Mathematical Physics Seminar

The monomer-dimer covering problem shows up in several areas of

physics and other fields.  While the partition function of the  pure dimer

covering problem on planar lattices was solved ages ago by Kasteleyn and

Fisher-Temperley in terms of Paffians (whose squares are easily computable

determinants), the inclusion of monomers (i.e., vertices uncovered by

dimers)  is an intractable problem.  Together with Alessandro Giuliani

and Ian Jauslin we have succeeded in utilizing an ancient theorem*

to write the partition function of a restricted monomer-dimer problem

as a Pfaffian. This is the model in which monomers are allowed on the

boundary vertices of an arbitrary planar graph.

*J. Combinatorial Theory  5, 313-319 (1968).