3 2 0 S t o c k a de S t ., L a w r e n c e , K S 66 0 4 9 • ( 7 8 5) 31 2 -8 0 1 2• mitch e ll @ m a t h .k u .edu Lon MitchellEducation

Monotonic surfaces spanning finite regions of Z d arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z 2 , and for bias λ > d 2 in Z d , when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.

No abstract

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/ nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.