Arctic curves of the octahedron equation

• Published in: Journal of Physics A: Mathematical and Theoretical
• Main Authors: Di Francesco, P., Soto-Garrido, R.
• Other Authors: Department of Mathematics Urbana, University of Illinois at Urbana-Champaign Urbana, University of Illinois System-University of Illinois System, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Physics Illinois at Urbana-Champaign, USA
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2014
• Subjects: arctic curve; dimer; tiling; exact solution; phase transition; [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]; [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]; [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]; Arctic
• Online Access: https://hal-cea.archives-ouvertes.fr/cea-01002519; https://doi.org/10.1088/1751-8113/47/28/285204

PACS numbers: 64.60.De, 64.70.qd, 02.10.Ox International audience We study the octahedron relation (also known as the $A_{\infty}$ $T$-system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a suitable class of doubly periodic initial conditions, we find exact solutions with a particularly simple factorized form. For these, we show that the density function that measures the average dimer occupation of a face of the Aztec graph, obeys a system of linear recursion relations with periodic coefficients. This allows us to explore the thermodynamic limit of the corresponding dimer models and to derive exact "arctic" curves separating the various phases of the system.