The domino shuffling algorithm and Anisotropic KPZ stochastic growth
Annales Henri Lebesgue, Volume 4 (2021) , pp. 1005-1034.

Metadata

Keywordsrandom tilings, stochastic interface growth, anisotropic KPZ

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 w (ρ) depends on the average interface slope ρ, as well as on the edge weights 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[D 2 v w (ρ)]<0 and the height fluctuations grow at most logarithmically in time. Moreover, we prove that Dv w (ρ) is discontinuous at each of the (finitely many) smooth (or “gaseous”) slopes ρ; 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 w (ρ). 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] Astala, Kari; Duse, Erik; Prause, István; Zhong, Xiao Dimer Models and Conformal Structures (2004) (https://arxiv.org/abs/2004.02599)

[BF14] Borodin, Alexei; Ferrari, Patrik L. Anisotropic Growth of Random Surfaces in 2+1 Dimensions, Commun. Math. Phys., Volume 325 (2014) no. 2, pp. 603-684 | Article | MR 3148098 | Zbl 1303.82015

[BS95] Barabási, Albert-László; Stanley, H. Eeugene Fractal Concepts in Surface Growth, Cambridge University Press, 1995 | Zbl 0838.58023

[BT18] Borodin, Alexei; Toninelli, Fabio Lucio Two-dimensional anisotropic KPZ growth and limit shapes, J. Stat. Mech. Theory Exp. (2018) no. 8, 083205 | Article | MR 3855490 | Zbl 1457.82283

[CCM20] Comets, Francis; Cosco, Clément; Mukherjee, Chiranjib Renormalizing the Kardar–Parisi–Zhang equation in d3 in weak disorder, J. Stat. Phys., Volume 179 (2020) no. 3, pp. 713-728 | Article | MR 4099995 | Zbl 1434.60278

[CD20] Chatterjee, Sourav; Dunlap, Alexander Constructing a solution of the (2+1)-dimensional KPZ equation, Ann. Probab., Volume 48 (2020) no. 2, pp. 1014-1055 | Article | MR 4089501 | Zbl 1434.60148

[CFT19] Chhita, Sunil; Ferrari, Patrik L.; Toninelli, Fabio Lucio 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 | Article | MR 4033679 | Zbl 1431.05036

[CKP01] Cohn, Henry; Kenyon, Richard; Propp, James A variational principle for domino tilings, J. Am. Math. Soc., Volume 14 (2001) no. 2, pp. 297-346 | Article | MR 1815214 | Zbl 1037.82016

[CLP98] Cohn, Henry; Larsen, Michael; Propp, James The shape of a typical boxed plane partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR 1641839 | Zbl 0908.60083

[CSZ20] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos The two-dimensional KPZ equation in the entire subcritical regime, Ann. Probab., Volume 48 (2020) no. 3, pp. 1086-1127 | Article | MR 4112709 | Zbl 1444.60061

[CT19] Chhita, Sunil; Toninelli, Fabio Lucio A (2+1)-dimensional anisotropic KPZ growth model with a smooth phase, Commun. Math. Phys., Volume 367 (2019) no. 2, pp. 483-516 | Article | MR 3936124 | Zbl 1428.82046

[DGRZ20] Dunlap, Alexander; Gu, Yu; Ryzhik, Lenya; Zeitouni, Ofer 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 | Article | MR 4087492 | Zbl 1445.35345

[EKLP92a] Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James Alternating-sign matrices and domino tilings. I, J. Algebr. Comb., Volume 1 (1992) no. 2, pp. 111-132 | Article | MR 1226347 | Zbl 0779.05009

[EKLP92b] Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James Alternating-sign matrices and domino tilings. II, J. Algebr. Comb., Volume 1 (1992) no. 3, pp. 219-234 | Article | MR 1194076 | Zbl 0788.05017

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

[GK13] Goncharov, Alexander B.; Kenyon, Richard Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 5, pp. 747-813 | Article | Numdam | MR 3185352 | Zbl 1288.37025

[GW95] Gates, David G.; Westcott, Mark Stationary states of crystal growth in three dimensions, J. Stat. Phys., Volume 81 (1995) no. 3-4, pp. 999-1012 | Zbl 1107.60325

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

[Ken09] Kenyon, Richard 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 1180.82001

[KO06] Kenyon, Richard; Okounkov, Andrei Planar dimers and Harnack curves, Duke Math. J., Volume 131 (2006) no. 3, pp. 499-524 | Article | MR 2219249 | Zbl 1100.14047

[KO07] Kenyon, Richard; Okounkov, Andrei Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | Article | MR 2358053 | Zbl 1156.14029

[KOS06] Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott Dimers and amoebae, Ann. Math., Volume 163 (2006) no. 3, pp. 1019-1056 | Article | MR 2215138 | Zbl 1154.82007

[KPZ86] Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng Dynamic scaling of growing interfaces, Phys. Rev. Lett., Volume 56 (1986) no. 9, pp. 889-892 | Article | Zbl 1101.82329

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

[LT19] Legras, Martin; Toninelli, Fabio Lucio 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 | Article | MR 3911895 | Zbl 1417.82024

[Mor66] Morrey, Charles B. Jr. Multiple integrals in the calculus of variations, Grundlehren der Mathematischen Wissenschaften, 130, Springer, 1966 | MR 0202511 | Zbl 0142.38701

[MU18] Magnen, Jacques; Unterberger, Jérémie The scaling limit of the KPZ equation in space dimension 3 and higher, J. Stat. Phys., Volume 171 (2018) no. 4, pp. 543-598 | Article | MR 3790153 | Zbl 1394.35508

[NR62] Nijenhuis, Albert; Richardson, Roger W. A theorem on maps with non-negative Jacobians, Mich. Math. J., Volume 9 (1962) no. 2, pp. 173-176 | MR 140030 | Zbl 0111.05902

[Pro03] Propp, James Generalized Domino–Shuffling, Theor. Comput. Sci., Volume 303 (2003) no. 2-3, pp. 267-301 | Article | MR 1990768 | Zbl 1052.68095

[PS97] Prähofer, Michaelm; Spohn, Herbert 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 | Article | MR 1478060 | Zbl 0945.8243

[PT80] Pokrovskiĭ, V. L.; Talapov, A. L. The theory of two-dimensional incommensurate crystals, Sov. Phys., JETP, Volume 51 (1980) no. 1, pp. 134-148 | MR 579253

[S10] de Silva, Daniela; Savin, Ovidiu Minimizers of convex functionals arising in random surfaces, Duke Math. J., Volume 151 (2010) no. 3, pp. 487-532 | Article | MR 2605868 | Zbl 1204.35080

[TFW92] Tang, Lei-Han; Forrest, Bruce M.; Wolf, Dietrich E. Kinetic surface roughening. II. Hypercube stacking models, Phys. Rev. A, Volume 45 (1992) no. 10, pp. 7162-7169 | Article

[Ton17] Toninelli, Fabio Lucio A (2+1)-dimensional growth process with explicit stationary measures, Ann. Probab., Volume 45 (2017) no. 5, pp. 2899-2940 | Article | MR 3706735 | Zbl 1383.82031

[Ton18] Toninelli, Fabio Lucio (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, ed.), World Scientific; Sociedade Brasileira de Matemática, 2018, pp. 2733-2758 | Zbl 1453.82046

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

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