Spectral properties of the renormalization group at infinite temperature

TL;DR

This paper analyzes the spectral properties of the renormalization group linearization at an infinite temperature fixed point for classical Ising systems, revealing dense point spectra with unusual characteristics.

Contribution

It explicitly determines the spectral properties of the RG linearization at a trivial fixed point for certain RG maps, highlighting unique spectral phenomena.

Findings

Spectral properties are explicitly worked out at the trivial fixed point.

The spectrum is dense point spectrum with no point spectrum for the adjoint operators.

Results suggest complex spectral behavior in RG linearizations at infinite temperature.

Abstract

The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations.