Self-affine morphology of random interfaces governs their functionalities across tribological, geological, (opto-)electrical and biological applications. However, the knowledge of how energy carriers or generally classical/quantum waves interact with structural irregularity is still incomplete. In this work, we study vibrational energy transport through random interfaces exhibiting different correlation functions on the two-dimensional hexagonal lattice. We show that random interfaces at the atomic scale are Cantor composites populated on geometrical fractals, thus multifractals, and calculate their quantized conductance using atomistic approaches. We obtain a universal scaling law, which contains self-similarity for mass perturbation, and exponential scaling of structural irregularity quantified by fractal dimension. The multifractal nature and Cantor-composite picture may also be extendable to charge and photon transport across random interfaces.
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