Alexander-Orbach conjecture

By substituting the Alexander-Orbach conjecture d = 4/3 or equivalently /i = [(3d-4)i/-/ ]/2 into eq. (4), we find that the upper boimd of Jaf of eq. (4) is unity, Our numerical results strongly support Saf — 1- We claim the value of Jaf to be very close to unity independent of the Euclidean dimension d. ... 

In the percolation problem one has for the spectral dimension approximately, d = 4/3, independent of the space dimensions according to the Alexander and Orbach conjecture. It is also the mean field value for lattice animals (branched structures defined on lattices) or Cayley tree-like structures. Hence the Cayley tree corresponds to the mean field solution to percolation. ... 

Alexander and Orbach [35] conjectured based on numerical evidence at hand that for fractal objects embedded in a two- or three-dimensional Euclidean space, d 4/3. We find similar values for the three proteins examined below. In fact the value of d depends on the size of the protein and approaches 2 for large proteins with thousands of residues [162]. 

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