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

Take the same segment length and a large common number of steps. The ideal walks have size proportional to $N^{1/2}$; excluded volume swells a three-dimensional chain to approximately $N^{0.59}$ and a two-dimensional chain to $N^{3/4}$. Projection changes an isotropic walk's prefactor, rather than its exponent. The projected three-dimensional ideal walk has a smaller planar size than an otherwise equal two-dimensional ideal walk. Therefore the asymptotic ordering isThe following subparts make the size and projection factors explicit. Comparing root-mean-square sizes gives the same ordering as comparing mean distance magnitudes here.