Friday, May 6, 2011 - 3:05pm
1 hour (actually 50 minutes)
Colloids are mixtures of molecules well-studied in material science that are not well-understood mathematically. Physicists model colloids as a system of two types of tiles (type A and type B) embedded on a region of the plane, where no two tiles can overlap. It is conjectured that at high density, the type A tiles tend to separate out and form large "clusters". To verify this conjecture, we need methods for counting these configurations directly or efficient algorithms for sampling. Local sampling algorithms are known to be inefficient. However, we provide the first rigorous analysis of a global "DK Algorithm" introduced by Dress and Krauth. We also examine the clustering effect directly via a combinatorial argument. We prove for a certain class of colloid models that at high density the configurations are likely to exhibit clustering, whereas at low density the tiles are all well-distributed. Joint work with Sarah Miracle and Dana Randall.