×

Airy point process at the liquid-gas boundary. (English) Zbl 1428.60063

Summary: Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function, we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Borodin, A. and Ferrari, P. L. (2014). Anisotropic growth of random surfaces in \(2+1\) dimensions. Comm. Math. Phys.325 603–684. · Zbl 1303.82015
[2] Borodin, A. and Rains, E. M. (2005). Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys.121 291–317. · Zbl 1127.82017
[3] Boutillier, C. (2007). Pattern densities in non-frozen planar dimer models. Comm. Math. Phys.271 55–91. · Zbl 1124.60080
[4] Boutillier, C., Bouttier, J., Chapuy, G., Corteel, S. and Ramassamy, S. (2017). Dimers on rail yard graphs. Ann. Inst. Henri Poincaré D4 479–539. · Zbl 1391.82008
[5] Bouttier, J., Chapuy, G. and Corteel, S. (2017). From Aztec diamonds to pyramids: Steep tilings. Trans. Amer. Math. Soc.369 5921–5959. · Zbl 1362.05033
[6] Breuer, J. and Duits, M. (2014). The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles. Adv. Math.265 441–484. · Zbl 1335.60031
[7] Bufetov, A. and Gorin, V. (2016). Fluctuations of particle systems determined by Schur generating functions. Available at arXiv:1604.01110. · Zbl 1400.82064
[8] Bufetov, A. and Knizel, A. (2016). Asymptotics of random domino tilings of rectangular Aztec diamonds. Available at arXiv:1604.01491.
[9] Chhita, S. and Johansson, K. (2016). Domino statistics of the two-periodic Aztec diamond. Adv. Math.294 37–149. · Zbl 1344.82021
[10] Chhita, S. and Young, B. (2014). Coupling functions for domino tilings of Aztec diamonds. Adv. Math.259 173–251. · Zbl 1288.05038
[11] Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc.14 297–346. · Zbl 1037.82016
[12] Corwin, I. and Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles. Invent. Math.195 441–508. · Zbl 1459.82117
[13] de Tilière, B. (2007). Scaling limit of isoradial dimer models and the case of triangular quadri-tilings. Ann. Inst. Henri Poincaré Probab. Stat.43 729–750. · Zbl 1176.82008
[14] Di Francesco, P. and Soto-Garrido, R. (2014). Arctic curves of the octahedron equation. J. Phys. A47 1751–8113. · Zbl 1296.05121
[15] Dubédat, J. (2015). Dimers and families of Cauchy–Riemann operators I. J. Amer. Math. Soc.28 1063–1167.
[16] Dubédat, J. and Gheissari, R. (2015). Asymptotics of height change on toroidal Temperleyan dimer models. J. Stat. Phys.159 75–100. · Zbl 1319.82009
[17] Duits, M. (2013). Gaussian free field in an interlacing particle system with two jump rates. Comm. Pure Appl. Math.66 600–643. · Zbl 1259.82091
[18] Duits, M. (2015). On global fluctuations for non-colliding processes. Available at arXiv:1510.08248. · Zbl 1429.60072
[19] Ferrari, P. L. and Spohn, H. (2003). Step fluctuations for a faceted crystal. J. Stat. Phys.113 1–46. · Zbl 1116.82331
[20] Gorin, V. (2017). Bulk universality for random lozenge tilings near straight boundaries and for tensor products. Comm. Math. Phys.354 317–344. · Zbl 1405.60018
[21] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys.242 277–329. · Zbl 1031.60084
[22] Johansson, K. (2005). The Arctic circle boundary and the Airy process. Ann. Probab.33 1–30. · Zbl 1096.60039
[23] Johansson, K. (2017). Edge fluctuations of limit shapes. Available at arXiv:1704.06035.
[24] Kenyon, R. (1997). Local statistics of lattice dimers. Ann. Inst. Henri Poincaré Probab. Stat.33 591–618. · Zbl 0893.60047
[25] Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab.28 759–795. · Zbl 1043.52014
[26] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab.29 1128–1137. · Zbl 1034.82021
[27] Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser.16 191–230. Amer. Math. Soc., Providence, RI. · Zbl 1180.82001
[28] Kenyon, R. and Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Math.199 263–302. · Zbl 1156.14029
[29] Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056. · Zbl 1154.82007
[30] Nienhuis, B., Hilhorst, H. J. and Blöte, H. W. J. (1984). Triangular SOS models and cubic-crystal shapes. J. Phys. A17 3559–3581.
[31] Okounkov, A. and Reshetikhin, N. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc.16 581–603. · Zbl 1009.05134
[32] Panova, G. (2015). Lozenge tilings with free boundaries. Lett. Math. Phys.105 1551–1586. · Zbl 1323.05032
[33] Petrov, L. (2014). Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes. Probab. Theory Related Fields160 429–487. · Zbl 1315.60013
[34] Petrov, L. (2015). Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Ann. Probab.43 1–43. · Zbl 1315.60062
[35] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys.108 1071–1106. · Zbl 1025.82010
[36] Thurston, W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly97 757–773. · Zbl 0714.52007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.