Computing Bounds on Critical Indices

Abstract

In this paper we give an outline of a computer assisted proof in which we use an extended version of the Wilson-Kadanoff renormalization group scheme to get rigorous bounds on a critical exponent that is universal for a class of one parameter families F _µ of hierarchical lattice systems in d = 3 dimensions. The parameter µ will be referred to as temperature. Assume that a given family F _µ undergoes a phase transition at µ = µ _c, then we define κ to be the exponent that describes the scaling behavior of the free energy as the temperature approaches the critical value µ _c. More precisely, if U _n (µ) is the free energy density coresponding to the system F _µ confined to a cube of volume 2^dn at temperature µ, then

$$ k = \mathop {\lim }\limits_{\mu \to {\mu _c}} {\text{ }}\frac{1}{{\log \left( {\left| \mu \right. - {\mu _c}} \right)}}\log \left( {\mathop {\lim }\limits_{n \to \infty } {\text{ }}{U_n}\left( \mu \right)} \right). $$

(1.1)

Supported by the National Science Foundation under Grant No. DMR-85-40879.

Supported by the National Science Foundation under Grants No. DMS-85-18622 and DMS-87-03539.