Antiferromagnetic Potts model on Sierpinski carpets

An approximate real-space renormalisation group method, based on the Migdal-Kadanoff recursion relations, is used to study the critical properties of the pure and diluted $q$-state antiferromagnetic Potts model on Sierpinski carpets. Fixed points and phase diagrams are calculated for both large and small lacunarity carpets family. The pure model has phase transitions at finite temperatures and critical behaviour is observed for $q$ less than a cutoff value $q_0$. We determine the value of $q_0$, the percolation concentration $p_{\rm c}$ for the diluted model, and the critical exponents, and we investigate their dependence on various geometrical characteristics of the fractal