Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

TL;DR

This paper rigorously analyzes the regularity and potential pathologies of the renormalization-group (RG) transformations, showing they are well-behaved in some cases but can become ill-defined and non-Gibbsian in others, especially near phase transitions.

Contribution

It provides rigorous theorems on the regularity and pathologies of RG maps, clarifying when RG transformations are well-defined or ill-defined, especially for the Ising model.

Findings

RG map is single-valued and Lipschitz continuous on its domain.

Pathologies occur near first-order phase transitions in certain RG transformations.

Renormalized measures can be non-Gibbsian, indicating ill-defined RG maps.

Abstract

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.