Extreme boundary conditions and random tilings

• Published in: SciPost Physics Lecture Notes
• Main Author: Stéphan, Jean-Marie
• Other Authors: Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan Villeurbanne (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
• Language: English
• Published: HAL CCSD 2020
• Subjects: [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]; Arctic
• Online Access: https://hal.archives-ouvertes.fr/hal-02879914; https://doi.org/10.21468/SciPostPhysLectNotes.26

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.]