I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a scale transformation (renormalization group transformation in the QFT case and dilation of the mesh/ruler when computing the perimeter of a fractal). Is there a more profound link between the two? I have't read much about the subject but it seems that for any statistical model at criticality the fractal dimension of the clusters becomes a function of the full dimension of the field. Is it a general rule?