Suppose one creates a random, closed, likely self-crossing polygon from $n$ unit-length vectors arranged head-to-tail, randomly oriented except for the requirement that their sum is zero (so the polygon closes). Here are some examples, for $n=36$:

Q. What is the expected diameter of such a polygon, as a function of $n$? In particular, what is the expected growth rate w.r.t. $n$?

Perhaps it is expected to be $\sim \sqrt{n}$? Continuing the example above, I see about $1.1 \sqrt{n}$. But that could also be about $0.2 n$ if the growth is linear instead. My limited simulations do not distinguish between square-root and linear growth.

To answer

*guest*'s query, I start with a closed regular polygon, and then reflect subchains across the line determined by randomly selected pairs of vertices, until thorough mixing is reached.

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