Infinitely ramified fractal lattices and percolation

A family of exact fractal lattices is presented. By adjusting two external parameters, a wide range of fractal and fracton dimensionalities can be achieved, including the fracton dimensionality of 2 which is critical for diffusion. These fractal lattices have an infinite ramification characterized by a ramification exponent p for which an inequality is derived. The infinite ramification makes the problem of percolation on these lattices a non-trivial one. We give numerical evidence for a percolation transition on these fractals. This transition is studied by a real-space renormalization group technique on lattices with fractal dimensionality d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.