A curiously slowly mixing Markov chain
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909977320357888 |
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| author | Diaconis, Persi Lin, Andrew Ram, Arun |
| author_facet | Diaconis, Persi Lin, Andrew Ram, Arun |
| contents | We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A curiously slowly mixing Markov chain Diaconis, Persi Lin, Andrew Ram, Arun Probability Combinatorics Representation Theory 60J10 (Primary) 05E18 (Secondary) We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors. |
| title | A curiously slowly mixing Markov chain |
| topic | Probability Combinatorics Representation Theory 60J10 (Primary) 05E18 (Secondary) |
| url | https://arxiv.org/abs/2511.01245 |