Scaling for vibrational modes of fractals tethered at the boundaries

Using a mapping between a scalar elastic network tethered at its boundaries and a diffusion problem with permanent traps, we study various vibrational properties of progressively tethered disordered fractals. Different scaling forms are proposed for different types of boundary tethering and numerical investigation of the scaling is performed by the approximate diagonalization of the corresponding large, sparse transition probability matrices. Rather different localization behaviors are found for the leading modes depending on the type of tethering. For example, while the largest nontrivial eigenvalue for the tethering of the hull appears to scale by a power law of the cluster size S in an essentially same manner as the untethered case except for the amplitude which depends on the number of tethered sites, the corresponding quantity for the all-boundary tethering seems to scale logarithmically with S.