Superlocalization of waves and electrons in hopping conductivity

We have found the two different densities of states d s = 4/3 and d v = 1.05 and two inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ(r) exp [-r do ] with values of d o1 = 1.73 and d o2 = 2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity a(t) exp [-(T 0 /T) β ] between impurities on a random fractal structure give the values of β = 2/5 for the scalar and of β = 1/2 for the vector interactions, respectively.