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Quadri-tilings of the plane


de Tilière, B (2007). Quadri-tilings of the plane. Probability Theory and Related Fields, 137(3-4):487-518.

Abstract

We introduce quadri-tilings and show that they are in bijection with dimer models on a family of graphs R * arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called triangular quadri-tilings, as an interface model in dimension 2+2. Assigning “critical" weights to edges of R *, we prove an explicit expression, only depending on the local geometry of the graph R *, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of Kenyon (Invent Math 150:409–439, 2002). We also show that when edges of R * are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of all triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.

Abstract

We introduce quadri-tilings and show that they are in bijection with dimer models on a family of graphs R * arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called triangular quadri-tilings, as an interface model in dimension 2+2. Assigning “critical" weights to edges of R *, we prove an explicit expression, only depending on the local geometry of the graph R *, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of Kenyon (Invent Math 150:409–439, 2002). We also show that when edges of R * are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of all triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Language:English
Date:2007
Deposited On:04 Jan 2010 13:24
Last Modified:23 Jan 2022 14:31
Publisher:Springer
ISSN:0178-8051
Additional Information:The original publication is available at www.springerlink.com
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s00440-006-0002-9
Related URLs:http://arxiv.org/abs/math/0403324
  • Content: Accepted Version
  • Description: Accepted manuscript, Version 3
  • Content: Accepted Version
  • Description: Accepted manuscript, Version 2
  • Content: Accepted Version
  • Description: Accepted manuscript, Version 1
  • Content: Published Version
  • Language: English
  • Description: Nationallizenz 142-005