Mixing Times of Plane Random Rhombus Tilings
We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2001-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2300/pdf |
| _version_ | 1827324147326255104 |
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| author | Nicolas Destainville |
| author_facet | Nicolas Destainville |
| author_sort | Nicolas Destainville |
| collection | DOAJ |
| description | We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate that the dynamics are rapidly mixing: the ergodic times are polynomial in the number of tiles up to logarithmic corrections. We use an inherent symmetry of tiling sets which enables to decompose them into smaller subsets where a technique from probability theory, the so-called coupling technique, can be applied. We also point out an interesting occurrence in this work of extreme-value statistics, namely Gumbel distributions. |
| first_indexed | 2024-04-25T02:07:23Z |
| format | Article |
| id | doaj.art-7d63aea4aa7e4aa0bf2f7eb35a48e263 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:07:23Z |
| publishDate | 2001-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-7d63aea4aa7e4aa0bf2f7eb35a48e2632024-03-07T14:27:43ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502001-01-01DMTCS Proceedings vol. AA,...Proceedings10.46298/dmtcs.23002300Mixing Times of Plane Random Rhombus TilingsNicolas Destainville0https://orcid.org/0000-0003-3867-5102Groupe de Physique Théorique (LPQ)We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate that the dynamics are rapidly mixing: the ergodic times are polynomial in the number of tiles up to logarithmic corrections. We use an inherent symmetry of tiling sets which enables to decompose them into smaller subsets where a technique from probability theory, the so-called coupling technique, can be applied. We also point out an interesting occurrence in this work of extreme-value statistics, namely Gumbel distributions.https://dmtcs.episciences.org/2300/pdfrandom tilingsdiscrete dynamical systemsmarkovian processesquasicrystals[info] computer science [cs][info.info-cg] computer science [cs]/computational geometry [cs.cg][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Nicolas Destainville Mixing Times of Plane Random Rhombus Tilings Discrete Mathematics & Theoretical Computer Science random tilings discrete dynamical systems markovian processes quasicrystals [info] computer science [cs] [info.info-cg] computer science [cs]/computational geometry [cs.cg] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Mixing Times of Plane Random Rhombus Tilings |
| title_full | Mixing Times of Plane Random Rhombus Tilings |
| title_fullStr | Mixing Times of Plane Random Rhombus Tilings |
| title_full_unstemmed | Mixing Times of Plane Random Rhombus Tilings |
| title_short | Mixing Times of Plane Random Rhombus Tilings |
| title_sort | mixing times of plane random rhombus tilings |
| topic | random tilings discrete dynamical systems markovian processes quasicrystals [info] computer science [cs] [info.info-cg] computer science [cs]/computational geometry [cs.cg] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/2300/pdf |
| work_keys_str_mv | AT nicolasdestainville mixingtimesofplanerandomrhombustilings |