Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A4 / a / i / Solution

For the planar random walk with independent steps of length $b$, cross terms vanish and $⟨R^2⟩=N{b}^2$. Thus $R_{\mathrm{rms}}=b{N}^{1/2}$. At large $N$, the planar Gaussian end distribution gives $⟨R⟩\simeq b\sqrt{\pi N}/2$.