Formation of facets for an effective model of crystal growth. (English) Zbl 1446.82080
Sidoravicius, Vladas (ed.), Sojourns in probability theory and statistical physics. I. Spin glasses and statistical mechanics, a festschrift for Charles M. Newman. Singapore: Springer; Shanghai: NYU Shanghai. Springer Proc. Math. Stat. 298, 199-245 (2019).
Summary: We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a \(2+1\) solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to “grow” through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.
For the entire collection see [Zbl 1429.60003].
For the entire collection see [Zbl 1429.60003].
MSC:
| 82D20 | Statistical mechanics of solids |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B30 | Statistical thermodynamics |
| 82B24 | Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics |
Keywords:
equilibrium crystal shapes; microscopic facets; SOS model with bulk fields; infinite sequence of first-order transitionsReferences:
| [1] | Biskup, M., Chayes, L., Kotecký, R.: Critical region for droplet formation in the two-dimensional Ising model. Comm. Math. Phys. 242(1-2), 137-183 (2003) · Zbl 1041.82004 · doi:10.1007/s00220-003-0946-x |
| [2] | Bodineau, T.: The Wulff construction in three and more dimensions. Comm. Math. Phys. 207(1), 197-229 (1999) · Zbl 1015.82005 · doi:10.1007/s002200050724 |
| [3] | Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41(3), 1033-1098 (2000). Probabilistic techniques in equilibrium and nonequilibrium statistical physics · Zbl 0969.00035 · doi:10.1063/1.533180 |
| [4] | Bodineau, T., Schonmann, R.H., Shlosman, S.: 3D crystal: how flat its flat facets are? Comm. Math. Phys. 255(3), 747-766 (2005) · Zbl 1076.74016 · doi:10.1007/s00220-004-1283-4 |
| [5] | Bonzel, H.P.: 3D equilibrium crystal shapes in the new light of STM and AFM. Phys. Rep. 385(1), 1-67 (2003) · doi:10.1016/S0370-1573(03)00273-4 |
| [6] | Bonzel, H.P., Yu, D.K., Scheffler, M.M.: The three-dimensional equilibrium crystal shape of Pb: recent results of theory and experiment. Appl. Phys. A 87, 391-397 (2007) · doi:10.1007/s00339-007-3951-7 |
| [7] | Bricmont, J., El Mellouki, A., Fröhlich, J.: Random surfaces in statistical mechanics: roughening, rounding, wetting. J. Statist. Phys. 42(5-6), 743-798 (1986) · doi:10.1007/BF01010444 |
| [8] | Bricmont, J., Fontaine, J.-R., Lebowitz, J.L.: Surface tension, percolation, and roughening. J. Statist. Phys. 29(2), 193-203 (1982) · doi:10.1007/BF01020782 |
| [9] | Campanino, M., Ioffe, D., Louidor, O.: Finite connections for supercritical Bernoulli bond percolation in 2D. Markov Process. Relat. Fields 16(2), 225-266 (2010) · Zbl 1198.82031 |
| [10] | Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, L.F.: Dynamics of \((2+1)\)-dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion. Ann. Probab. 42(4), 1516-1589 (2014) · Zbl 1311.60114 · doi:10.1214/13-AOP836 |
| [11] | Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, L.F.: Scaling limit and cube-root fluctuations in SOS surfaces above a wall. J. Eur. Math. Soc. (JEMS) 18(5), 931-995 (2016) · Zbl 1344.60091 · doi:10.4171/JEMS/606 |
| [12] | Caputo, P., Martinelli, F., Toninelli, F.L.: On the probability of staying above a wall for the \((2+1)\)-dimensional SOS model at low temperature. Probab. Theor. Relat. Fields 163, 803-831 (2015) · Zbl 1345.60112 · doi:10.1007/s00440-015-0658-0 |
| [13] | Cerf, R.: Large deviations of the finite cluster shape for two-dimensional percolation in the Hausdorff and \(L^1\) metric. J. Theoret. Probab. 13(2), 491-517 (2000) · Zbl 0974.60089 · doi:10.1023/A:1007841407417 |
| [14] | Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Comm. Math. Phys. 222(1), 147-179 (2001) · Zbl 1013.82010 · doi:10.1007/s002200100505 |
