Planar dimers and Harnack curves

By Richard Kenyon, A. Okounkov. 2003

168 citations recorded by the source corpus.

In this paper we study the connection between dimers and Harnack curves discovered in math-ph/0311005. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curve of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions.

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/3e8a7f5ef328d919338404e407646f1b761a84b7

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In Applied Mathematics

1. Richard Kenyon
2. A. Okounkov
3. Scott Sheffield
4. Mikko Parviainen
5. Jason Miller
6. Juan J. Manfredi
7. Julio D. Rossi
8. Nikos Katzourakis

In Geometry and Topology

1. Richard Kenyon
2. A. Okounkov
3. Scott Sheffield
4. R. Pandharipande
5. Matt Bainbridge
6. Davesh Maulik
7. Paul D. Johnson
8. Marcos Mariño

In Radiology, Nuclear Medicine and Imaging

1. Richard Kenyon
2. A. Okounkov

Citations in the corpus

• Dimers and amoebae (2003, background): “It is also shown in [12] that all Harnack curves arise as spectral curves of some dimer model.”
• Limit shapes and the complex Burgers equation (2005, unknown): “with A,B,C2R and where the real numbers a1,...,an, b1,...,bn and c1,...,cn occur in cyclic order around R[1 (that is, all the a’s come first, then the b’s and then the c’s); see [ 11 ].”
• Discrete conformal maps and ideal hyperbolic polyhedra (2010, unknown): “The same Ronkin function also appears in the work of Kenyon, Okounkov, and Sheffield on the dimer model [15] [16] as the Legendre dual of a “surface tension” in a variational principle governing the limit shape of random surfaces.”
• Lectures on Dimers (2009, methodology): “In [15] it was also shown, using dimer techniques, that the areas of complementary components of A(P ) and distances between tentacles are global coordinates for the space of Harnack curves with a given Newton polygon.”