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

A three-dimensional self-avoiding walk has swollen size $R\sim b{N}^{\nu }$, with $\nu \simeq 0.59$; the three-dimensional Flory approximation gives the nearby estimate $3/5$. Isotropic projection gives $⟨R_{\mathrm{proj}}^2⟩=(2/3)⟨R^2⟩$, preserving this exponent. For the mean distance rather than its root mean square, the orientation average instead gives $⟨R_{\mathrm{proj}}⟩=(\pi /4)⟨R⟩$. Neither prefactor changes its position between the ideal $N^{1/2}$ walks and the planar swollen $N^{3/4}$ walk.