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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2505.12926 |
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| _version_ | 1866908496079880192 |
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| author | Barbour, Andrew Brightwell, Graham Luczak, Malwina |
| author_facet | Barbour, Andrew Brightwell, Graham Luczak, Malwina |
| contents | We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its equilibrium distribution~$π^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its equilibrium distribution~$π^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence to equilibrium for density dependent Markov jump processes Barbour, Andrew Brightwell, Graham Luczak, Malwina Probability 60J75, 60C05, 60F15 We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its equilibrium distribution~$π^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its equilibrium distribution~$π^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size. |
| title | Convergence to equilibrium for density dependent Markov jump processes |
| topic | Probability 60J75, 60C05, 60F15 |
| url | https://arxiv.org/abs/2505.12926 |