On the Fractal dimension and correlations in percolation theory

We discuss the fractal dimension of the infinite cluster at the percolation threshold. Using sealing theory and renormalization group we present an explicit expression for the two-point correlation function within percolation clusters. The fractal dimension is given by direct integration of this function.See especially Ref. 1 for a discussion of the general aspects of percolation.     

Axial and diagonal anisotropy crossover exponents for cubic systems

Cubic n-component spin systems exhibit two distinct quadratic spin anisotropy crossover exponents, appropriate for axial and diagonal anisotropies. Results for n→∞ and to order ?², and for nm-component systems are given, and relevance to multi-critical points in various systems is discussed.     

Magnetic correlations on fractals

The critical behavior of magnetic spin models on various fractal structures is reviewed, with emphasis on branching and nonbranching Koch curves and Sierpiriski gaskets and carpets. The spin correlation function is shown to have unusual exponential decays, e.g., of the form exp[-(r/gx) ^x ], and to crossover to other forms at larger distancesr. The various fractals are related to existing models for the backbone of the infinite incipient cluster at the percolation threshold, and conclusions are drawn regarding the behavior of spin correlations on these models.     

Flory approximant for self-avoiding walks on fractals

A Flory approximant for the exponent describing the end-to-end distance of a self-avoiding walk (SAW) on fractals is derived. The approximant involves the fractal dimensionalities of the backbone and of the minimal path, and the exponent describing the resistance of the fractal. The approximant yields values which are very close to those available from exact and numerical calculations.