%0 Book Section %1 statphys23_0998 %A Simoi, J. De %A Marmi, S. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K group hierarchical julia lee-yang model potts renormalization set statphys23 topic-2 %T Dynamics of the renormalization group for Hierarchical Potts models %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=998 %X Hierarchical lattices are a class of lattices for which it is possible to write an exact block renormalization procedure, therefore producing an exact Renormalization Group map. In the '80s a work by Derrida, De Seze and Itzykson showed that for hierarchical graphs the RG map can be written as an endomorphism of the Riemann sphere $\mathbbC$ and, using a (then) recent result by M.Lyubich, they found the measure supported on the Fisher zeros as the invariant measure supported on the Julia (i.e. unstable) set of the rational (RG) map. In this work we naturally extend this idea to a broader class of models, namely hierarchical hypergraphs. This allows not only to solve exactly Potts models on a larger number of lattices, but also to obtain exact thermodynamical functions in presence of a nonzero external magnetic field. The RG map becomes a rational map on a complex multiprojective space of appropriate dimensions. Getting rigorous results in this setting is harder due to the lack of a fully developed theory for iterations of rational maps in more than one dimension, but this does not prevent a numerical study of the dynamics, keeping an eye on the recent partial developments on the subject.

@incollection{statphys23_0998, abstract = {Hierarchical lattices are a class of lattices for which it is possible to write an exact block renormalization procedure, therefore producing an exact Renormalization Group map. In the '80s a work by Derrida, De Seze and Itzykson showed that for hierarchical graphs the RG map can be written as an endomorphism of the Riemann sphere $\hat{\mathbb{C}}$ and, using a (then) recent result by M.Lyubich, they found the measure supported on the Fisher zeros as the invariant measure supported on the Julia (i.e. unstable) set of the rational (RG) map. In this work we naturally extend this idea to a broader class of models, namely hierarchical hypergraphs. This allows not only to solve exactly Potts models on a larger number of lattices, but also to obtain exact thermodynamical functions in presence of a nonzero external magnetic field. The RG map becomes a rational map on a complex multiprojective space of appropriate dimensions. Getting rigorous results in this setting is harder due to the lack of a fully developed theory for iterations of rational maps in more than one dimension, but this does not prevent a numerical study of the dynamics, keeping an eye on the recent partial developments on the subject.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Simoi, J. De and Marmi, S.}, biburl = {https://www.bibsonomy.org/bibtex/2125a662a9de45e76644f2455adb949fb/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {50d5ed966900eb81ba51ce82fd29a650}, intrahash = {125a662a9de45e76644f2455adb949fb}, keywords = {group hierarchical julia lee-yang model potts renormalization set statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:36.000+0200}, title = {Dynamics of the renormalization group for Hierarchical Potts models}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=998}, year = 2007 }