Newton polygons and zig-zag path correspondences are extended from bipartite to non-bipartite dimer models on torus graphs, with real-rootedness proven for marginal polynomials of several graph families.
Graphical Condensation Generalizations Involving Pfaffians and Determinants
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abstract
Graphical condensation is a technique used to prove combinatorial identities among numbers of perfect matchings of plane graphs. Propp and Kuo first applied this technique to prove identities for bipartite graphs. Yan, Yeh, and Zhang later applied graphical condensation to nonbipartite graphs to prove more complex identities. Here we generalize some of the identities of Yan, Yeh, and Zhang. We also describe the latest generalization of graphical condensation in which the number of perfect matchings of a plane graph is expressed as a Pfaffian or a determinant where the entries are also numbers of perfect matchings of subgraphs.
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Newton polygons for the non-bipartite dimer model
Newton polygons and zig-zag path correspondences are extended from bipartite to non-bipartite dimer models on torus graphs, with real-rootedness proven for marginal polynomials of several graph families.