Domino tilings of the Aztec diamond with doubly periodic weightings
In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a non-intersecting pat...
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ftdatacite:10.48550/arxiv.1911.01250 2023-05-15T15:05:06+02:00 Domino tilings of the Aztec diamond with doubly periodic weightings Berggren, Tomas 2019 https://dx.doi.org/10.48550/arxiv.1911.01250 https://arxiv.org/abs/1911.01250 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Probability math.PR Mathematical Physics math-ph FOS Mathematics FOS Physical sciences Article CreativeWork article Preprint 2019 ftdatacite https://doi.org/10.48550/arxiv.1911.01250 2022-03-10T16:29:05Z In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a non-intersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel we obtain the local behavior in the smooth and rough regions as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture such as the arctic curves and in particular the number of smooth regions is deduced. Moreover we compute the limit of the height function and as a consequence we confirm, in the setting of this paper, that the limit in the rough region fulfills the complex Burgers' equation, as stated by Kenyon and Okounkov. : 57 pages, 12 figures, typos corrected, minor clarifications in some of the arguments Article in Journal/Newspaper Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic Kenyon ENVELOPE(-174.867,-174.867,-85.167,-85.167) |
institution |
Open Polar |
collection |
DataCite Metadata Store (German National Library of Science and Technology) |
op_collection_id |
ftdatacite |
language |
unknown |
topic |
Probability math.PR Mathematical Physics math-ph FOS Mathematics FOS Physical sciences |
spellingShingle |
Probability math.PR Mathematical Physics math-ph FOS Mathematics FOS Physical sciences Berggren, Tomas Domino tilings of the Aztec diamond with doubly periodic weightings |
topic_facet |
Probability math.PR Mathematical Physics math-ph FOS Mathematics FOS Physical sciences |
description |
In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a non-intersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel we obtain the local behavior in the smooth and rough regions as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture such as the arctic curves and in particular the number of smooth regions is deduced. Moreover we compute the limit of the height function and as a consequence we confirm, in the setting of this paper, that the limit in the rough region fulfills the complex Burgers' equation, as stated by Kenyon and Okounkov. : 57 pages, 12 figures, typos corrected, minor clarifications in some of the arguments |
format |
Article in Journal/Newspaper |
author |
Berggren, Tomas |
author_facet |
Berggren, Tomas |
author_sort |
Berggren, Tomas |
title |
Domino tilings of the Aztec diamond with doubly periodic weightings |
title_short |
Domino tilings of the Aztec diamond with doubly periodic weightings |
title_full |
Domino tilings of the Aztec diamond with doubly periodic weightings |
title_fullStr |
Domino tilings of the Aztec diamond with doubly periodic weightings |
title_full_unstemmed |
Domino tilings of the Aztec diamond with doubly periodic weightings |
title_sort |
domino tilings of the aztec diamond with doubly periodic weightings |
publisher |
arXiv |
publishDate |
2019 |
url |
https://dx.doi.org/10.48550/arxiv.1911.01250 https://arxiv.org/abs/1911.01250 |
long_lat |
ENVELOPE(-174.867,-174.867,-85.167,-85.167) |
geographic |
Arctic Kenyon |
geographic_facet |
Arctic Kenyon |
genre |
Arctic |
genre_facet |
Arctic |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1911.01250 |
_version_ |
1766336849270276096 |