Six-vertex model, dimers, shapes, and all that: March 14 – 18, 2016

Organized by: Pavel Bleher, Vladimir Korepin and Bernard Nienhuis

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Dates: March 14-18, 2016

The workshop will be devoted to recent results on the six-vertex and related models. The bulk free energy in the six-vertex model was explicitly calculated for periodic boundary conditions by Lieb. Then other boundary conditions were studied. A very interesting case is the domain wall boundary conditions (DWBC). The six-vertex model provides an important ‘counterexample’ in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions. In particular, it is different for periodic and domain wall boundary conditions. The dimer model on the Aztec diamond is a special case of the six-vertex model, as well as the square ice model, ferroelectric and antiferroelectric models, ensembles of alternating sign matrices, and others.

The partition function of the six-vertex model with DWBC in a finite box has been expressed, via the Yang-”Baxter equations, in terms of a Hankel determinant. It can be furthermore expressed as a partition function of an ensemble of random matrices with a non-polynomial interaction. This expression was used, via the powerful Riemann-Hilbert approach, to obtain the asymptotic behavior of the partition function of the six-vertex model with DWBC in different phase regions.

The remarkable observation by Razumov and Stroganov which links some observables on the square grid to those on the half-infinite cylinder is a subject of active research. It appears to admit generalizations to other geometries and other models