Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy
Hubert Lacoin
Journal de l'École polytechnique — Mathématiques, Volume 7  (2020), p. 1-62

We consider the Solid-on-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field (φ(x)) x 2 , and the energy functional

V(φ)=βxy|φ(x)-φ(y)|-xh1{φ(x)=0}-1{φ(x)<0}.

We prove that for β sufficiently large, there exists a decreasing sequence (h n * (β)) n0 , satisfying lim n h n * (β)=h w (β), and such that: (A) The free energy associated with the system is infinitely differentiable on {h n * } n1 h w (β), and not differentiable on {h n * } n1 . (B) For each n0 within the interval (h n+1 * ,h n * ) (with the convention h 0 * =), there exists a unique translation invariant Gibbs state which is localized around height n, while at a point of non-differentiability, at least two ergodic Gibbs states coexist. The respective typical heights of these two Gibbs states are n-1 and n. The value h n * corresponds thus to a first order layering transition from level n to level n-1. These results combined with those obtained in [28] provide a complete description of the wetting and layering transition for SOS.

Nous considérons le modèle « Solid-On-Solid » (SOS) incluant une interaction avec une paroi. Il s’agit du modèle de mécanique statistique associé au champ à valeurs entières (φ(x)) x 2 et à la fonctionnelle d’énergie

V(φ)=βxy|φ(x)-φ(y)|-xh1{φ(x)=0}-1{φ(x)<0}.

Nous démontrons que pour des valeurs de β suffisamment grandes, il existe une suite décroissante (h n * (β)) n0 , satisfaisant lim n h n * (β)=h w (β), et telle que : (A) l’énergie libre associée au système est infiniment dérivable sur {h n * } n1 h w (β), et n’admet pas de dérivée aux points {h n * } n1  ; (B) pour tout entier n0, pour les valeurs de h dans l’intervalle (h n+1 * ,h n * ) (avec la convention h 0 * =), il existe une unique mesure de Gibbs correspondant à une hauteur de localisation n, alors qu’aux points de non-dérivabilité il y a multiplicité des états de Gibbs, en particulier il en existe deux correspondant aux hauteurs de localisation n-1 et n respectivement. La valeur h n * marque donc une transition de niveau entre la hauteur n et la hauteur n-1. Ces résultats et ceux prouvés dans [28] fournissent une description complète des transitions de niveau et de la transition de mouillage pour le modèle SOS.

Received : 2018-09-18
Accepted : 2019-10-15
Published online : 2019-11-08
DOI : https://doi.org/10.5802/jep.110
Classification:  60K35,  60K37,  82B27,  82B44
Keywords: Solid-on-Solid, wetting, layering transitions, Gibbs states
@article{JEP_2020__7__1_0,
     author = {Hubert Lacoin},
     title = {Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {1-62},
     doi = {10.5802/jep.110},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2020__7__1_0}
}
Lacoin, Hubert. Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy. Journal de l'École polytechnique — Mathématiques, Volume 7 (2020) , pp. 1-62. doi : 10.5802/jep.110. https://jep.centre-mersenne.org/item/JEP_2020__7__1_0/

[1] D. Ahlberg, V. Tassion & A. Teixeira - “Existence of an unbounded vacant set for subcritical continuum percolation”, Electron. Comm. Probab. 23 (2018), article ID 63, 8 pages | Article | MR 3863919 | Zbl 1401.60173

[2] M. Aizenman - “Translation invariance and instability of phase coexistence in the two-dimensional Ising system”, Comm. Math. Phys. 73 (1980) no. 1, p. 83-94 | Article | MR 573615

[3] K. S. Alexander, Personal communication

[4] K. S. Alexander, F. Dunlop & S. Miracle-Solé - “Layering and wetting transitions for an SOS interface”, J. Statist. Phys. 142 (2011) no. 3, p. 524-576 | Article | MR 2771044 | Zbl 1209.82011

[5] K. Armitstead & J. M. Yeomans - “A series approach to wetting and layering transitions. II. Solid-on-solid models”, J. Phys. A 21 (1988) no. 1, p. 159-171 | Article | MR 939725

[6] A. G. Basuev - “Ising model in half-space: a series of phase transitions in low magnetic fields.”, Theoret. and Math. Phys. 153 (2007) no. 2, p. 1539-1574 | Article | MR 2388585 | Zbl 1139.82309

[7] R. Bissacot, R. Fernández & A. Procacci - “On the convergence of cluster expansions for polymer gases”, J. Statist. Phys. 139 (2010) no. 4, p. 598-617 | Article | MR 2638929 | Zbl 1196.82135

[8] R. Brandenberger & C. E. Wayne - “Decay of correlations in surface models”, J. Statist. Phys. 27 (1982) no. 3, p. 425-440 | Article | MR 659803

[9] J. Bricmont, A. El Mellouki & J. Fröhlich - “Random surfaces in statistical mechanics: roughening, rounding, wetting, ...”, J. Statist. Phys. 42 (1986) no. 5-6, p. 743-798 | Article | MR 833220

