zbMATH — the first resource for mathematics

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)

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.