Hierarchical interfaces in random media. II: The Gibbs measures. (English) Zbl 1101.82317
Summary: We continue the analysis of hierarchical interfaces in random media started in earlier work [A. Bovier and C. Külske, J. Stat. Phys. 69, No. 1–2, 79–110 (1992; Zbl 0893.60071), A. Bovier and P. Picco, J. Stat. Phys. 62, No. 1–2, 177–199 (1991), doi:10.1007/BF01020865]. We show that from the estimates on the renormalized random variables established in that work, it follows that these models possess unique Gibbs states describing mostly flat interfaces in dimension \(D > 3\), if the disorder is weak and the temperature low enough. In the course of the proof we also present very explicit formulas for expectations of local observables.
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B24 | Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
Keywords:
Disordered systems; interfaces; hierarchical model; Gibbs states; renormalization of stochastic sequencesCitations:
Zbl 0893.60071References:
| [1] | A. Bovier and C. K?lske, Stability of hierarchical interfaces in a random field model,J. Stat. Phys. 69:79 (1992). · Zbl 0893.60071 · doi:10.1007/BF01053784 |
| [2] | A. Bovier and P. Picco, Stability of interfaces in random environments: A renormalization group analysis of a hierarchical model,J. Stat. Phys. 62:177 (1991). · doi:10.1007/BF01020865 |
| [3] | H. O. Georgii, Gibbs measures and phase transitions, inStudies in Mathematics, Vol. 9 (de Gruyter, Berlin, 1988). · Zbl 0657.60122 |
| [4] | Ch. K?lske, Ph.D. Thesis, Ruhr-Universit?t Bochum, (1993). |
| [5] | Ya. G. Sinai,Theory of Phase Transitions: Rigorous Results (Pergamon Press, Oxford, 1982). |
| [6] | W. Stout,Almost Sure Convergence (Academic Press, New York, 1974). · Zbl 0321.60022 |
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