# zbMATH — the first resource for mathematics

Limit shapes and the complex Burgers equation. (English) Zbl 1156.14029
Dimer models are statistical models defined on periodic edge-weighted planar bipartite graphs. A dimer configuration is a perfect matching, i.e., a set of edges such that each vertex is incident with exactly one edge. An alternative description is via height functions, defined on the set of faces and determined by the matching uniquely up to an additive constant. For dimer configurations associated with tilings of the plane, height functions are well-known tools to recognize tilability of a region.
The paper under review concerns limit shapes of height functions with prescribed boundary conditions. The most interesting phenomenon is the formation of facets, typical for crystal surfaces. In previous works by the same authors [jointly with S. Sheffield, Ann. Math. (2) 163, No. 3, 1019–1056 (2006; Zbl 1154.82007); Duke Math. J. 131, No. 3, 499–524 (2006; Zbl 1100.14047)], a spectral curve $$P(z,w) = 0$$ of a given periodic edge-weighted planar bipartite graph has been introduced and shown to have the Harnack maximality property. Also the basic classification of frozen/liquid/gaseous phases of a dimer model has been given. Finally, the limit shapes $$h$$ of the height functions have been shown to minimize the functional $$\int \sigma(\nabla h(x,y))\,dx\,dy$$, where the “surface tension” $$\sigma = \mathcal R^\vee$$ is the Legendre dual of the Ronkin function $$\mathcal R$$ of the spectral curve $$P(z,w) = 0$$.
Continuing the previous research, the authors concentrate on the liquid region, where the surface tension is analytic and strictly convex, so that the minimizer $$h$$ is analytic as well. Theorem 1 says that the Euler–Lagrange equation of the volume-constrained minimizing problem can be rewritten as $$\nabla h = (\arg w, -\arg z)/\pi$$, where $$z,w$$ are complex functions satisfying $$z_x/z + w_y/w = c$$ and $$P(z,w) = 0$$. It follows that the equation is exactly solvable and its solutions are parametrized by an analytic function $$Q$$ of two variables. The next step is to prove that the function $$Q$$ is algebraic for a dense set of boundary conditions.
This is done in Theorem 2 in the case of $$P(z,w) = z + w - 1$$ (the height functions are stepped surfaces). A number of details of independent interest (the winding property and its consequences) are established in the course of the proof. Duals of winding curves are called cloud curves. An equivalence between feasibility of a polygonal contour and (unique) existence of an inscribed cloud curve is established in Theorem 3.
Several explicitly computed examples crown the work.

##### MSC:
 14H81 Relationships between algebraic curves and physics 35F30 Boundary value problems for nonlinear first-order PDEs 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
Full Text:
##### References:
  Abanov, A. G., Hydrodynamics of correlated systems, in Applications of Random Matrices in Physics, NATO Sci. Ser. II Math. Phys. Chem., 221, pp. 139–161. Springer, Dordrecht, 2006. · Zbl 1135.82018  Cohn, H., Kenyon, R. & Propp, J., A variational principle for domino tilings. J. Amer. Math. Soc., 14 (2001), 297–346. · Zbl 1037.82016 · doi:10.1090/S0894-0347-00-00355-6  Cohn, H. & Pemantle, R., Private communication, 2001.  Fournier, J.-C., Pavage des figures planes sans trous par des dominos: fondement graphique de l’algorithme de Thurston et parallélisation. C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 107–112. · Zbl 0838.05092  Fulton, W. & Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, pp. 45–96. Amer. Math. Soc., Providence, RI, 1997. · Zbl 0898.14018  Griffiths, P. & Harris, J., Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York, 1994. · Zbl 0836.14001  Guionnet, A., First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models. Comm. Math. Phys., 244 (2004), 527–569. · Zbl 1076.82026 · doi:10.1007/s00220-003-0992-4  Harris, J. & Morrison, I., Moduli of Curves. Graduate Texts in Mathematics, 187. Springer, New York, 1998. · Zbl 0913.14005  Itenberg, I. & Viro, O., Patchworking algebraic curves disproves the Ragsdale conjecture. Math. Intelligencer, 18 (1996), 19–28. · Zbl 0876.14017 · doi:10.1007/BF03026748  Kenyon, R., Height fluctuations in honeycomb dimers. Preprint, 2004. arXiv:math-ph/0405052.  Kenyon, R. & Okounkov, A., Planar dimers and Harnack curves. Duke Math. J., 131 (2006), 499–524. · Zbl 1100.14047 · doi:10.1215/S0012-7094-06-13134-4  Kenyon, R., Okounkov, A. & Sheffield, S., Dimers and amoebae. Ann. of Math., 163 (2006), 1019–1056. · Zbl 1154.82007 · doi:10.4007/annals.2006.163.1019  Kenyon, R., Okounkov, A. & Vafa, C., In preparation.  Matytsin, A., On the large-N limit of the Itzykson–Zuber integral. Nuclear Phys. B, 411 (1994), 805–820. · Zbl 1049.81631 · doi:10.1016/0550-3213(94)90471-5  Mikhalkin, G., Amoebas of algebraic varieties and tropical geometry, in Different Faces of Geometry, Int. Math. Ser. (N.Y.), 3, pp. 257–300. Kluwer/Plenum, New York, NY, 2004. · Zbl 1072.14013  Mikhalkin, G., Enumerative tropical algebraic geometry in R 2. J. Amer. Math. Soc., 18 (2005), 313–377. · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7  Morrey, C. B., Jr., Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966. · Zbl 0142.38701  Okounkov, A., Random surfaces enumerating algebraic curves, in European Congress of Mathematics (Stockholm, 2004), pp. 751–768. Eur. Math. Soc., Zürich, 2005. · Zbl 1078.14082  Okounkov, A., Reshetikhin, N. & Vafa, C., Quantum Calabi–Yau and classical crystals, in The Unity of Mathematics, Progr. Math., 244, pp. 597–618. Birkhäuser, Boston, MA, 2006.  Passare, M. & Rullgård, H., Amoebas, Monge–Ampère measures, and triangulations of the Newton polytope. Duke Math. J., 121 (2004), 481–507. · Zbl 1043.32001 · doi:10.1215/S0012-7094-04-12134-7  Pokrovskii, V. & Talapov, A., The theory of two-dimensional incommensurate crystals. Zh. Èksper. Teoret. Fiz., 78 (1980), 269–295 (Russian). English translation in Soviet Phys. JETP, 51 (1980), 134–148.  Speyer, D. E., Horn’s problem, Vinnikov curves, and the hive cone. Duke Math. J., 127 (2005), 395–427. · Zbl 1069.14037 · doi:10.1215/S0012-7094-04-12731-0  Vinnikov, V., Selfadjoint determinantal representations of real plane curves. Math. Ann., 296 (1993), 453–479. · Zbl 0789.14029 · doi:10.1007/BF01445115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.