Arctic curves of the T-system with Slanted Initial Data

Abstract

We study the T-system of type $A_\infty$, also known as the octahedron recurrence/equation, viewed as a 2+1-dimensional discrete evolution equation. Generalizing the study of [P. Di Francesco and R. Soto-Garrido. Arctic curves of the octahedron equation. J. Phys. A, 47(28):285204, 34, 2014], we consider initial data along parallel ``slanted" planes perpendicular to an arbitrary admissible direction $(r,s,t)\in {\mathbb Z}_+^3$. The solution of the T-system is interpreted as the partition function of a dimer model on some suitable ``pinecone" graph introduced in [M. Bousquet-M\'elou, J. Propp, and J. West. Perfect matchings for the three-term Gale-Robinson sequences. Electron. J. Combin., 16(1):Research Paper 125, 37, 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.