# Normalization in the $O(n)$ model

The $$O(n)$$ model is the model of $$n$$-dimensional spins $$\vec{s}_i$$ at each lattice $$i$$ which are restricted to be on the $$n-1$$ dimensional sphere of radius $$S$$ with Hamiltionian $$H(\vec{s}_1,\dots,\vec{s}_N)=\sum_{\langle i,j\rangle}\vec{s}_i\cdot\vec{s}_j.$$ Usually $$S$$ is taken to be $$\sqrt{n}$$. Is there some physical reasoning behind this? In McGreevy's notes this is important to relate the $$O(n\rightarrow 0)$$ model to self-avoiding walks. I did some thorough note on this found here. It is also used in the treatment of Stanley on the $$O(n\rightarrow\infty)$$ model.