Arvind Ayyer: The Monopole-Dimer Model

The monomer-dimer model in statistical physics, unlike the dimer model, is not believed to have an "easy" solution in terms of Pfaffians. We develop a new signed model of monopoles and loops whose partition function is a determinant for any planar graph. We then apply this to the rectangular grid and obtain a generalization of Kasteleyn's miraculous product formula.

Similar to the dimer model, the partition function turns out to be a square for special graphs, and we find a combinatorial interpretation of the square root. Lastly, we study the thermodynamic limit and obtain formulas for the free energy, the entropy and certain correlations.