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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.

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
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