Multifractals and critical phenomena in percolating networks: Fixed point, gap scaling, and universality

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Abstract
Analogies between critical phenomena and the continuous spectrum of scaling exponents associated with fractal measures are pointed out. The analogies are based first on the Hausdorff-Bernstein reconstruction theorem, which states that the positive integer moments suffice to characterize a probability distribution function with finite support, and second on the joint probability distribution for the positive integer moments. This joint probability distribution, which can be considered as a fixed point, is universal and exhibits both gap scaling and the infinite set of exponents. Monte Carlo simulations of the electrical properties of percolation clusters on the square and triangular lattices support this general result. Extensions to other fields where infinite sets of exponents have arisen, such as diffusion-limited aggregation and localization, should be straight-forward.