Renormalization-Group Transformations Under Strong Mixing Conditions: Gibbsianness and Convergence of Renormalized Interactions

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite-range lattice gases, when suitable strong mixing conditions are satisfied. Using a block decimation procedure, cluster expansion, and detailed comparison between statistical ensembles, we are able to prove Gibbsianness and convergence to a trivial (i.e., Gaussian and product) fixed point. Our results apply to the 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.

[1]  M. Aizenman,et al.  The phase transition in a general class of Ising-type models is sharp , 1987 .

[2]  Pierre Picco,et al.  Cluster expansion ford-dimensional lattice systems and finite-volume factorization properties , 1990 .

[3]  FINITE VOLUME MIXING CONDITIONS FOR LATTICE SPIN SYSTEMS AND EXPONENTIAL APPROACH TO EQUILIBRIUM OF GLAUBER DYNAMICS , 1993 .

[4]  R. Dobrushin,et al.  Completely Analytical Gibbs Fields , 1985 .

[5]  A. V. Enter.  Ill-defined block-spin transformations at arbitrarily high temperatures , 1996 .

[6]  Weakly Gibbsian Measures and Quasilocality: A Long-Range Pair-Interaction Counterexample , 1999 .

[7]  J. Bricmont,et al.  Renormalization Group Pathologies and the Definition of Gibbs States , 1998 .

[8]  E. Olivieri,et al.  One-dimensional random Ising systems with interaction decayr−(1+ɛ): A convergent cluster expansion , 1987 .

[9]  Variational Principle for Some Renormalized Measures , 1999 .

[10]  A. Enter.  On the possible failure of the Gibbs property for measures on lattice systems , 1996 .

[11]  Instability of renormalization-group pathologies under decimation , 1995 .

[12]  Absence of renormalization group pathologies near the critical temperature. Two examples , 1996 .

[13]  S. Shlosman,et al.  Complete Analyticity of the 2D Potts Model above the Critical Temperature , 1997 .

[14]  F. Martinelli,et al.  On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results , 1996 .

[15]  S. Shlosman,et al.  (Almost) Gibbsian Description of the Sign Fields of SOS Fields , 1998 .

[16]  W. Sullivan.  Potentials for almost Markovian random fields , 1973 .

[17]  F. Martinelli,et al.  For 2-D lattice spin systems weak mixing implies strong mixing , 1994 .

[18]  E. Olivieri,et al.  Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem , 1981 .

[19]  R. Dobrushin.  Prescribing a System of Random Variables by Conditional Distributions , 1970 .

[20]  Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions , 1993 .

[21]  E. Olivieri.  On a cluster expansion for lattice spin systems: A finite-size condition for the convergence , 1988 .

[22]  E. Olivieri,et al.  Analyticity for one-dimensional systems with long-range superstable interactions , 1983 .

[23]  E. Olivieri,et al.  On the Ising Model with Strongly Anisotropic External Field , 1999 .

[24]  P. Pearce,et al.  Mathematical properties of position-space renormalization-group transformations , 1979 .

[25]  F. Martinelli,et al.  Approach to equilibrium of Glauber dynamics in the one phase region , 1994 .

[26]  A. G. Basuev.  Hamiltonian of the phase separation border and phase transitions of the first kind. I , 1985 .

[27]  M. Cassandro,et al.  The Lavoisier law and the critical point , 1975 .

[28]  A. Mazel,et al.  Layering transition in SOS model with external magnetic field , 1994 .

[29]  A. Sokal,et al.  Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory , 1991, hep-lat/9210032.

[30]  Some numerical results on the block spin transformation for the 2D Ising model at the critical point , 1994, cond-mat/9404030.

[31]  “Non-Gibbsian” States and their Gibbs Description , 1999 .

[32]  F. Martinelli,et al.  Approach to equilibrium of Glauber dynamics in the one phase region , 1994 .

[33]  Pathological behavior of renormalization-group maps at high fields and above the transition temeprature , 1994, hep-lat/9409016.

[34]  H. Kunz,et al.  General properties of polymer systems , 1971 .

[35]  H. Yau.  Logarithmic Sobolev inequality for lattice gases with mixing conditions , 1996 .

[36]  David Preiss,et al.  Cluster expansion for abstract polymer models , 1986 .

[37]  The large block spin interaction , 1986 .

[38]  S. Shlosman.  Uniqueness and half-space nonuniqueness of gibbs states in Czech models , 1986 .

[39]  G. Gallavotti,et al.  Block-spins interactions in the Ising model , 1974 .

[40]  Christian Maes,et al.  Relative Energies for Non-Gibbsian States , 1997 .

[41]  R. Dobrushin,et al.  Large and Moderate Deviations in the Ising Model , 1994 .

[42]  A note on the projecton of Gibbs measures , 1994 .

[43]  K. Leuven,et al.  Almost Gibbsian versus weakly Gibbsian measures , 1999 .

[44]  R. Dobrushin,et al.  Completely analytical interactions: Constructive description , 1987 .

[45]  Justification of the renormalization-group method , 1980 .

[46]  R. Dobrushin.  Probability contributions to statistical mechanics , 1994 .

[47]  Some remarks on pathologies of renormalization-group transformations for the Ising model , 1993 .

[48]  Roberto H. Schonmann,et al.  Complete analyticity for 2D Ising completed , 1995 .

[49]  Charles M. Newman,et al.  Normal fluctuations and the FKG inequalities , 1980 .

[50]  Renormalization group at criticality and complete analyticity of constrained models: A numerical study , 1996, hep-th/9603098.

[51]  R. Dobrushin,et al.  The central limit theorem and the problem of equivalence of ensembles , 1977 .

[52]  J. V. Leeuwen,et al.  Renormalization Theory for Ising Like Spin Systems , 1976 .

[53]  E. Olivieri,et al.  One dimensional spin glasses with potential decay 1/r1+g. Absence of phase transitions and cluster properties , 1987 .

[54]  R. Dobrushin,et al.  Constructive Criterion for the Uniqueness of Gibbs Field , 1985 .

[55]  R. L. Dobrushin,et al.  Perturbation methods of the theory of Gibbsian fields , 1996 .

[56]  Weakly gibbsian measures for lattice spin systems , 1997 .