Physics and combinatorics of the octahedron equation: from cluster algebras to arctic curves

Philippe di Francesco

IPhT

Mon, Mar. 17th 2014, 11:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

The octahedron equation is a non-linear system of recursion relations in discrete 2+1 dimensions, that first appeared in the context of generalized Heisenberg quantum spin chains (T-system). Some solutions of this equation also have a purely combinatorial interpretation as generalized Coxeter-Conway Frieze patterns.par This same equation plays a central role in the seemingly unrelated purely combinatorial problem of enumeration of domino tilings of a plane square-shaped domain called the Aztec diamond. For large size, the tiling configurations display some drastic change of typical behavior between the corners of the domain, where it freezes in certain tiling patterns, and a central region in the bulk, with disorder. In the continuum limit of large size and small mesh, the separation of phases is along the so-called ``arctic circle'' inscribed inside the square domain. par In this talk, we shall show that the octahedron equation is part of a combinatorial structure called cluster algebra, and how its exact solution may be rephrased in terms of flat connexions and non-intersecting lattice paths, and eventually domino tilings. Using the octahedron equation explicitly, we will proceed and determine various arctic curves depending on initial data. As predicted by Kenyon and Okounkov, we find in general new ``facet'' phases of the dimer model within some connected components of these curves.

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