Sampling Biased Lattice Configurations using Exponential Metrics

Abstract

Monotonic surfaces spanning finite regions of ℤd 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 ℤ2, and for bias λ > d2 in ℤd, 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.