In the first half of the talk, I will define the dimer model on planar graphs and prove Kasteleyn’s groundbreaking result expressing the partition function (i.e. the generating function) of the model as a Pfaffian. I will then survey various results arising as a consequence, culminating in the beautiful limit shape theorems of Kenyon, Okounkov and coworkers.

In the second half, I will define a variant of the monomer-dimer model on planar graphs and prove that the partition function of this model can be expressed as a determinant. I will use this result to calculate various quantities of interest to statistical physicists and end with some open questions.