- What is the average time a random walker takes to get from A to B on a fractal structure and how does this mean time scale with the size of the system and the distance between source and target? We take a nonprobabilistic approach toward this problem and show how the solution is readily obtained using an analysis of thermal vibrations on fractals. Invariance under scaling and continuity with respect to the spectral dimension are shown to be emergent properties of the solution obtained via vibrational analysis. Our result emphasizes the duality between diffusion and vibrations on fractal structures. Applications to biological systems are discussed.