Potts ferromagnet: Transformations and critical exponents in planar hierarchical lattices

We prove that the duality transformation for a Potts ferromagnet on two-rooted planar hierarchical lattices (HL) preserves the thermal eigenvalue. This leads to a relation between the correlation length critical exponents of a HL and its corresponding dual lattice. Using hyperscaling, we show that their specific heat critical exponents coincide. For a smaller class of HL—namely of diamond and tress types—we prove that another transformation also preserves and .