Two-point statistics of multifractal measures

The relationship between the ƒ(α) function of a multifractal and the spatial correlations of its singularity strengths is examined. An expression to compute the probability of observing two different singularities α’ and α″ within a distance r is derived for measures arising from isotropic random multiplicative processes. The correlation of α's is shown to decay logarithmically with distance. Possible applications to turbulence models are discussed, and the results are illustrated for a binomial measure, where the scaling of two-point correlation functions is shown to exhibit a phase transition.