Scaling transformation of random walk distributions in a lattice.

Abstract

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution R(q)(x)=[1+b(q-1)x(2)](1/(1-q)) (q>1), which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map q(')=f(q) from the RGT and we show that this map has two fixed points: q=1, attractor, and q=2, repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.