Gaps in dimer systems and beyond

What Mathematisches Kolloquium 2011-06-08 16:15 2011-06-08 17:00 2011-06-08 from 16:15 to 17:00 C209 Rindler, Krattenthaler vCal iCal

Mihai Ciucu (Indiana University, Bloomington)

Abstract: The prototypical example of interaction of gaps in dimer coverings was formulated by Fisher and Stephenson in 1963. Suppose we have a huge $2n\times 2n$ chessboard, and we remove a white and a black unit square from around its center. How does the number of domino tilings of the leftover board change, as the two removed unit squares move around the center of the board?

We extend the set-up to dimer coverings on planar lattices, and focus on the interaction of a finite number of gaps in the dimer covering. A number of qualitatively different behaviors arise in this way. These include two-dimensional electrostatics, for bipartite lattices with critical weighting, and a radically different interaction for critical non-bipartite lattice s. Effects of the boundary of lattice graphs will also be considered, as well as connections to some classical problems in enumerative combinatorics involving plane partitions.

Throughout the discussion there is a helpful interplay between combinatorics and physical intuition. The talk is intended for a general audience.

Von 15:45 bis 16:15 Uhr, Kaffeejause im Common Room.