Characteristics of Fractons: from Specific Realizations to Ensemble Averages

Specific realizations due to computer experiments show that fracton excitations possess a core in which the vibrational amplitudes fall off sharply at their edges. When constructing the ensemble average of the fracton wave functions, we find \(φfr{~}exp[-(r/(ω))^{dφ}]\), where \((ω){~}ω-0.71\). The exponent, -0.71, is in close agreement with the prediction of the fracton dispersion law for a two-dimensional percolating network, -0.705, and the geometrical exponent d φ is much larger than current theoretical limits for impurity electronic states. This suggests that the two problems are different.