Extreme boundary conditions and random tilings
Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for...
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ftdatacite:10.48550/arxiv.2003.06339 2023-05-15T15:06:00+02:00 Extreme boundary conditions and random tilings Stéphan, Jean-Marie 2020 https://dx.doi.org/10.48550/arxiv.2003.06339 https://arxiv.org/abs/2003.06339 unknown arXiv https://dx.doi.org/10.21468/scipostphyslectnotes.26 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Statistical Mechanics cond-mat.stat-mech Mathematical Physics math-ph FOS Physical sciences article-journal Article ScholarlyArticle Text 2020 ftdatacite https://doi.org/10.48550/arxiv.2003.06339 https://doi.org/10.21468/scipostphyslectnotes.26 2022-03-10T16:02:57Z Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.] : Expanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 66 pages. v2: various minor improvements Article in Journal/Newspaper Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic |
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DataCite Metadata Store (German National Library of Science and Technology) |
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topic |
Statistical Mechanics cond-mat.stat-mech Mathematical Physics math-ph FOS Physical sciences |
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Statistical Mechanics cond-mat.stat-mech Mathematical Physics math-ph FOS Physical sciences Stéphan, Jean-Marie Extreme boundary conditions and random tilings |
topic_facet |
Statistical Mechanics cond-mat.stat-mech Mathematical Physics math-ph FOS Physical sciences |
description |
Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.] : Expanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 66 pages. v2: various minor improvements |
format |
Article in Journal/Newspaper |
author |
Stéphan, Jean-Marie |
author_facet |
Stéphan, Jean-Marie |
author_sort |
Stéphan, Jean-Marie |
title |
Extreme boundary conditions and random tilings |
title_short |
Extreme boundary conditions and random tilings |
title_full |
Extreme boundary conditions and random tilings |
title_fullStr |
Extreme boundary conditions and random tilings |
title_full_unstemmed |
Extreme boundary conditions and random tilings |
title_sort |
extreme boundary conditions and random tilings |
publisher |
arXiv |
publishDate |
2020 |
url |
https://dx.doi.org/10.48550/arxiv.2003.06339 https://arxiv.org/abs/2003.06339 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
https://dx.doi.org/10.21468/scipostphyslectnotes.26 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.2003.06339 https://doi.org/10.21468/scipostphyslectnotes.26 |
_version_ |
1766337671575109632 |