Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality. (English) Zbl 1266.60161
One of the most fundamental problems (not yet solved) in the study of the stochastic Ising model is the understanding of the system’s behaviour in the phase-coexistence region under homogeneous boundary condition. Among all related questions is the determination of upper bound on the mixing at the phase-coexistence region under the all-plus boundary. Some estimates have been previously derived and here one improves these results into an upper bound of logarithm order \((\log L)\) on the mixing-time (i.e. quasi-polynomial in the side-length \(L\)). The framework refers mainly to the Glauber dynamics for the Ising model.
Reviewer: Guy Jumarie (Montréal)
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |
References:
| [1] | Abraham, D. B., Reed, P.: Phase separation in the two-dimensional Ising ferromagnet. Phys. Rev. Lett. 33, 377-379 (1974) |
| [2] | Alexander, K. S.: The spectral gap of the 2-D stochastic Ising model with nearly single-spin boundary conditions. J. Statist. Phys. 104, 59-87 (2001) · Zbl 0990.82016 · doi:10.1023/A:1010301525937 |
| [3] | Alexander, K. S., Yoshida, N.: The spectral gap of the 2-D stochastic Ising model with mixed boundary conditions. J. Statist. Phys. 104, 89-109 (2001) · Zbl 0990.82015 · doi:10.1023/A:1010382316368 |
| [4] | Bianchi, A.: Glauber dynamics on nonamenable graphs: boundary conditions and mixing time. Electron. J. Probab. 13, 1980-2012 (2008) · Zbl 1187.82073 |
| [5] | Bodineau, T., Martinelli, F.: Some new results on the kinetic Ising model in a pure phase. J. Statist. Phys. 109, 207-235 (2002) · Zbl 1027.82028 · doi:10.1023/A:1019939712267 |
| [6] | Bricmont, J., El Mellouki, A., Fröhlich, J.: Random surfaces in statistical mechanics: rough- ening, rounding, wetting,. . . . J. Statist. Phys. 42, 743-798 (1986) |
| [7] | Campanino, M., Ioffe, D., Velenik, Y.: Ornstein-Zernike theory for finite range Ising models above Tc. Probab. Theory Related Fields 125, 305-349 (2003) · Zbl 1032.60093 · doi:10.1007/s00440-002-0229-z |
| [8] | Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36, 1287-1321 (2008) · Zbl 1160.60026 · doi:10.1214/07-AOP359 |
| [9] | Caputo, P., Martinelli, F., Simenhaus, F., Toninelli, F. L.: “Zero” temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion. Comm. Pure Appl. Math. 64, 778-831 (2011) · Zbl 1219.82112 · doi:10.1002/cpa.20359 |
| [10] | Caputo, P., Martinelli, F., Toninelli, F. L.: Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach. Comm. Math. Phys. 311, 157-189 (2012) · Zbl 1276.82034 · doi:10.1007/s00220-012-1425-z |
| [11] | Cesi, F., Guadagni, G., Martinelli, F., Schonmann, R. H.: On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point. J. Statist. Phys. 85, 55-102 (1996) · Zbl 0937.82004 · doi:10.1007/BF02175556 |
| [12] | Chayes, L., Schonmann, R. H., Swindle, G.: Lifshitz’ law for the volume of a two- dimensional droplet at zero temperature. J. Statist. Phys. 79, 821-831 (1995) · Zbl 1081.82544 · doi:10.1007/BF02181205 |
| [13] | Dobrushin, R., Hryniv, O.: Fluctuations of the phase boundary in the 2D Ising ferromagnet. Comm. Math. Phys. 189, 395-445 (1997) · Zbl 0888.60083 · doi:10.1007/s002200050209 |
| [14] | Dobrushin, R., Kotecký, R., Shlosman, S.: Wulff Construction. A Global Shape from Lo- cal Interaction. Transl. Math. Monogr. 104, Amer. Math. Soc., Providence, RI (1992) · Zbl 0917.60103 |
| [15] | Fisher, D. S., Huse, D. A.: Dynamics of droplet fluctuations in pure and random Ising systems. Phys. Rev. B 35, 6841-6846 (1987) |
| [16] | Fontes, L. R., Schonmann, R. H., Sidoravicius, V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys. 228, 495-518 (2002) · Zbl 1004.82013 · doi:10.1007/s002200200658 |
| [17] | Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22, 89-103 (1971) · Zbl 0346.06011 · doi:10.1007/BF01651330 |
| [18] | Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Comm. Math. Phys. 27, 103-136 (1972) |
| [19] | Glauber, R. J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294-307 (1963) · Zbl 0145.24003 · doi:10.1063/1.1703954 |
| [20] | Greenberg, L., Ioffe, D.: On an invariance principle for phase separation lines. Ann. Inst. Henri Poincaré Probab. Statist. 41, 871-885 (2005) · Zbl 1115.60039 · doi:10.1016/j.anihpb.2005.05.001 |
| [21] | Griffiths, R. B.: Correlations in Ising ferromagnets. III. A mean-field bound for binary corre- lations. Comm. Math. Phys. 6, 121-127 (1967) |
| [22] | Higuchi, Y.: On some limit theorems related to the phase separation line in the two- dimensional Ising model. Z. Wahrsch. Verw. Gebiete 50, 287-315 (1979) · Zbl 0406.60084 · doi:10.1007/BF00534152 |
| [23] | Hryniv, O.: On local behaviour of the phase separation line in the 2D Ising model. Probab. Theory Related Fields 110, 91-107 (1998) · Zbl 0897.60038 · doi:10.1007/s004400050146 |
| [24] | Ioffe, D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Statist. Phys. 74, 411-432 (1994) · Zbl 0946.82502 · doi:10.1007/BF02186818 |
| [25] | Ioffe, D.: Exact large deviation bounds up to Tc for the Ising model in two dimensions. Probab. Theory Related Fields 102, 313-330 (1995) · Zbl 0830.60018 · doi:10.1007/BF01192464 |
| [26] | Kelly, D. G., Sherman, S.: General Griffiths’ inequalities on correlations in Ising ferromag- nets. J. Math. Phys. 9, 466-484 (1968) |
| [27] | Levin, D. A., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. Amer. Math. Soc. (2009) · Zbl 1160.60001 |
| [28] | Lifshitz, I. M.: Kinetics of ordering during second-order phase transitions. Soviet J. Experi- ment. Theor. Phys. 15, 939 (1962) |
| [29] | Liggett, T. M.: Interacting Particle Systems. Classics in Math., Springer, Berlin (2005) · Zbl 1103.82016 |
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