Dimers and amoebae

By Richard Kenyon, A. Okounkov, Scott Sheffield. 2003

566 citations recorded by the source corpus.

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff731012e2226d9dfea642779cbca4ebbfb312e9

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Citations in the corpus

The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability (2008, background): “ the celebrated Lee-Yang theorem [31] (locating the zeros of the partition function of the ferromagnetic Ising model on the imaginary axis in the complex fugacity plane) and its many modern relatives [1, 3, 4, 5, 22, 27, 33, 37, 38, 45, 48, 51]. This also illustrates Hinkkanen’s observations about z”
Brane dimers and quiver gauge theories (2005, methodology): “This is indeed the case, as noted by [41] and used extensively in [24].”
Stochastic six-vertex model (2014, unknown): “ely dierent methods to prove this theorem. Note that in a related context of random height function arising from tilings and dimer models, the variational approach is much more developed, see [CKP], [KOS], [KO]. We will revisit the idea of variational problems later from a point of view of interacti”
Dimer models from mirror symmetry and quivering amoebae (2005, background): “ed, C is now known as a Harnack curve. These are in some sense the “best” or most generic real curves. Recalling the deﬁnition of the spectral curve an edge weights of a dimer model, a main result of [25] is that For any choice of real edge weights subject to a certain sign constraint, the spectral ”
Amoebas, nonnegative polynomials and sums of squares supported on circuits (2014, background): “having their nature in complex algebraic geometry with applications in various mathematical subjects including complex analysis [10,11], the topology of real algebraic curves [24], dynamical systems [8], dimers/crystal shapes [18], and in particular with strong connections to tropical geometry, see ”
Dimer models and toric diagrams (2005, methodology): “ between quiver gauge theories and toric geometries as their moduli space. Dimers turn out to be the right tool to study both in a simple and eﬃcient fashion. 3. Dimer models This section is based on [20, 25]. A bipartite graph is a graph with the property that all vertices can be coloured black or ”
Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, background): “It was shown by [36, 32] that a Harnack curve can always be realized as the limit shape of a crystal model, which makes the link with combinatorics, but we shall not use that here.”
Brownian Gibbs property for Airy line ensembles (2011, background): “In [51], it is shown that models based on perfect matchings (on any weighted doubly-periodic bipartite graph G in the plane) are exactly solvable in a rather strong sense.”
Minimizers of convex functionals arising in random surfaces (2008, background): “More generally, one can construct random surfaces corresponding to random perfect matchings (also called dimer congurations ) on a weighted, bipartite, periodic, planar graph G. Kenyon, Okounkov and Sheeld [ KOS ] showed that these random surfaces converge a.s.”
Limit shapes and the complex Burgers equation (2005, unknown): “(see [ 12 ]). The Newton polygon is defined by |x|+|y|61.”
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.”
Quantum Calabi-Yau and Classical Crystals (2003, background): “We will now quote some results of [24, 25] and indicate their relevance in our setting.”
Nonintersecting paths and the Hahn orthogonal polynomial ensemble (2008, methodology): “The latter measures were classified by Sheffield in his thesis [She], and in the article [ KOS ] correlation kernels for the measures µ� were computed.”