The statistical geometry of scale-free random trees

Abstract

The properties of random trees (Galton-Watson trees) with scale-free (power-like) probability distribution of coordinations are investigated in the thermodynamic limit. The scaling form of volume probability is found, and the connectivity dimensions are determined and compared with other exponents which describe the growth. The (local) spectral dimension is also determined through the study of the massless limit of the Gaussian model on such trees.