Scaling for vibrational modes of fractals tethered at the boundaries

TL;DR

This paper investigates how different boundary tethering methods affect vibrational modes and localization behaviors in disordered fractals, using a novel mapping to diffusion with traps and numerical matrix diagonalization.

Contribution

It introduces a new mapping approach to study vibrational properties of tethered fractals and explores their scaling and localization behaviors under various boundary conditions.

Findings

Different boundary tethering types lead to distinct localization behaviors.

Proposed scaling forms vary with boundary conditions.

Numerical analysis confirms diverse vibrational mode properties.

Abstract

Using a recently introduced 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 types of tethering.