Coalescence in Markov chains
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917350710706176 |
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| author | Grimmett, Geoffrey R. Holmes, Mark |
| author_facet | Grimmett, Geoffrey R. Holmes, Mark |
| contents | A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are $|S|$ trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number $k(μ)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(μ)$, as the coupling $μ$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors ('Non-coupling from the past', $\textit{In and Out of Equilibrium 3}$, Springer, 2021) on these two fundamental questions, which have special importance for the 'coupling from the past' algorithm.
We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coalescence in Markov chains Grimmett, Geoffrey R. Holmes, Mark Probability 60J10, 60J22 A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are $|S|$ trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number $k(μ)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(μ)$, as the coupling $μ$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors ('Non-coupling from the past', $\textit{In and Out of Equilibrium 3}$, Springer, 2021) on these two fundamental questions, which have special importance for the 'coupling from the past' algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties. |
| title | Coalescence in Markov chains |
| topic | Probability 60J10, 60J22 |
| url | https://arxiv.org/abs/2510.13572 |