The double dimer model with quaternions

Опубликовано 17 августа 2016, 21:56

The 'dimer model' is the study of the set of perfect matchings (dimer coverings) of a graph. For planar graphs, it is possible to count the number of dimer coverings using determinants. This technique has led to a very rich theory of the dimer coverings of Z^2 and other periodic planar graphs. The 'double dimer model' is obtained by superimposing two independent uniform dimer coverings, resulting in a set of loops (and doubled edges). Typical probability questions in this setting are: how long are the loops and where do they go? By introducing quaternionic variables in this model one can measure certain 'topological' quantities, and in particular answer this second question for planar graphs. We show how to use these ideas to prove the conformal invariance of double-dimer loops in Z^2 (in the scaling limit of small mesh size).