![]() (the factor 2 appears because each nearest neighbour coupling is generated by two different sums on adjacent white sites). Needless to say, countless more accurate solutions are available in the literature, but our selection is motivated mainly by sake of technical simplicity, rather than sophistication. Among the various possible lines of attack, we consider here the simple and very instructive suggestion given by H. This step is a key point of renormalization procedure, and can be solved (at least approximately) with the help of physical intuition and appropriate technical procedures. We have thus to make some appropriate manipulations on the form of H ( 1 ), before proceeding to a new renormalization, otherwise interactions of rapidly increasing complexity would be generated. The transformed Hamiltonian (17.60) does not have the same form as the original one (17.58), because of the presence of the sum over second nearest neighbors and the sum over the sites of the square. The number of preserved sites is half of the original number of sites, and the length scale is now a ( 1 ) = 2 a. In the reduced Hamiltonian (17.60), the subscript 〈 ij 〉 denotes nearest neighbor pairs (of preserved sites) interacting with coupling parameter K 1, the subscript denotes next nearest neighbor pairs interacting with parameter K 2, and the last term indicates spins on squares interacting with K 3. (the factor 2 in the equality K 1 = 2 B ( K ) appears because each nearest neighbor coupling is generated by two different sums on adjacent empty dots). (17.60) H ( 1 ) = - K 1 ∑ 〈 ij 〉 s i s j - K 2 ∑ s i s j - K 3 ∑ square s i s j s k s l, K 1 = 2 B ( K ), K 2 = B ( K ), K 3 = C ( K ) (1985) examined discrete diffusion and showed that it involves a fractal diffusion front that can be modeled by the hull and the perimeter of a percolation cluster. Duplantier that the perimeter's dimension is 4/3. Duplantier that the hull's dimension is 7/4.Ī more demanding definition of the boundary yields the perimeter. The hull of a spanning cluster is its boundary. The backbone is the path followed by a fluid diffusing through the lattice. Numerical estimates suggest the backbone has dimension 1.61. The backbone is the subset of the spanning cluster that remains after removing all parts that can be separated from both spanned sides by removing a single filled cell from the spanning cluster. Many fractals are defined as part of a percolation cluster. In addition, spanning clusters have holes of all sizes they are statistically self-similar fractals. This d is the mass dimension of Section II.C. Percolation lattices well below, near, and well above the percolation threshold.Īt p = p c the masses of the spanning clusters scale with the lattice size L as L d, independently of the lattices.
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