Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble
We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508624000167/type/journal_article |
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| author | Amol Aggarwal Jiaoyang Huang |
| author_facet | Amol Aggarwal Jiaoyang Huang |
| author_sort | Amol Aggarwal |
| collection | DOAJ |
| description | We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics. |
| format | Article |
| id | doaj-art-b69a4630992e47a8b7dc130a6caf2826 |
| institution | Kabale University |
| issn | 2050-5086 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-b69a4630992e47a8b7dc130a6caf28262025-02-12T03:45:03ZengCambridge University PressForum of Mathematics, Pi2050-50862025-01-011310.1017/fmp.2024.16Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line EnsembleAmol Aggarwal0Jiaoyang Huang1Columbia University, NY Clay Mathematics Institute; E-mail:University of Pennsylvania, PAWe consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics.https://www.cambridge.org/core/product/identifier/S2050508624000167/type/journal_article60F0560B20 |
| spellingShingle | Amol Aggarwal Jiaoyang Huang Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble Forum of Mathematics, Pi 60F05 60B20 |
| title | Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble |
| title_full | Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble |
| title_fullStr | Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble |
| title_full_unstemmed | Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble |
| title_short | Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble |
| title_sort | edge statistics for lozenge tilings of polygons ii airy line ensemble |
| topic | 60F05 60B20 |
| url | https://www.cambridge.org/core/product/identifier/S2050508624000167/type/journal_article |
| work_keys_str_mv | AT amolaggarwal edgestatisticsforlozengetilingsofpolygonsiiairylineensemble AT jiaoyanghuang edgestatisticsforlozengetilingsofpolygonsiiairylineensemble |