Arctic curves of the T-system with slanted initial data

• Published in: Journal of Physics A: Mathematical and Theoretical
• Main Authors: Di Francesco, Philippe, Vu, Hieu Trung
• Other Authors: Morris and Gertrude Fine Endowment, David G. Bourgin Mathematics Fellowship and the University of Illinois at Urbana-Champaign Campus Research Board, Simons Foundation, National Science Foundation Division of Mathematical Sciences
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: IOP Publishing 2024
• Subjects: Arctic
• Online Access: http://dx.doi.org/10.1088/1751-8121/ad65a5; https://iopscience.iop.org/article/10.1088/1751-8121/ad65a5; https://iopscience.iop.org/article/10.1088/1751-8121/ad65a5/pdf

Abstract We study the T -system of type A ∞ , also known as the octahedron recurrence/equation, viewed as a 2 + 1 -dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with ‘flat’ initial data, we consider initial data along parallel ‘slanted’ planes perpendicular to an arbitrary admissible direction ( r , s , t ) ∈ Z + 3 . The corresponding solutions of the T -system are interpreted as partition functions of dimer models on some suitable ‘pinecone’ graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T -system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function.