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Arctic circles, domino tilings and random square Young tableaux

Mathematical Physics & Probability

Speaker: Dan Romik, UC Davis
Location: 1147 MSB
Start time: Wed, Oct 28 2009, 4:10PM

If you tile a large checkerboard randomly with dominoes, the resulting tiling has local statistics that are more or less spatially homogeneous. However, if you choose a random domino tiling of a domain that looks like a (discrete version of a) square rotated by 45 degrees, called the "Aztec diamond", Jockusch, Propp and Shor proved that the tiling is far from being homogeneous: In fact, there will be a "frozen" region consisting of dominoes that align in a perfect non- random brickwork pattern adjacent to each of the 4 corners of the diamond, and a "temperate" region where the true randomness is, and the shape of the temperate region is asymptotically exactly the disk inscribed in the diamond. Such an "arctic circle" shape has been observed in several other random combinatorial models, one of them arising in the study of random square Young tableaux. In this talk I will describe a new proof of a more detailed version of the arctic circle theorem that also describes accurately the domino statistics inside the temperate region, and use it to explain why the arctic circle phenomenon for random domino tilings and the one for random square Young tableaux are more closely related than was previously thought.