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On homogenization and scaling limit of some gradient perturbations of a massless free field. (English) Zbl 0871.35010
The authors study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians \(H(\varphi)\) which are gradient perturbations of a massless free field. They prove, under suitable assumptions of positivity of the Hessian of the Hamiltonian, a central limit theorem for these models, and show that their long distance behavior is identical to a new homogeneized continuum massless free field. They obtain also new bounds on the 2-point correlation functions of these models.
Reviewer: B.Helffer (Orsay)

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
81T25 Quantum field theory on lattices
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[1] Brascamp, H.J. and Lieb, E.: On extensions of the Brunn-Minkowski and Prekopa-Leinler theorems.J. Funct. Anal. 22, 366–389(1976) · Zbl 0334.26009
[2] Brydges, D. and Yau, H-T.: Grad {\(\phi\)} perturbations of massless Gaussian fields.Commun. Math. Phys. 129, 351–392(1990) · Zbl 0705.60101
[3] Carlen, E.A., Kusuoka, S. and Stroock, D.W.: Upper bounds for symmetric Markov transition functions.Ann. Inst. H. Poincaré. Probab. Statist., Sup. au no 2, 245–287(1987) · Zbl 0634.60066
[4] Gawedzki K. and Kupiainen, A.: Renormalization group study of a critical lattice model, I and II.Commun. Math. Phys. 82, 407–433(1981)
[5] Gilbarg, D. and Trudinger, N.S.:Elliptic Partial Differential Equations of Second Order. Berlin-Heidelberg-New York: Springer Verlag, 1977 · Zbl 0361.35003
[6] Guo, M. Z. and Papanicolaou, G.: Self-diffusion of interacting Brownian particles.Probabilistic methods in mathematical physics (Katata/Kyoto, 1985) Boston, MA: Academic Press, 1987, pp. 113–151
[7] Helffer, B.: Spectral properties of the Kac operator in large dimensions.Preprint, LMENS-93-17, 1993
[8] Helffer, B. and Sjöstrand, J.: On the correlation for Kac-like models in the convex case.J. Statist. Phys. 74(1/2), 349–409(1994) · Zbl 0946.35508
[9] Künnemann, R.: The diffusion limit for reversible jump processes on \(\mathbb{Z}\) d with ergodic random bond conductivities.Commun. Math. Phys. 90, 27–68(1983) · Zbl 0523.60097
[10] Magnen, J. and Sénéor, R.: The infrared behavior of () 3 4 .Ann. Physics 152, 136–202 (1984) · Zbl 0534.60097
[11] Newman, C.: Normal fluctuations and the FKG inequalities.Commun. Math. Phys. 74, 119–128(1980) · Zbl 0429.60096
[12] Sjöstrand, J.: Correlation asymptotics and Witten Laplacians.To appear. · Zbl 0877.35084
[13] Funaki, T. and Spohn, H.: Motion by mean curvature from Ginzburg-Landau interface model.Preprint · Zbl 0884.58098
[14] Stein, Elias M.:Singular Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton University Press, 1970 · Zbl 0207.13501
[15] Papanicolaou, G.C. and Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. Volume 2 ofColl. Math. Soc. Janos Bolya, 27. Random fields, Amsterdam, North Holland Publ. Co., 1981, pp. 835–873 · Zbl 0499.60059
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