Word Problem and decimation procedure in the Ising model on infinite hyperbolic group lattices

摘要

An elaborate counting procedure for solving the Word Problem yields the free-energy of the Ising model defined over infinite hyperbolic group lattices Lambda(p). Such structures are characterized by a local 2D geometry (2p-sided faces connected by bonds) and a tree-like skeleton responsible of macroscopic boundary effects. The comparison of the above solution with the one derived by decimation gives a simple geometrical interpretation for the renormalized bond activities in terms of a particular dimer covering generating function. The infinite order of the group-lattices and the absence of a thermodynamic limit procedure, leads to an anomalous entropic behaviour at low temperature which can be interpreted in terms of a recursive appearance of decoupled infinite regions.