Some basic information about domino tilings of Aztec diamonds can be found on this page.
A cardioid inside the Aztec diamond. Tilings of an Aztec diamond are in bijection with certain of types interlacing/dual interlacing sequences of partitions. Random tilings become random such sequences called Schur processes, introduced by Okounkov-Reshetikhin [OR03]. For the case of the Aztec diamond, this observation is due to Johansson [Joh05]; see [BCC17] for an explicit and expository account. These Schur processes (measures on the sets of all tilings) depend on two sequences of parameters: \(x_1, \dots, x_n\) and \(y_1, \dots, y_n\) (where \(n\) is the size of the Aztec diamond). If all parameters are equal to 1, one recovers uniform tilings as above. If all \(y\)'s are 1 but \(x\)'s alternate periodically between \(a\) and \(a^{-1}\) for a fixed positive parameter \(a\), the measure stops being uniform if \(a \ne 1\). Interestingly, the arctic circle then acquires a cusp and becomes a cardioid, as in the example below. Nota bene: all samples below, unless otherwise stated, have been generated using the RSK bijection and conventions described in [BBB+18].
Here is a size 100 Aztec diamond from the non-uniform measure described, with \(a = 4\), displaying an arctic cardioid.
A tacnode inside the Aztec diamond. In the same setting as above, if the \(y\) sequence, instead of being constantly 1, also alternates between the same values of \(a\) and \(a^{-1}\), the limit shape acquires two cusps. If one sends \(a\) to 0 or \(\infty\), the two cusps will merge, and - the author conjectures - form the tacnode process previously encountered by Borodin-Duits [BD11].
Here is a size 100 Aztec diamond from the non-uniform measure described, with \(a = 4\), displaying two cusps coming together in a tacnode.
Aztec diamonds and conjugate semi-standard Young tableaux. Still in the setting of the Schur process from above, one can take the first half of the \(x\)'s and of the \(y\)'s to be equal to 1, while the others are 0. This results in an embedding, inside the Aztec diamond, of the oriented digital boiling phenomenon of Gravner-Tracy-Widom [GTW01]. The limit shape consists then of two large semi-standard Young tableaux of conjugate shape. Each can be seen as half a plane partition (notice each half of the Aztec diamond right below only contains three of the four types of dominoes). Note the (mild) singularity in the middle. If one varies the proportion of the number of 1's aforedescribed to either above or below \(1/2\), one can obtain either two such smaller tableaux separated by a frozen phase (second example below), or two large tableaux (not conjugate anymore) separated by the Aztec liquid phase in the middle.
Here is a picture of two large semi-standard Young tableaux of conjugate shape inside the Aztec diamond of size 100.
Here is a picture of two large semi-standard Young tableaux of conjugate shape, separated by a frozen phase, inside the Aztec diamond of size 100.
Here is a picture of two large semi-standard Young tableaux, separated by the ``usual'' Aztec liquid phase, inside the Aztec diamond of size 200.
Waterfalls inside the Aztec diamond. Still in the setting of the Schur process, if \(x_i = y_i = q^i\) (using the conventions of [BBB+18]), and for \(q\) close to but not-too-close to 1, one can see---this author conjectures---the waterfall phenomenon of Borodin--Gorin--Rains [BGR10] in domino tilings. Notice the two ``extreme'' liquid sides in the size 200 example below (each containing only three of the four types of dominoes) are joined by an almost frozen phase showing separation between the yellow and green dominoes.
Here is an example of the (conjectural) waterfalls phenomenon inside a size 200 Aztec diamond for \(q = 0.95\).
Random tilings of squares inside the Aztec diamond. Unlike randomly tiling an Aztec diamond, tiling a square does not reveal any limit shapes. The whole region is ``liquid''. One can nevertheless do it efficiently by first centrally embedding the square inside an Aztec diamond (of size 99 for a 100-by-100 square in the examples below), freezing the dimers in the four corners in the obvious way, and using generalized shuffling (as described in [JdlRV06]) to make a uniform sample. Below is its embedding inside the Aztec diamond.
[BBB+18] D. Betea, C. Boutillier, J. Bouttier, G. Chapuy, S. Corteel, and M. Vuletic, Perfect sampling algorithms for Schur processes, Markov Process. Related Fields 24 (2018), no. 3, 381-418.
[BCC17] J. Bouttier, G. Chapuy, and S. Corteel, From Aztec diamonds to pyramids: steep tilings, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5921-5959.
[BD11] A. Borodin and M. Duits, Limits of determinantal processes near a tacnode, Ann. Inst. H. Poincar Probab. Statist., 47 (2011) no. 1, 243-258.
[BGR10] A. Borodin, V. Gorin, and E. M. Rains, q-distributions on boxed plane partitions, Selecta Mathematica 16 (2010), no. 4, 731-789.
[GTW01] J. Gravner, C. A. Tracy, and H. Widom, Limit theorems for height uctuations in a class of discrete space and time growth models, Journal of Statistical Physics 102 (2001), no. 5, 1085-1132.
[JdlRV06] E. Janvresse, T. de la Rue, and Y. Velenik, A note on domino shuffling., The Electronic Journal of Com- binatorics [electronic only] 13 (2006), no. 1, Research paper R30, 20 p.
[Joh05] K. Johansson, The arctic circle boundary and the Airy process, Ann. Probab. 33 (2005), no. 1, 1-30.
[OR03] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), no. 3, 581-603 (electronic).
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