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Domino statistics of the two-periodic Aztec diamond
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-2943-7006
2016 (English)In: Advances in Mathematics, Vol. 294, 37-149 p.Article in journal (Refereed) Published
Abstract [en]

Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

Place, publisher, year, edition, pages
Elsevier, 2016. Vol. 294, 37-149 p.
Keyword [en]
Domino tiling; Two-periodic Aztec diamond; Kasteleyn matrix; Asymptotics; Local statistics; Airy kernel
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-203634DOI: 10.1016/j.aim.2016.02.025ISI: 000374207300002ScopusID: 2-s2.0-84960395062OAI: oai:DiVA.org:kth-203634DiVA: diva2:1082249
Note

QC 20170316

Available from: 2017-03-16 Created: 2017-03-16 Last updated: 2017-03-16Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
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Output format
  • html
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  • asciidoc
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