Phase transitions on fractals. III. Infinitely ramified lattices

For pt.II see vol.17, p.435 (1984). In the first two papers of this series the authors considered self-similar fractal lattices with a finite order of ramification R. In the present paper they study physical models defined on a family of fractals with R= infinity. In order to characterise the geometry of these systems, they need the connectivity Q and the lacunarity L, in addition to the fractal dimensionality D. It is found that discrete-symmetry spin models on these lattices undergo a phase transition at T_c>0. An approximate renormalisation group scheme is constructed and used to find the dependence of T_c and the critical exponents on the geometrical factors. They also solve the problem of resistor networks on these fractals, and discuss its consequences concerning spin models with continuous symmetry.