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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.

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