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 ${\alpha }_{\min <\mathrm{\alpha }<{\alpha }_{\max }}$. 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 ${\alpha }_{\max =\mathrm{\infty }}$. 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.