| [15] | Cerf, R., Pisztora, Á.: On the Wulff crystal in the Ising model. Ann. Probab. 28(3), 947-1017 (2000) · Zbl 1034.82006 |
| [16] | Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Amer. Math. Soc. 14(2), 297-346 (2001) · Zbl 1037.82016 · doi:10.1090/S0894-0347-00-00355-6 |
| [17] | Dobrushin, R.L.: Gibbs states describing a coexistence of phases for the three-dimensional ising model. Th. Prob. Appl. 17(3), 582-600 (1972) · Zbl 0275.60119 |
| [18] | Dobrushin, R.L., Kotecký, R., Shlosman, S.: Wulff construction, volume 104 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1992) A global shape from local interaction, Translated from the Russian by the authors · Zbl 0917.60103 |
| [19] | Dobrushin, R.L., Shlosman, S.B.: Droplet condensation in the Ising model: moderate deviations point of view. In Probability and phase transition (Cambridge, 1993), volume 420 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pp. 17-34. Kluwer Acadamic Publisation, Dordrecht (1994) · Zbl 0832.60032 |
| [20] | Dobrushin, R. L., Shlosman, S. B.: Large and moderate deviations in the Ising model. In: Probability Contributions to Statistical Mechanics, vol. 20 of Adv. Soviet Math., pp. 91-219. Amer. Math. Soc., Providence, RI (1994) · Zbl 0815.60024 |
| [21] | Emundts, A., Bonzel, H.P., Wynblatt, P., Thürmer, K., Reutt-Robey, J., Williams, E.D.: Continuous and discontinuous transitions on 3D equilibrium crystal shapes: a new look at Pb and Au. Surf. Sci. 481(1-3), 13-24 (2001) · doi:10.1016/S0039-6028(01)01055-X |
| [22] | Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Statist. Phys. 113(1-2), 1-46 (2003) · Zbl 1116.82331 · doi:10.1023/A:1025703819894 |
| [23] | Ferrari, P.L., Spohn, H.: Constrained Brownian motion: fluctuations away from circular and parabolic barriers. Ann. Probab. 33(4), 1302-1325 (2005) · Zbl 1082.60071 · doi:10.1214/009117905000000125 |
| [24] | Ioffe, D., Shlosman, S.: Ising model fog drip: the first two droplets. In In and out of equilibrium. 2, vol. 60 of Progr. Probab., pp. 365-381. Birkhäuser, Basel (2008) · Zbl 1173.82311 |
| [25] | Ioffe, D., Shlosman, S., Toninelli, F.L.: Interaction versus entropic repulsion for low temperature Ising polymers. J. Stat. Phys. 158(5), 1007-1050 (2015) · Zbl 1321.82014 · doi:10.1007/s10955-014-1153-1 |
| [26] | Ioffe, D., Shlosman, S., Velenik, Y.: An invariance principle to Ferrari-Spohn diffusions. Comm. Math. Phys. 336(2), 905-932 (2015) · Zbl 1323.60055 · doi:10.1007/s00220-014-2277-5 |
| [27] | Ioffe, D., Velenik, Y.: Ballistic phase of self-interacting random walks. In: Penrose, M., Schwetlick, H., Mörters, P., Moser, R., Zimmer, J. (eds.) Analysis and Stochastics of Growth Processes and Interface Models, pp. 55-79. Oxford University Press, Oxford (2008) · Zbl 1255.60168 · doi:10.1093/acprof:oso/9780199239252.003.0003 |
| [28] | Ioffe, D., Velenik, Y.: Low temperature interfaces: prewetting, layering, faceting and Ferrari-Spohn diffusions. Markov Process. Relat. Fields 24, 487-537 (2018) · Zbl 1414.60079 |
| [29] | Ioffe, D., Velenik, Y., Wachtel, V.: Dyson Ferrari-Spohn diffusions and ordered walks under area tilts. Probab. Theor. Relat. Fields 170(1), 11-47 (2017) · Zbl 1405.60146 |
| [30] | Kenyon, R.: Height fluctuations in the honeycomb dimer model. Comm. Math. Phys. 281(3), 675-709 (2008) · Zbl 1157.82028 · doi:10.1007/s00220-008-0511-8 |
| [31] | Miracle-Sole, S.: Facet shapes in a Wulff crystal. In: Mathematical Results in Statistical Mechanics (Marseilles, 1998), pp. 83-101. World Scientific Publishing, River Edge, NJ (1999) · Zbl 1055.82512 |
| [32] | Okounkov, A.: Limit shapes real and imagined. Bull. Amer. Math. Soc. (N.S.) 53(2), 187-216 (2016) · Zbl 1348.60053 · doi:10.1090/bull/1512 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
