Arctic curves of the T-system with slanted initial data

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+. The solution of the T-system is interpreted as the partition function of a dimer model on some suitable “pinecone" graph introduced in [Mireille Bousquet-Mélou, James Propp, and Julian West. The corresponding solutions of the $T$-system are interpreted as partition functions of 
dimer models on some suitable ``pinecone" graphs introduced by Bousquet-Melou, 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.