Interesting Esoterica

An arctic circle theorem for groves

Article by Petersen, T. K. and Speyer, D.
  • Published in 2004
  • Added on
In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone' in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds. Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.

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pages
25

BibTeX entry

@article{Petersen2004,
	abstract = {In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone' in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds.   Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.},
	author = {Petersen, T. K. and Speyer, D.},
	month = {jul},
	pages = 25,
	title = {An arctic circle theorem for groves},
	url = {http://arxiv.org/abs/math/0407171 http://arxiv.org/pdf/math/0407171v1},
	year = 2004,
	archivePrefix = {arXiv},
	eprint = {math/0407171},
	primaryClass = {math.CO},
	urldate = {2014-07-01}
}