• DOI:10.1103/PHYSREVA.36.2352
• Corpus ID: 27031109

Anomalies in the multifractal analysis of self-similar resistor networks.

@article{Fourcade1987AnomaliesIT,
  title={Anomalies in the multifractal analysis of self-similar resistor networks.},
  author={Fourcade and Tremblay},
  journal={Physical review. A, General physics},
  year={1987},
  volume={36 5},
  pages={
          2352-2358
        }
}

• Fourcade, Tremblay
• Published 1 September 1987
• Mathematics, Materials Science
• Physical review. A, General physics

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(\ensuremath{\alpha}). In general f(\ensuremath{\alpha}) has a fixed convexity, has a maximum value equal to the usual fractal dimension, is continuous, is positive, and has a finite support ${\ensuremath{\alpha}}_{\mathrm{min}l\mathrm{\ensuremath{\alpha}}l{\ensuremath{\alpha}}_{\mathrm{max}}}$. Also, it usually characterizes the asymptotic form of the current distribution. Here, explicit examples of physically acceptable exceptions to the behavior of f(\ensuremath{\alpha}) 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(\ensuremath{\alpha}) which has an apparent maximum at a finite value of \ensuremath{\alpha} while the true maximum is at ${\ensuremath{\alpha}}_{\mathrm{max}=\mathrm{\ensuremath{\infty}}}$. In the second example, it is shown that f(\ensuremath{\alpha}) 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(\ensuremath{\alpha}) as a continuous set of fractal dimensions.