Extreme boundary conditions and random tilings
Expanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 62 pages Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Ra...
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ftinsalyonhal:oai:HAL:hal-02879914v1 2024-06-02T08:02:22+00:00 Extreme boundary conditions and random tilings Stéphan, Jean-Marie Probabilités, statistique, physique mathématique (PSPM) Institut Camille Jordan (ICJ) École Centrale de Lyon (ECL) Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon) Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL) Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS) ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018) 2020-06-24 https://hal.science/hal-02879914 https://doi.org/10.21468/SciPostPhysLectNotes.26 en eng HAL CCSD info:eu-repo/semantics/altIdentifier/arxiv/2003.06339 info:eu-repo/semantics/altIdentifier/doi/10.21468/SciPostPhysLectNotes.26 hal-02879914 https://hal.science/hal-02879914 ARXIV: 2003.06339 doi:10.21468/SciPostPhysLectNotes.26 https://hal.science/hal-02879914 2020 [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] info:eu-repo/semantics/preprint Preprints, Working Papers, . 2020 ftinsalyonhal https://doi.org/10.21468/SciPostPhysLectNotes.26 2024-05-07T03:07:08Z Expanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 62 pages 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.] Report Arctic INSA Lyon HAL (Institut National des Sciences Appliquées) Arctic SciPost Physics Lecture Notes |
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INSA Lyon HAL (Institut National des Sciences Appliquées) |
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ftinsalyonhal |
language |
English |
topic |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] |
spellingShingle |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] Stéphan, Jean-Marie Extreme boundary conditions and random tilings |
topic_facet |
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] [PHYS.COND.CM-SCE]Physics [physics]/Condensed Matter [cond-mat]/Strongly Correlated Electrons [cond-mat.str-el] |
description |
Expanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 62 pages 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.] |
author2 |
Probabilités, statistique, physique mathématique (PSPM) Institut Camille Jordan (ICJ) École Centrale de Lyon (ECL) Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL) Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon) Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL) Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS) ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018) |
format |
Report |
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 |
HAL CCSD |
publishDate |
2020 |
url |
https://hal.science/hal-02879914 https://doi.org/10.21468/SciPostPhysLectNotes.26 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_source |
https://hal.science/hal-02879914 2020 |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/2003.06339 info:eu-repo/semantics/altIdentifier/doi/10.21468/SciPostPhysLectNotes.26 hal-02879914 https://hal.science/hal-02879914 ARXIV: 2003.06339 doi:10.21468/SciPostPhysLectNotes.26 |
op_doi |
https://doi.org/10.21468/SciPostPhysLectNotes.26 |
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SciPost Physics Lecture Notes |
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