Anomalies in the multifractal analysis of self-similar resistor networks

Each of the moments of the current distribution in self-similar networks scales with a different exponent. The Legendre transform of these exponents as a function of the order of the moment is called f({α}). In general f({α}) has a fixed convexity, has a maximum value equal to the usual fractal dimension, is continuous, is positive, and has a finite support α_min<α<αₘₐₓ. Also, it usually characterizes the asymptotic form of the current distribution. Here, explicit examples of physically acceptable exceptions to the behavior of f({α}) are exhibited. In the first example, the moments near the zeroth one do not converge uniformly in the large-size limit, leading to an f({α}) which has an apparent maximum at a finite value of {α} while the true maximum is at α_max=∞. In the second example, it is shown that f({α}) can take negative values in domains which are relevant for a full characterization of the current distribution. Disorder seems essential to obtain the latter behavior which for these systems compromises the interpretation of f({α}) as a continuous set of fractal dimensions.