Communications in Mathematical Physics

, Volume 91, Issue 1, pp 75–80 | Cite as

A general central limit theorem for FKG systems

  • Charles M. Newman
Article

Abstract

A central limit theorem is given which is applicable to (not necessarily monotonic) functions of random variables satisfying the FKG inequalities. One consequence is convergence of the block spin scaling limit for the magnetization and energy densities (jointly) to the infinite temperature fixed point of independent Gaussian blocks for a large class of Ising ferromagnets whenever the susceptibility is finite. Another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models.

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References

  1. [CG]
    Cox, J.T., Grimmett, G.: Central limit theorems for percolation models. J. Stat. Phys.25, 237–251 (1981)Google Scholar
  2. [FKG]
    Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)Google Scholar
  3. [IS]
    Iagolnitzer, D., Souillard, B.: Lee-Yang theory and normal fluctuations. Phys. Rev. B19, 1515–1518 (1979)Google Scholar
  4. [L]
    Lebowitz, J.: Bounds on correlations and analyticity properties of ferromanetic Ising spin systems. Commun. Math. Phys.28, 313–321 (1972)Google Scholar
  5. [N1]
    Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.74, 119–128 (1980)Google Scholar
  6. [N2]
    Newman, C.M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. Proceedings of the symposium on inequalities in statistics and probability held at the University of Nebraska, October, 1982Google Scholar
  7. [NS]
    Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Stat. Phys.26, 613–628 (1981)Google Scholar
  8. [S]
    Simon, B.: Correlation inequalities and the mass gap inP(φ)2 I. Domination by the two point function. Commun. Math. Phys.31, 127–136 (1973)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Charles M. Newman
    • 1
    • 2
  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA
  2. 2.Institute of Mathematics and Computer ScienceThe Hebrew University of JerusalemJerusalemIsrael

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