Arctic curves of the octahedron equation

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.

39 pages, 21 figures; typos fixed, four references and an appendix added

References in corpus (3)

• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Algebraic arctic curves in the domain-wall six-vertex model
• T-systems, networks and dimers

Cited by in corpus (10)

• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Arctic curve of the free-fermion six-vertex model in an L-shaped domain
• From Aztec diamonds to pyramids: steep tilings
• On the domino shuffle and matrix refactorizations
• Double tangent method for two-periodic Aztec diamonds
• Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit
• Breakdown of the thermodynamic limit in quantum spin and dimer models
• Wiener-Hopf factorizations and matrix-valued orthogonal polynomials
• The free-fermionic $C_2^{(1)}$ loop model, double dimers and Kashaev's recurrence
• On $\lambda$-determinants and tiling problems