Asymptotic Geometry of Discrete Interlaced Patterns: Part II
Annales de l'Institut Fourier, Volume 70 (2020) no. 1, pp. 375-436.

We study the asymptotic boundary of the liquid region of large random lozenge tiling models defined by uniformly random interlacing particle systems. In particular, we study a non-phase separating part of the boundary, i.e., a part of the boundary that does not border to a frozen phase. This is called the singular part of the boundary. We prove that isolated components of this boundary are lines and classify four different cases. Moreover, we show that the singular part of the boundary can have infinite one-dimensional Hausdorff measure. This has implications to the study of the free boundary problem arising in the variational problem studied by Kenyon and Okounkov and in a related work by De Silva and Savin.

Nous étudions la limite asymptotique de la région liquide de grands modèles aléatoires de tuiles rhombiques définies par des systèmes de particules uniformément et aléatoirement entrelacés. En particulier nous étudions une partie séparatrice, non phasée, de la limite, i.e. la partie de la limite qui ne touche pas une phase gelée. Cela s’appelle la partie singulière de la limite. Nous prouvons que les composantes isolées de cette limite sont des droites qui se classent en quatre cas différents. De plus, nous montrons que la partie singulière de la limite peut avoir une mesure de Hausdorff infinie et unidimensionnelle. Cela a des implications pour l’étude du problème de la limite libre découlant du problème variationnel étudié par Kenyon et Okounkov et un travail relié de De Silva et Savin.

Received: 2017-12-15
Revised: 2018-09-26
Accepted: 2019-01-17
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3315
Classification: 42AXX,  42B20,  60G55
Keywords: Interlaced patterns, liquid region, boundary.
@article{AIF_2020__70_1_375_0,
     author = {Duse, Erik and Metcalfe, Anthony},
     title = {Asymptotic Geometry of Discrete Interlaced Patterns: Part II},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {1},
     year = {2020},
     pages = {375-436},
     doi = {10.5802/aif.3315},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_1_375_0/}
}
Duse, Erik; Metcalfe, Anthony. Asymptotic Geometry of Discrete Interlaced Patterns: Part II. Annales de l'Institut Fourier, Volume 70 (2020) no. 1, pp. 375-436. doi : 10.5802/aif.3315. https://aif.centre-mersenne.org/item/AIF_2020__70_1_375_0/

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