Scale-invariant random-walks and optimization of non-extensive entropy

As an illustration of the unfolding of scale invariance in stochastic and statistical mechanical systems we consider some properties of random-walks with long-tailed step distributions. We analyze how a renormalization group (RG) transformation flows along maximum entropy step distributions under a given constrained moment and ends at a fixed-point distribution that is identified as that for the Weierstrass walk. This walk displays a transition from Gaussian to fractal be A. Robledo, J. Quintana havior and it is appropriate to employ the generalized Tsallis entropy for the latter regime. Along the RG flow the entropy decreases monotonically and reaches a minimum at the fixed point.