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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ftdoajarticles:oai:doaj.org/article:edcff4130c8f455d8f01ee365f81070a 2023-05-15T15:04:00+02:00 Extreme boundary conditions and random tilings Jean-Marie Stéphan 2021-03-01T00:00:00Z https://doi.org/10.21468/SciPostPhysLectNotes.26 https://doaj.org/article/edcff4130c8f455d8f01ee365f81070a EN eng SciPost https://scipost.org/SciPostPhysLectNotes.26 https://doaj.org/toc/2590-1990 2590-1990 doi:10.21468/SciPostPhysLectNotes.26 https://doaj.org/article/edcff4130c8f455d8f01ee365f81070a SciPost Physics Lecture Notes, p 26 (2021) Physics QC1-999 article 2021 ftdoajarticles https://doi.org/10.21468/SciPostPhysLectNotes.26 2022-12-31T12:26:26Z 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.] Article in Journal/Newspaper Arctic Directory of Open Access Journals: DOAJ Articles Arctic SciPost Physics Lecture Notes |
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Physics QC1-999 |
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Physics QC1-999 Jean-Marie Stéphan Extreme boundary conditions and random tilings |
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Physics QC1-999 |
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.] |
format |
Article in Journal/Newspaper |
author |
Jean-Marie Stéphan |
author_facet |
Jean-Marie Stéphan |
author_sort |
Jean-Marie Stéphan |
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 |
SciPost |
publishDate |
2021 |
url |
https://doi.org/10.21468/SciPostPhysLectNotes.26 https://doaj.org/article/edcff4130c8f455d8f01ee365f81070a |
geographic |
Arctic |
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Arctic |
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Arctic |
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Arctic |
op_source |
SciPost Physics Lecture Notes, p 26 (2021) |
op_relation |
https://scipost.org/SciPostPhysLectNotes.26 https://doaj.org/toc/2590-1990 2590-1990 doi:10.21468/SciPostPhysLectNotes.26 https://doaj.org/article/edcff4130c8f455d8f01ee365f81070a |
op_doi |
https://doi.org/10.21468/SciPostPhysLectNotes.26 |
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SciPost Physics Lecture Notes |
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