Correlated volumes for extended wave functions on a random-regular graph

We study the metallic phase of the Anderson model in a random -regular graph, specifically the degree of ergodicity of the high-energy wave functions. We use the multifractal formalism to analyze numerical data for unprecedented large system sizes, obtaining a set of correlated volumes Nq which control the finite -size effects of the wave function q moment. Those volumes grow very fast, ln[ln(Nq)] <^> W, with disorder strength but show no tendency to diverge, at least in an intermediate metallic regime. Close to the Anderson transitions, we characterize the crossover to system sizes much smaller than the first correlated volume. Once this crossover has taken place, we obtain evidence of a scaling in which the derivative of the first fractal dimension behaves critically with an exponent nu = 1.