Exact Renormalization of Super-Diffusion on the Tower-of-Hanoi Network

We propose the Tower-of-Hanoi network as a hierarchical, small-world network possessing both, geometric and long-range links. Modeling diffusion via a random walk on this network provides a mean-square displacement with an exact, anomalous exponent $d_{w}=2-\ln(\phi)/\ln(2)=1.30576\ldots$. Here, $\phi=\left(1+\sqrt{5}\right)/2$ is the ``golden ratio'' that is intimately related to Fibonacci sequences. This may be the first solvable model with super-diffusion for any fractal structure. This appears to be also the first known instance of any physical exponent containing $\phi$. It originates from an unusual renormalization group fixed point with a subtle boundary layer. The connection between network geometry and the emergence of $\phi$ in this context is still elusive.