Spectral Properties of the Renormalization Group

Abstract

The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work investigates various spectral properties of the RG map for Ising-type classical lattice systems. It consists of four parts. The first part carries out some explicit calculations of the spectrum of the linearization of the RG at infinite temperature, and discovers that it is of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, but only residual spectrum. The second part presents a rigorous justification of the existence and differentiability of the RG map in the infinite volume limit at high temperature by a cluster expansion approach. The third part continues the theme of the third part, and shows that the matrix of partial derivatives of the RG map displays an approximate band property for finite-range and translation-invariant Hamiltonians at high temperature. The last part justifies the differentiability of the RG map in the infinite volume limit at the critical temperature under a certain condition. In summary, the first part deals with special cases where exact computations can be done, whereas the remaining parts are concerned with a general theory and provide a mathematically sound base.