Fig. 4

Top: Tail index \(\alpha\) normalized by 3, and R, as measured from the different distributions \(P_n(r)\). Continuous lines are just a guide for the eyes. In the inboxes, the histograms represent the \(P_n(r)\) for \(n=30,70,200\), as obtained from the simulation of 200,000 different walks. The continuous lines are the corresponding best estimated Rayleigh distributions. Bottom: typical values of the \(P_n(r)\) as a function of n. As a measure of typical value we use the geometric mean, which better characterizes the distributions when they are close to power laws. For large n we can observe that \(r^{t.v.}_n\sim n^{1/2}\) (continuous line)