With B. Goncalves, I have introduced two new classes of networks that resemble small-world properties. These networks are recursively constructed but retain a fixed, regular degree. They consist of a one-dimensional lattice backbone overlayed by a hierarchical sequence of long-distance links. Both types of networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular networks are planar, have a diameter growing as N, and lead to super-diffusion with an exact, anomalous exponent, but possesses only a trivial fixed point T for the Ising ferromagnet. In turn, the 4-regular networks are non-planar, have a diameter growing as _{c}=0~2, exhibit "ballistic" diffusion (^{√[log N]}d), and a non-trivial ferromagnetic transition, _{w}=1T. It suggest that the 3-regular networks are still quite "geometric", while the 4-regular networks qualify as true small-world networks with mean-field properties. Synchronization of processors on these networks is an example of an engineering application. _{c}>0 |

We modeled diffusion on these networks by studying the mean-square displacement of random walks with time, d. It is an instance of a physical exponent containing the ``golden ratio''_{w}=2-log_{2}(f)=1.30576... f=(1+√5)/2 that is intimately related to Fibonacci sequences and since Euclid's time has been found to be fundamental throughout geometry, architecture, art, and Nature itself. It originates from a singular renormalization group fixed point with a subtle boundary layer, for whose resolution f is the main protagonist. The origin of this rare singularity is easily understood in terms of the physics of the process. Currently, we are studying a number of non-equilibrium processes on these networks, for which they may provide non-trivial but potentially solvable instances, such as self-avoiding walks, contact and exclusion processes, sandpile models, etc. |

Early on, in collaboration with Carl Bender, I have devised a new random walk model
on a hyper-spherical lattice. With this model I can analytically
investigate the behavior of dissipative systems and
their interaction with boundaries in arbitrary dimensions. It
provides an analytical continuation for a discrete random walk
to non-integer dimensions by reducing the |

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