Random Tilings - NSF Proposal

• Main Author: Jim Propp
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Subjects: Arctic
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.3769; http://www-math.mit.edu/~propp/proposal.ps

this document shows a random domino-tiling of the Aztec diamond of order 64, generated by domino-shuffling software written in 1993 by my undergraduate assistant Sameera Iyengar. This picture, and others like it, suggested that there is a qualitative difference between the behavior of a random tiling near the corners of the Aztec diamond and the behavior near the middle: near the corners, the tiles tend to line up in one particular direction, forming a "brick-wall pattern," while in the middle of the region there is a mixture of orientations (with the precise composition of the mixture varying in different places). I conjectured that this phenomenon can be described in terms of the circle inscribed in the Aztec diamond: with probability tending to 1 as n goes to infinity, a random tiling of the Aztec diamond of order n will exhibit a mix of orientations at locations inside the circle and will exhibit a brick-wall pattern at locations outside the circle (where our notions of "inside" and "outside" incorporate a thickening of the circle by nffl, for some fixed ffl ? 0 that can be as small as desired). In 1994, I was able to prove this conjecture (now called the "arctic circle theorem") using ideas from the theory of interacting particle systems in one dimension [J1]. In the meantime, my undergraduate students and I followed a different line that proved to be equally productive. We sought a way of calculating, for any two adjoining squares in an Aztec diamond, the probability that, under random tiling, the two squares would be covered by the same domino. It turned out that domino-shuffling gives rise to recurrence relations that allow one to iteratively calculate these probabilities (which we call "domino-probabilities" or, using the dual picture, "edge-probabilities") a.