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Notes
The rotations of \({\mathbb {R}}^{3N}\) make what is called the orthogonal group O(3N) of dimension \(\frac{3N(3N-1)}{2}\) and there are also 3N parallel translations.
They are synthesized by ribosomes in the cells.
The very applicability of the concept “random”, accompanied by the physicist’s intuition attached to it, remains problematic for this model.
Madras N. (2014) A lower bound for the end-to-end distance of self-avoiding walk. Canad. Math. Bull. 57, 113–118.
It may seem obvious that \(\mathbb E\,{\hbox {diam}}_N^2\) can’t be significantly greater than \(E ||x_1-x_N||^2\), but, apparently, this also remains only conjectural.
Hugo Duminil-Copin, Alan Hammond (2012), Self-avoiding walk is sub-ballistic. arXiv:1205.0401v1.
This space is more complicated for real proteins.
The distance between an S in this component and (001, 002, ...00N) may serve as an upper bound on the folding time of S in some model.
I didn’t try to solve this exercise, but it seems not very difficult.
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Gromov, M. Mathematics of life spaces: continuation of the 2018 large dimensions course. Theory Biosci. 141, 59–63 (2022). https://doi.org/10.1007/s12064-022-00362-0
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DOI: https://doi.org/10.1007/s12064-022-00362-0