Renormalization Group, Entropy Optimization, and Nonextensivity at Criticality

The scaling properties of pair correlations at criticality are reproduced through an equivalence between random walk distributions and order parameter correlations. The shift from Gaussian to fractal walks with self-similar clusters corresponds to the changeover from a Gaussian to a nontrivial fixed point with nonvanishing dimensional anomaly. We show that the renormalization group trajectories lead to fixed points of minimum entropy, and use the Tsallis entropy index $q$ to measure nonextensivity as behavior departs from Gaussian.