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
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| Graduation Date |
2022-11
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| Campus | |
| Scholarly Level |
Graduate
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International