An introduction to the dimer model

By Richard Kenyon. 2003

191 citations recorded by the source corpus.

A perfect matching of a graph is a subset of edges which covers every vertex exactly once, that is, for every vertex there is exactly one edge in the set with that vertex as endpoint. The dimer model is the study of the set of perfect matchings of a (possibly infinite) graph. The most well-known example is when the graph is Z, for which perfect matchings are equivalent (via a simple duality) to domino tilings, that is, tilings of the plane with 2× 1 and 1× 2 rectangles. In the first three sections we study domino tilings of the plane and of finite polygonal regions, or equivalently, perfect matchings on Z and subgraphs of Z. In the last two sections we study the FK-percolation model and the dimer model on a more general family of planar graphs.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7c66ff59834233f8ea7857e4e0b249f1a7c98f87

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Citations in the corpus

Dimer models from mirror symmetry and quivering amoebae (2005, background): “edges of the graph indeed traces out a zig-zag pattern. For an arbitrary bipartite graph on T2, a zig-zag path can intersect itself. But for a dimer model describing a toric gauge theory, it is shown [26, 29, 30] that these zig-zag paths are simple closed curves in T2 and never intersect themselves.”
Dimer models and toric diagrams (2005, background): “ 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 ”
Dimers and amoebae (2003, background): “o a product of the 2 corresponding edge weights of G) from the set of all perfect matchings on G. These so-called dimer models are the subject of an extensive physics and mathematics literature. (See [9] for a survey.) Since the set of perfect matchings of G is also in one-to-one correspondence with”
CONSISTENCY CONDITIONS FOR DIMER MODELS (2011, methodology): “The notion of a zigzag path is based on work by Kenyon in [16] and Kenyon and Schlenker in [17].”
Brane tilings and their applications (2008, methodology): “The concept of isoradial embedding is known in mathematics since long ago [11, 71, 72], but their physical meaning is first proposed in [70].”