Arctic curves of the octahedron equation

Authors :: Philippe Di Francesco
Rodrigo Soto-Garrido
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]
Source :: Journal of Physics A: Mathematical and Theoretical, Journal of Physics A: Mathematical and Theoretical, 2014, 47 (28), pp.285204. ⟨10.1088/1751-8113/47/28/285204⟩
Publication Year :: 2014
Publisher :: IOP Publishing, 2014.
Abstract: 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.<br />Comment: 39 pages, 21 figures; typos fixed, four references and an appendix added