[10] W. K. Burton, N. Cabrera & F. C. Frank - “The growth of crystals and the equilibrium structure of their surfaces”, Philos. Trans. Roy. Soc. London Ser. A 243 (1951), p. 299-358 | Article | MR 43005 | Zbl 0043.23402

[11] P. Caputo, E. Lubetzky, F. Martinelli, A. Sly & F. L. Toninelli - “Scaling limit and cube-root fluctuations in SOS surfaces above a wall”, J. Eur. Math. Soc. (JEMS) 18 (2016) no. 5, p. 931-995 | Article | MR 3500829 | Zbl 1344.60091

[12] P. Caputo, F. Martinelli & F. L. Toninelli - “Entropic repulsion in |φ| p surfaces: a large deviation bound for all p1”, Boll. Un. Mat. Ital. 10 (2017) no. 3, p. 451-466 | Article | Zbl 1375.60134

[13] F. Cesi & F. Martinelli - “On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results”, J. Statist. Phys. 82 (1996) no. 3-4, p. 823-913 | Article | MR 1372429 | Zbl 1042.82512

[14] F. Cesi & F. Martinelli - “On the layering transition of an SOS surface interacting with a wall. II. The Glauber dynamics”, Comm. Math. Phys. 177 (1996) no. 1, p. 173-201 | Article | MR 1382225 | Zbl 0901.60076

[15] J. T. Chalker - “The pinning of an interface by a planar defect”, J. Phys. A 15 (1982) no. 9, p. L481-L485 | Article | MR 673472

[16] L. Coquille & Y. Velenik - “A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model”, Probab. Theory Related Fields 153 (2012) no. 1-2, p. 25-44 | Article | MR 2925569 | Zbl 1246.82013

[17] E. I. Dinaburg & A. E. Mazel - “Layering transition in SOS model with external magnetic field”, J. Statist. Phys. 74 (1994) no. 3-4, p. 533-563 | Article | MR 1263384 | Zbl 0827.60099

[18] R. L. Dobrushin - “Gibbs states describing a coexistence of phases for the three-dimensional Ising model”, Theor. Probability Appl. 17 (1972) no. 4, p. 582-600 | Article | Zbl 0275.60119

[19] H. von Dreifus, A. Klein & J. F. Perez - “Taming Griffiths’ singularities: infinite differentiability of quenched correlation functions”, Comm. Math. Phys. 170 (1995) no. 1, p. 21-39 | Article | MR 1331689 | Zbl 0820.60086

[20] J. Fröhlich & T. Spencer - “The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas”, Comm. Math. Phys. 81 (1981) no. 4, p. 527-602 | Article | MR 634447

[21] G. Giacomin & H. Lacoin - “Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger”, Ann. Appl. Probab. 28 (2018) no. 1, p. 577-606 | Article | MR 3770884 | Zbl 1388.60161

[22] J.-B. Gouéré - “Subcritical regimes in the Poisson Boolean model of continuum percolation”, Ann. Probab. 36 (2008) no. 4, p. 1209-1220 | Article | MR 2435847 | Zbl 1148.60077

[23] P. Hall - “On continuum percolation”, Ann. Probab. 13 (1985) no. 4, p. 1250-1266 | Article | MR 806222 | Zbl 0588.60096

[24] Y. Higuchi - “On some limit theorems related to the phase separation line in the two-dimensional Ising model”, Z. Wahrsch. Verw. Gebiete 50 (1979) no. 3, p. 287-315 | Article | MR 554548 | Zbl 0406.60084

[25] R. Holley - “Remarks on the FKG inequalities”, Comm. Math. Phys. 36 (1974), p. 227-231 | Article | MR 341552

[26] D. Ioffe & Y. Velenik - “Low-temperature interfaces: prewetting, layering, faceting and Ferrari-Spohn diffusions”, Markov Process. Related Fields 24 (2018) no. 3, p. 487-537 | MR 3821253 | Zbl 1414.60079

[27] R. Kotecký & D. Preiss - “Cluster expansion for abstract polymer models”, Comm. Math. Phys. 103 (1986) no. 3, p. 491-498 | Article | MR 832923 | Zbl 0593.05006

[28] H. Lacoin - “Wetting and layering for solid-on-solid I: Identification of the wetting point and critical behavior”, Comm. Math. Phys. 362 (2018) no. 3, p. 1007-1048 | Article | MR 3845294 | Zbl 1398.82023

[29] R. H. Swendsen - “Roughening transition in the solid-on-solid model”, Phys. Rev. B 15 (1977) no. 2, p. 689-692 | Article

[30] H. N. V. Temperley - “Statistical mechanics and the partition of numbers. II. The form of crystal surfaces”, Math. Proc. Cambridge Philos. Soc. 48 (1952), p. 683-697 | Article | MR 53036

[31] J. D. Weeks, G. H. Gilmer & H. J. Leamy - “Structural Transition in the Ising-Model Interface”, Phys. Rev. Lett. 31 (1973) no. 8, p. 549-551 | Article