Gaussian field theories, random Cantor sets and multifractality

The computation of multifractal scaling properties associated with a critical field theory involves non-local operators and remains an open problem using conventional techniques of field theory. We propose a new description of Gaussian field theories in terms of random Cantor sets and show how universal multifractal scaling exponents can be calculated. We use this approach to characterize the multifractal critical wave function of Dirac fermions interacting with a random vector potential in two spatial dimensions. We show that the multifractal scaling exponents are self-averaging.