Robustness of surface edge states in the presence of weak, localized disorder

Edge states are expected to allow for robust, nearly lossless transport along surfaces and interfaces in the presence of disorder. Here we show that this expectation can break down when localized random surface perturbation enables coupling between surface waves and bulk modes, even when it is a weak perturbation. We study the transmission of edge states in a conservative, prototypical spring-mass lattice half plane whose boundary is perturbed by finitely many random variations of surface masses or surface springs. We analyze how an incident surface wave is reflected, transmitted, and partially radiated into the bulk. Using a modal expansion on the orthonormal eigenfunctions of the unperturbed system and a multiscale analysis of the resulting random scattering problem, we apply diffusion approximation theory to characterize the statistics of reflectivity, transmissivity, and radiative loss. This approach yields explicit expressions for the moments of these quantities in the regime of weak disorder and large propagation distances. We identify structural and frequency regimes in which radiative losses into the bulk become substantial, reaching 20-80% of the incident energy flux carried by the edge state in the high-frequency range. Our results demonstrate that ballistic edge transport is not guaranteed by the mere existence of an edge mode but depends sensitively on disorder characteristics and unperturbed surface structure. These findings suggest alternative possibilities for engineering surface properties to control wave transport, with implications for phononic, photonic, and electronic systems, including graphene-based structures and nanoscale materials.