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Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions...

Diophantine approximation on Veech surfaces

Pascal Hubert, Thomas A. Schmidt (2012)

Bulletin de la Société Mathématique de France

We show that Y. Cheung’s general Z -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let ∑ be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in ∑ , we define discrete analogs of the 2 2 g Dirac operators on ∑ . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair Γ ⊂ ∑ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ . As a consequence, if these conditions are met, the partition function of the dimer...

Dynamics of dianalytic transformations of Klein surfaces

Ilie Barza, Dorin Ghisa (2004)

Mathematica Bohemica

This paper is an introduction to dynamics of dianalytic self-maps of nonorientable Klein surfaces. The main theorem asserts that dianalytic dynamics on Klein surfaces can be canonically reduced to dynamics of some classes of analytic self-maps on their orientable double covers. A complete list of those maps is given in the case where the respective Klein surfaces are the real projective plane, the pointed real projective plane and the Klein bottle.

Currently displaying 181 – 200 of 887