Title: Background of "Magnetic model" fractals Post by: gamma on May 03, 2009, 10:26:47 PM We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map. http://arxiv.org/abs/0708.0616 We obtain the exact renormalization map and plots of Lee-Yang and Fisher zeros distributions for Potts models on a number of hierarchical lattices: the diamond hierarchical lattice, a lattice we call spider web, the Sierpinski gasket and cylinders. Such models are only examples among the ones we can study in the general framework of hierarchical lattices, developed in a previous paper. http://arxiv.org/abs/0711.4610 (Jacopo De Simoi, Stefano Marmi) |