Faithfulness of Real-Space Renormalization Group Maps

Abstract

The behavior of \(b=2\) real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical \(q=2,3,4\) Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice \(L_g\) and allowing the source configuration size \(L_0\) to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that \(b=2\) weighted majority-like RSRG maps acting on the \(q=2\) Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.