9500936 Propp Abstract The investigator studies two-dimensional structures such as tilings of finite regions with the goal of understanding how constraints at the boundary propagate inward in the presence of randomness. He uses methods from the mathematical fields of combinatorics, dynamical systems, and interacting particle systems in order to prove results that quantify the way boundary conditions temper randomness. His main objects of study are currently dimer models, which can also be interpreted as random tiling models. Crystals are ordinarily thought of as the ultimate in orderly arrangement, but at an atomic scale there is often structural disorder with measurable consequences for the bulk properties of matter. Statistical physicists have made much progress towards modeling these sorts of structure, usually by making some simplifying assumptions. One of these is the supposition that the structure is two-dimensional (an appropriate assumption if one is studying surfaces of crystals); another is the supposition that the structure has no boundary. The investigator is relaxing this second assumption and studying what happens in certain standard models of crystalline matter in which there is a boundary of a specified shape having specified structure, and in which all the randomness is confined to the interior.
