Conjecture of Alexander and Orbach.

The dynamical properties of fractal networks have received wide range of attention. Works on this area by several pioneering authors$^{1-2}$ have led to the introduction of the \textit{spectral dimension} that dictates the \textit{dynamic} properties on a fractal lattice. Most of the studies involving spectral dimension have been performed on a type of fractal lattice known as \textit{percolation} network. Alexander and Orbach$^{2}$ conjectured that the spectral dimension might be exactly 4/3 for percolation networks with Euclidean dimension $d_{e }\ge $ 2. Recent numerical simulations, however, could not decisively prove or disprove this conjecture, although there are other indirect evidences that it is true. We apply a stochastic approach$^{3}$ to determine the spectral dimension of percolation network for d$_{e }\ge $ 2 $a$nd check the validity of the Alexander-Orbach conjecture. Our preliminary results on 2- and 3-dimensional percolation networks indeed show that Alexander-Orbach conjecture is true, resolving a long-standing debate. References: 1. P. G. deGennes, La Recherche 7 (1976) 919. 2. S. Alexander and R. Orbach, J. Phys. Lett. (Paris) 43 (1982) L625. 3. J. Rudra and J. Kozak, Phys. Lett A 151 (1990) 429.