Renormalization group, entropy optimization, and nonextensivity at criticality

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