Geometrical phase transitions on hierarchical lattices and universality

In order to examine the validity of the principle of universality for phase transitions on hierarchical lattices, we have studied percolation on a variety of hierarchical lattices, within exact position-space renormalization-group schemes. It is observed that the percolation critical exponent ${\nu }_p$ strongly depends on the topology of the lattices, even for lattices with the same intrinsic dimensions and connectivities. These results support some recent similar results on thermal phase transitions on hierarchical lattices and point out the possible violation of universality in phase transitions on hierarchical lattices.