An arctic circle theorem for groves

• Published in 2004
• Added on 2014-07-01

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.

Links

• http://arxiv.org/abs/math/0407171
• http://arxiv.org/pdf/math/0407171v1

Other information

BibTeX entry

@article{Petersen2004,
	key = {Petersen2004},
	type = {article},
	title = {An arctic circle theorem for groves},
	author = {Petersen, T. K. and Speyer, D.},
	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.},
	comment = {},
	date_added = {2014-07-01},
	date_published = {2004-07-01},
	urls = {http://arxiv.org/abs/math/0407171,http://arxiv.org/pdf/math/0407171v1},
	collections = {},
	month = {jul},
	pages = 25,
	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}
}