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Wetting and layering for solid-on-solid. II: Layering transitions, Gibbs states, and regularity of the free energy. (Mouillage et stratification pour le modèle SOS II : transitions de niveau, états de Gibbs et régularité de l’énergie libre.) (English. French summary) Zbl 07128376
Summary: We consider the Solid-on-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $$(\phi (x))_{x\in \mathbb{Z}^2}$$, and the energy functional $V(\phi )=\beta \sum _{x\sim y}|\phi (x)-\phi (y)|-\sum _x\left(h\mathbf{1}_{\{ \phi (x)=0\} }-\infty \mathbf{1}_{\{ \phi (x)<0\} } \right).$ We prove that for $$\beta$$ sufficiently large, there exists a decreasing sequence $$(h^*_n(\beta ))_{n\ge 0}$$, satisfying $$\lim_{n\rightarrow \infty }h^*_n(\beta )=h_w(\beta)$$, and such that: $$(A)$$ The free energy associated with the system is infinitely differentiable on $$\mathbb{R} \smallsetminus \left(\{ h^*_n\} _{n\ge 1}\cup h_w(\beta )\right)$$, and not differentiable on $$\{ h^*_n\} _{n\ge 1}. (B)$$ For each $$n\ge 0$$ within the interval $$(h^*_{n+1},h^*_n)$$ (with the convention $$h^*_0=\infty)$$, 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 Part I [the author, Commun. Math. Phys. 362, No. 3, 1007–1048 (2018; Zbl 1398.82023)] provide a complete description of the wetting and layering transition for SOS.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K37 Processes in random environments 82B27 Critical phenomena in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
##### Keywords:
solid-on-solid; wetting; layering transitions; Gibbs states
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##### References:
 [1] Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto, Existence of an unbounded vacant set for subcritical continuum percolation, Electron. Comm. Probab., 23 (2018) · Zbl 1401.60173 [2] Aizenman, Michael, Translation invariance and instability of phase coexistence in the two-dimensional Ising system, Comm. Math. Phys., 73, 1, 83-94 (1980) [3] Alexander [4] Alexander; Dunlop, François; Miracle-Solé, Salvador, Layering and wetting transitions for an SOS interface, J. Statist. Phys., 142, 3, 524-576 (2011) · Zbl 1209.82011 [5] Armitstead, K.; Yeomans, A series approach to wetting and layering transitions. II. Solid-on-solid models, J. Phys. A, 21, 1, 159-171 (1988) [6] Basuev, Ising model in half-space: a series of phase transitions in low magnetic fields., Theoret. and Math. Phys., 153, 2, 1539-1574 (2007) · Zbl 1139.82309 [7] Bissacot, Rodrigo; Fernández, Roberto; Procacci, Aldo, On the convergence of cluster expansions for polymer gases, J. Statist. Phys., 139, 4, 598-617 (2010) · Zbl 1196.82135 [8] Brandenberger, R.; Wayne, Decay of correlations in surface models, J. Statist. Phys., 27, 3, 425-440 (1982) [9] Bricmont, J.; El Mellouki, A.; Fröhlich, J., Random surfaces in statistical mechanics: roughening, rounding, wetting, ..., J. Statist. Phys., 42, 5-6, 743-798 (1986) [10] Burton; Cabrera, N.; Frank, The growth of crystals and the equilibrium structure of their surfaces, Philos. Trans. Roy. Soc. London Ser. A, 243, 299-358 (1951) · Zbl 0043.23402 [11] Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, 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 [12] Caputo, Pietro; Martinelli, Fabio; Toninelli, Entropic repulsion in $$|\nabla \phi |^p$$ surfaces: a large deviation bound for all $$p\ge 1$$, Boll. Un. Mat. Ital., 10, 3, 451-466 (2017) · Zbl 1375.60134 [13] Cesi, Filippo; Martinelli, Fabio, On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results, J. Statist. Phys., 82, 3-4, 823-913 (1996) · Zbl 1042.82512 [14] Cesi, Filippo; Martinelli, Fabio, On the layering transition of an SOS surface interacting with a wall. II. The Glauber dynamics, Comm. Math. Phys., 177, 1, 173-201 (1996) · Zbl 0901.60076 [15] Chalker, The pinning of an interface by a planar defect, J. Phys. A, 15, 9, L481-L485 (1982) [16] Coquille, Loren; Velenik, Yvan, A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model, Probab. Theory Related Fields, 153, 1-2, 25-44 (2012) · Zbl 1246.82013 [17] Dinaburg; Mazel, Layering transition in SOS model with external magnetic field, J. Statist. Phys., 74, 3-4, 533-563 (1994) · Zbl 0827.60099 [18] Dobrushin, Gibbs states describing a coexistence of phases for the three-dimensional Ising model, Theor. Probability Appl., 17, 4, 582-600 (1972) · Zbl 0275.60119 [19] von Dreifus, Henrique; Klein, Abel; Perez, Taming Griffiths’ singularities: infinite differentiability of quenched correlation functions, Comm. Math. Phys., 170, 1, 21-39 (1995) · Zbl 0820.60086 [20] Fröhlich, Jürg; Spencer, Thomas, The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys., 81, 4, 527-602 (1981) [21] Giacomin, Giambattista; Lacoin, Hubert, Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger, Ann. Appl. Probab., 28, 1, 577-606 (2018) · Zbl 1388.60161 [22] Gouéré, Jean-Baptiste, Subcritical regimes in the Poisson Boolean model of continuum percolation, Ann. Probab., 36, 4, 1209-1220 (2008) · Zbl 1148.60077 [23] Hall, Peter, On continuum percolation, Ann. Probab., 13, 4, 1250-1266 (1985) · Zbl 0588.60096 [24] Higuchi, Yasunari, On some limit theorems related to the phase separation line in the two-dimensional Ising model, Z. Wahrsch. Verw. Gebiete, 50, 3, 287-315 (1979) · Zbl 0406.60084 [25] Holley, Richard, Remarks on the $${\rm FKG}$$ inequalities, Comm. Math. Phys., 36, 227-231 (1974) [26] Ioffe, D.; Velenik, Y., Low-temperature interfaces: prewetting, layering, faceting and Ferrari-Spohn diffusions, Markov Process. Related Fields, 24, 3, 487-537 (2018) · Zbl 1414.60079 [27] Kotecký, R.; Preiss, D., Cluster expansion for abstract polymer models, Comm. Math. Phys., 103, 3, 491-498 (1986) · Zbl 0593.05006 [28] Lacoin, Hubert, Wetting and layering for solid-on-solid I: Identification of the wetting point and critical behavior, Comm. Math. Phys., 362, 3, 1007-1048 (2018) · Zbl 1398.82023 [29] Swendsen, Roughening transition in the solid-on-solid model, Phys. Rev. B, 15, 2, 689-692 (1977) [30] Temperley, Statistical mechanics and the partition of numbers. II. The form of crystal surfaces, Math. Proc. Cambridge Philos. Soc., 48, 683-697 (1952) · Zbl 0048.19802 [31] Weeks; Gilmer; Leamy, Structural Transition in the Ising-Model Interface, Phys. Rev. Lett., 31, 8, 549-551 (1973)
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