Georgia Institute of Technology College of Sciences

We analyze the mixing time of a natural local Markov Chain (Gibbs sampler) for twocommonly studied models of random surfaces: (i) discrete monotone surfaces in Z3 with ``almostplanar" boundary conditions and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model.In both cases we prove the first almost optimal bounds O(L^2 polylog(L)) where L is the natural size of the system. Our proof is inspired by the so-called ``mean curvature" heuristic: on a large scale, the dynamics should approximate a deterministic motion in which each point of the surface moves according to a drift proportional to the local inverse mean curvature radius. Key technical ingredients are monotonicity, coupling and an argument due to D.Wilson in the framework of lozenge tiling Markov Chains together with Kenyon's results on the free Gaussian field approximation of monotone surfaces. The novelty of our approach with respect to previous results consists in proving that, with high probability, the dynamics is dominated by a deterministic evolution which, apart from polylog(L) corrections, follows the mean curvature prescription. Our method works equally well for both models despite the fact that their equilibrium maximal deviations from the average height profile occur on very different scales (log(L) for monotone surfaces and L^{1/2} for the SOS model).This is work in collaboration with PIETRO CAPUTO and FABIO LUCIO TONINELLI

We study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be nonautonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative perturbations.

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.