Field Theory And Second Renormalization Group For Multifractals In Percolation

TL;DR

This paper reformulates the field theory of multifractals in percolation, revealing that multifractal exponents are eigenvalues of a second renormalization group linked to electrical properties, providing new insights into their observability.

Contribution

It introduces a second renormalization group framework that explicitly connects multifractal exponents to symmetry-breaking fields, clarifying their physical significance in percolation.

Findings

Multifractal exponents are eigenvalues of a second renormalization group.

Electrical properties and noise cumulants are described by the second RG.

Multifractal exponents are 'dominant' rather than 'relevant' in scale analysis.

Abstract

The field-theory for multifractals in percolation is reformulated in such a way that multifractal exponents clearly appear as eigenvalues of a second renormalization group. The first renormalization group describes geometrical properties of percolation clusters, while the second-one describes electrical properties, including noise cumulants. In this context, multifractal exponents are associated with symmetry-breaking fields in replica space. This provides an explanation for their observability. It is suggested that multifractal exponents are ''dominant'' instead of ''relevant'' since there exists an arbitrary scale factor which can change their sign from positive to negative without changing the Physics of the problem.