Three views on the thinned Bernoulli field on the line
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914190181007360 |
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| author | Kuelske, Christof Schubert, Niklas |
| author_facet | Kuelske, Christof Schubert, Niklas |
| contents | This paper investigates the thinned Bernoulli field (TBF) on the one-dimensional integer lattice, where isolated occupied sites are removed from a standard Bernoulli configuration with density $p$. Our present work complements previous findings in higher dimensions and on trees by focusing on the detailed behavior on the line, particularly as $p$ approaches $1.$ First we show that while the TBF on the line is always quasilocally Gibbs, it displays a growing sensitivity to boundary conditions as $p$ increases, indicating an incipient loss of quasilocality. We provide precise asymptotics for this phenomenon, which is an echo of non-quasilocality happening in higher dimensions. Second, we turn to the one-sided point of view and prove that the TBF is a g-measure in the sense of dynamical systems and ergodic theory. The corresponding g-function is quasilocal but becomes long-range again for large $p$. From that we finally develop our third view, in which we provide a transparent construction of the process in terms of a driving Markov chain on the integers of generalized house of cards type, offering a novel perspective on the TBF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three views on the thinned Bernoulli field on the line Kuelske, Christof Schubert, Niklas Probability Mathematical Physics 82B26, 60K35 This paper investigates the thinned Bernoulli field (TBF) on the one-dimensional integer lattice, where isolated occupied sites are removed from a standard Bernoulli configuration with density $p$. Our present work complements previous findings in higher dimensions and on trees by focusing on the detailed behavior on the line, particularly as $p$ approaches $1.$ First we show that while the TBF on the line is always quasilocally Gibbs, it displays a growing sensitivity to boundary conditions as $p$ increases, indicating an incipient loss of quasilocality. We provide precise asymptotics for this phenomenon, which is an echo of non-quasilocality happening in higher dimensions. Second, we turn to the one-sided point of view and prove that the TBF is a g-measure in the sense of dynamical systems and ergodic theory. The corresponding g-function is quasilocal but becomes long-range again for large $p$. From that we finally develop our third view, in which we provide a transparent construction of the process in terms of a driving Markov chain on the integers of generalized house of cards type, offering a novel perspective on the TBF. |
| title | Three views on the thinned Bernoulli field on the line |
| topic | Probability Mathematical Physics 82B26, 60K35 |
| url | https://arxiv.org/abs/2512.08800 |