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Bipartite partial duals and circuits in medial graphs
It is well known that a plane graph is Eulerian if and only if its geometric
dual is bipartite. We extend this result to partial duals of plane graphs. We
then characterize all bipartite partial duals of a plane graph in terms of
oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices
Let S be a flat surface of genus g with cone type singularities. Given a
bipartite graph G isoradially embedded in S, we define discrete analogs of the
2^{2g} Dirac operators on S. These discrete objects are then shown to converge
to the continuous ones, in some appropriate sense. Finally, we obtain necessary
and sufficient conditions on the pair (S,G) for these discrete Dirac operators
to be Kasteleyn matrices of the graph G. As a consequence, if these conditions
are met, the partition function of the dimer model on G can be explicitly
written as an alternating sum of the determinants of these 2^{2g} discrete
Dirac operators.Comment: 39 pages, minor change
Critical Ising model and spanning trees partition functions
We prove that the squared partition function of the two-dimensional critical
Ising model defined on a finite, isoradial graph , is equal to
times the partition function of spanning trees of the graph
, where is the graph extended along the boundary; edges
of are assigned Kenyon's [Ken02] critical weights, and boundary edges of
have specific weights. The proof is an explicit construction,
providing a new relation on the level of configurations between two classical,
critical models of statistical mechanics.Comment: 38 pages, 26 figure
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