### Metadata

### Abstract

The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a $(2+1)$-dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth ${v}_{\mathtt{w}}\left(\rho \right)$ depends on the average interface slope $\rho $, as well as on the edge weights $\mathtt{w}$, that are assumed to be periodic in space. We show that this growth model belongs to the Anisotropic KPZ class [Ton18, Wol91]: one has $det\left[{D}^{2}{v}_{\mathtt{w}}\left(\rho \right)\right]<0$ and the height fluctuations grow at most logarithmically in time. Moreover, we prove that $D{v}_{\mathtt{w}}\left(\rho \right)$ is discontinuous at each of the (finitely many) smooth (or “gaseous”) slopes $\rho $; at these slopes, fluctuations do not diverge as time grows. For a special case of spatially $2-$periodic weights, analogous results have been recently proven [CT19] via an explicit computation of ${v}_{\mathtt{w}}\left(\rho \right)$. In the general case, such a computation is out of reach; instead, our proof goes through a relation between the speed of growth and the limit shape of domino tilings of the Aztec diamond.

### References

[ADPZ04] Dimer Models and Conformal Structures (2004) (https://arxiv.org/abs/2004.02599)

[BF14] Anisotropic Growth of Random Surfaces in $2+1$ Dimensions, Commun. Math. Phys., Volume 325 (2014) no. 2, pp. 603-684 | DOI | MR | Zbl

[BS95] Fractal Concepts in Surface Growth, Cambridge University Press, 1995 | Zbl

[BT18] Two-dimensional anisotropic KPZ growth and limit shapes, J. Stat. Mech. Theory Exp. (2018) no. 8, 083205 | DOI | MR | Zbl

[CCM20] Renormalizing the Kardar–Parisi–Zhang equation in $d\ge 3$ in weak disorder, J. Stat. Phys., Volume 179 (2020) no. 3, pp. 713-728 | DOI | MR | Zbl

[CD20] Constructing a solution of the $(2+1)$-dimensional KPZ equation, Ann. Probab., Volume 48 (2020) no. 2, pp. 1014-1055 | DOI | MR | Zbl

[CFT19] Speed and fluctuations for some driven dimer models, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD), Volume 6 (2019) no. 4, pp. 489-532 | DOI | MR | Zbl

[CKP01] A variational principle for domino tilings, J. Am. Math. Soc., Volume 14 (2001) no. 2, pp. 297-346 | DOI | MR | Zbl

[CLP98] The shape of a typical boxed plane partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR | Zbl

[CSZ20] The two-dimensional KPZ equation in the entire subcritical regime, Ann. Probab., Volume 48 (2020) no. 3, pp. 1086-1127 | DOI | MR | Zbl

[CT19] A $(2+1)$-dimensional anisotropic KPZ growth model with a smooth phase, Commun. Math. Phys., Volume 367 (2019) no. 2, pp. 483-516 | DOI | MR | Zbl

[DGRZ20] Fluctuations of the solutions to the KPZ equation in dimensions three and higher, Probab. Theory Relat. Fields, Volume 176 (2020) no. 3-4, pp. 1217-1258 | DOI | MR | Zbl

[EKLP92a] Alternating-sign matrices and domino tilings. I, J. Algebr. Comb., Volume 1 (1992) no. 2, pp. 111-132 | DOI | MR | Zbl

[EKLP92b] Alternating-sign matrices and domino tilings. II, J. Algebr. Comb., Volume 1 (1992) no. 3, pp. 219-234 | DOI | MR | Zbl

[Foc15] Inverse spectral problem for GK integrable system (2015) (https://arxiv.org/abs/1503.00289)

[GK13] Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 5, pp. 747-813 | DOI | Numdam | MR | Zbl

[GW95] Stationary states of crystal growth in three dimensions, J. Stat. Phys., Volume 81 (1995) no. 3-4, pp. 999-1012 | Zbl

[HH12] $2+1$-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions, Phys. Rev. Lett., Volume 109 (2012) no. 17, 170602 | DOI

[Ken09] Lectures on dimers, Statistical mechanics. Papers based on the presentations at the IAS/PCMI summer conference, Park City, UT, USA, July 1–21, 2007 (Scheffield, Scott, ed.) (IAS/Park City Mathematics Series), American Mathematical Society; Institute for Advanced Study, 2009, pp. 191-230 | Zbl

[KO06] Planar dimers and Harnack curves, Duke Math. J., Volume 131 (2006) no. 3, pp. 499-524 | DOI | MR | Zbl

[KO07] Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | DOI | MR | Zbl

[KOS06] Dimers and amoebae, Ann. Math., Volume 163 (2006) no. 3, pp. 1019-1056 | DOI | MR | Zbl

[KPZ86] Dynamic scaling of growing interfaces, Phys. Rev. Lett., Volume 56 (1986) no. 9, pp. 889-892 | DOI | Zbl

[Kuc17] Limit shapes for the dimer model (2017) (https://arxiv.org/abs/1712.08396)

[LT19] Hydrodynamic limit and viscosity solutions for a two-dimensional growth process in the anisotropic KPZ class, Commun. Pure Appl. Math., Volume 72 (2019) no. 3, pp. 620-666 | DOI | MR | Zbl

[Mor66] Multiple integrals in the calculus of variations, Grundlehren der Mathematischen Wissenschaften, 130, Springer, 1966 | MR | Zbl

[MU18] The scaling limit of the KPZ equation in space dimension 3 and higher, J. Stat. Phys., Volume 171 (2018) no. 4, pp. 543-598 | DOI | MR | Zbl

[NR62] A theorem on maps with non-negative Jacobians, Mich. Math. J., Volume 9 (1962) no. 2, pp. 173-176 | MR | Zbl

[Pro03] Generalized Domino–Shuffling, Theor. Comput. Sci., Volume 303 (2003) no. 2-3, pp. 267-301 | DOI | MR | Zbl

[PS97] An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime, J. Stat. Phys., Volume 88 (1997) no. 5-6, pp. 999-1012 | DOI | MR | Zbl

[PT80] The theory of two-dimensional incommensurate crystals, Sov. Phys., JETP, Volume 51 (1980) no. 1, pp. 134-148 | MR

[S10] Minimizers of convex functionals arising in random surfaces, Duke Math. J., Volume 151 (2010) no. 3, pp. 487-532 | DOI | MR | Zbl

[TFW92] Kinetic surface roughening. II. Hypercube stacking models, Phys. Rev. A, Volume 45 (1992) no. 10, pp. 7162-7169 | DOI

[Ton17] A $(2+1)$-dimensional growth process with explicit stationary measures, Ann. Probab., Volume 45 (2017) no. 5, pp. 2899-2940 | DOI | MR | Zbl

[Ton18] (2+ 1)-dimensional interface dynamics: mixing time, hydrodynamic limit and Anisotropic KPZ growth, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures (Sirakov, Boyan et al., eds.), World Scientific; Sociedade Brasileira de Matemática, 2018, pp. 2733-2758 | Zbl

[Wol91] Kinetic roughening of vicinal surfaces, Phys. Rev. Lett., Volume 67 (1991), pp. 1783-1786 | DOI

[Zha18] Domino shuffling height process and its hydrodynamic limit (2018) (https://arxiv.org/abs/1808.07409)