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Stability of the uniqueness regime for ferromagnetic Glauber dynamics under non-reversible perturbations. (English) Zbl 1405.82017
Authors’ abstract: We prove a general result concerning finite-range, attractive interacting particle systems on $$\{-1,1\}^{\mathbb Z^d}$$. If the particle system has a unique stationary measure and, in a precise sense, relaxes to this stationary measure at an exponential rate, then any small perturbation of the dynamics also has a unique stationary measure to which it relaxes at an exponential rate. To augment this result, we study the particular case of Glauber dynamics for the Ising model. We show that, for any non-zero external field, the dynamics converges to its unique invariant measure at an exponential rate. Previously, this was only known for $$\beta < \beta_c$$ and $$\beta$$ sufficiently large. As a consequence, Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase.

##### MSC:
 82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics 82C22 Interacting particle systems in time-dependent statistical mechanics 82D40 Statistical mechanical studies of magnetic materials
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##### References:
 [1] Achahbar, A; Alonso, JJ; Muñoz, MA, Simple nonequilibrium extension of the Ising model, Phys. Rev. E, 54, 4838-4843, (1996) [2] Aizenman, M, Geometric analysis of $$ϕ ^4$$ fields and Ising models, Commun. Math. Phys., 86, 1-48, (1982) · Zbl 0533.58034 [3] Basuev, AG, Ising model in half-space: a series of phase transitions in low magnetic fields, Theor. Math. Phys., 153, 1539-1574, (2007) · Zbl 1139.82309 [4] Crawford, N; Roeck, W; Schütz, M, Uniqueness regime for Markov dynamics on quantum lattice spin systems, J. Phys. A Math. Theor., 48, 425203, (2015) · Zbl 1327.82009 [5] Ueltschi, D.: Private communication (2016) · Zbl 0821.60097 [6] Maere, A, Phase transition and correlation decay in coupled map lattices, Commun. Math. Phys., 297, 229-264, (2010) · Zbl 1200.37073 [7] Dobrushin, RL, Markov processes with a large number of locally interacting components-existence of the limiting process and its ergodicity, Probl. Pereda. Inform., 7, 70-87, (1971) [8] Dobrushin, RL; Shlosman, SB, Completely analytical interactions: constructive description, J. Stat. Phys., 46, 983-1014, (1987) · Zbl 0683.60080 [9] Dobrushin, R. L., Shlosman, S. B.: Completely analytical Gibbs fields. In: Statistical Physics and Dynamical Systems, Springer, pp. 371-403 (1985) · Zbl 0569.46043 [10] Durlauf, SN, How can statistical mechanics contribute to social science?, Proc. Natl. Acad. Sci. USA, 96, 10582-10584, (1999) · Zbl 0984.91024 [11] Fernández, R., Toom, A.: Non-Gibbsianness of the invariant measures of non-reversible cellular automata with totally asymmetric noise. arXiv preprint arXiv:math-ph/0101014 (2001) · Zbl 0739.60096 [12] Gács, P, Reliable cellular automata with self-organization, J. Stat. Phys., 103, 45-267, (2001) · Zbl 0973.68158 [13] Gray, LF, The positive rates problem for attractive nearest neighbor spin systems on $$\mathbb{Z}$$, Z. Wahrscheinlichkeitstheorie Verwandte Geb., 61, 389-404, (1982) · Zbl 0479.60098 [14] Gray, LF, A reader’s guide to gacs’s “positive rates” paper, J. Stat. Phys., 103, 1-44, (2001) · Zbl 1135.82317 [15] Gross, L, Absence of second-order phase transitions in the dobrushin uniqueness region, J. Stat. Phys., 25, 57-72, (1981) [16] Higuchi, Y, Coexistence of infinite (*)-clusters ii. Ising percolation in two dimensions, Probab. Theory Relat. Fields, 97, 1-33, (1993) · Zbl 0794.60102 [17] Holley, R, Possible rates of convergence in finite range, Part. Syst. Random Media Large Deviat., 41, 215, (1985) · Zbl 0577.60099 [18] Israel, R .B.: Convexity in the Theory of Lattice Gases. Princeton University Press, Princeton (2015) [19] Künsch, H, Time reversal and stationary Gibbs measures, Stoch. Process. Appl., 17, 159-166, (1984) · Zbl 0536.60096 [20] Lacoin, H; Simenhaus, F; Toninelli, FL, Zero-temperature 2d stochastic Ising model and anisotropic curve-shortening flow, J. Eur. Math. Soc., 16, 2557-2615, (2014) · Zbl 1320.60158 [21] Lacoin, H; Simenhaus, F; Toninelli, F, The heat equation shrinks Ising droplets to points, Commun. Pure Appl. Math., 68, 1640-1681, (2015) · Zbl 1337.82017 [22] Lebowitz, JL; Maes, C; Speer, ER, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys., 59, 117-170, (1990) · Zbl 1083.82522 [23] Liggett, T .M.: Interacting Particle Systems, vol. 276. Springer, Berlin (1985) · Zbl 0559.60078 [24] Liggett, TM; Schonmann, RH; Stacey, AM, Domination by product measures, Ann. Probab., 25, 71-95, (1997) · Zbl 0882.60046 [25] Louis, P-Y; etal., Ergodicity of pca: equivalence between spatial and temporal mixing conditions, Electron. Commun. Probab., 9, 119-131, (2004) · Zbl 1059.60098 [26] Lubetzky, E; Sly, A, Information percolation and cutoff for the stochastic Ising model, J. Am. Math. Soc., 29, 729-774, (2016) · Zbl 1342.60173 [27] Martinelli, F; Olivieri, E; Scoppola, E, Metastability and exponential approach to equilibrium for low-temperature stochastic Ising models, J. Stat. Phys., 61, 1105-1119, (1990) · Zbl 0739.60096 [28] Martinelli, F; Olivieri, E, Approach to equilibrium of Glauber dynamics in the one phase region. i. the attractive case, Commun. Math. Phys., 161, 447-486, (1994) · Zbl 0793.60110 [29] Martinelli, F; Olivieri, E; Schonmann, RH, For 2-d lattice spin systems weak mixing implies strong mixing, Commun. Math. Phys., 165, 33-47, (1994) · Zbl 0811.60097 [30] Martirosyan, DG, Theorems on strips in the classical Ising ferromagnetic model, Sov. J. Contemp. Math., 22, 59-83, (1987) [31] Preston, CJ, An application of the ghs inequalities to show the absence of phase transition for Ising spin systems, Commun. Math. Phys., 35, 253-255, (1974) [32] Propp, J; Wilson, D, Coupling from the past: a users guide, Microsurv. Discrete Probab., 41, 181-192, (1998) · Zbl 0916.65147 [33] Schonmann, R; Shlosman, S, Complete analyticity of the 2d Ising model completed, Commun. Math. Phys., 170, 453-482, (1995) · Zbl 0821.60097 [34] Stroock, D; Zegarlinski, B, The equivalence of the logarithmic Sobolev inequality and the dobrushin-shlosman mixing condition, Commun. Math. Phys., 144, 303-323, (1992) · Zbl 0745.60104 [35] Toom, AL, Nonergodic multidimensional system of automata, Probl. Inf. Transm., 10, 70-79, (1974) · Zbl 0315.94053 [36] Berg, J; Steif, JE, On the existence and nonexistence of finitary codings for a class of random fields, Ann. Probab., 27, 1501-1522, (1999) · Zbl 0968.60091 [37] Yarotsky, DA, Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems, J. Stat. Phys., 118, 119-144, (2005) · Zbl 1130.82007
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