Calculation of the Critical Exponentηvia Renormalization-Group Recursion Formulas

Abstract

This paper presents an extension of %'ilson s renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent q. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero q. They are intended to demonstrate, qualitatively, how nonzero values for q are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson s earlier calculation of the exponents v and y.The equations are solved both by e expansion about four dimensions and by numerical integration in three dimensions. To order e~w e obtain q = 0.05c2. Numerical results in three dimension are q = 0.058, v = 0.588, and y = 1.14. The relation y = (2 -q) pis confirmed.

Towards functional flows for hierarchical models

“…Based on their construction, we expect that physical observables depend on the averaging procedure, on the measure factor ' , and on the decimation parameter '. It has proven difficult to systematically include wave function renormalizations and higher order operators in hierarchical models, and it is therefore not known whether the scheme dependence vanishes upon higher order corrections [18,19]. Still, the scheme dependence should give a reasonable estimate for the underlying error in the model assumptions, in particular, in comparison with functional methods.…”

“…Numerical stability and reliability in the results is ensured through powerful control and optimization techniques [11][12][13][14]. A different implementation of Wilson's idea is realised in hierarchical models of lattice scalar theories [15][16][17][18][19]. Hierarchical renormalization group transformations are often discrete rather than continuous.…”

Wilsonian approximated renormalization group for matrix and vector models in 2 < d < 4

“…To incorporate its contribution we need to differentiate it by the external momentum p 2 at p = 0, which can not be treated in the original ultra-local approximation. Following Golner's modification [14] justified on the dimensional ground, we approximate this procedure simply by replacing with multiplication of a propagator d dp 2 p=0 approx.…”

Osiris Wavelets in Three Dimensions

“…With 5) we know that the Sobolev inner product is zero unless the support of the s-level wavelet is contained in one of the basic cubes comprising the support of the mother wavelet. We may assume with no essential loss of generality that 6) or equivalently,…”

“…It suggested other modifications of the fluctuations that would admit a non-zero value for ' and yet preserve the hierarchical nature of the approximation, which was so essential to the tractability of the computational approach. This program was pioneered by Golner [6] with some degree of success. Our mathematical work is inspired by his achievement.…”