Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x_1$, label the rest $x_2, \dots, x_n$ in clockwise order. What is the expected value of $$ \sum_{1\le i\le n} |x_{i}-x_{i-1}|^2 $$ (where $x_0$ is interpreted as $x_n$)?

(As written, the expression $|x_{i}-x_{i-1}|$ represents distance in the Euclidean plane; I'd prefer to use the distance along the circumference of the circle, which isn't that different when $n$ is large.)

Cauchy's inequality trivially gives (using distance along the circumference of the circle) $$ 1 = \bigg( \sum_{1\le i\le n} |x_{i}-x_{i-1}| \bigg)^2 \le n \sum_{1\le i\le n} |x_{i}-x_{i-1}|^2, $$ and so the sum in question is at least $1/\sqrt n$. I'm mostly interested in whether this is the correct order of magnitude (although the exact constant is an interesting question as well).

Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x_1$, label the rest $x_2, \dots, x_n$ in clockwise order. What is the expected value of $$ \max |x_{i}-x_{i-1}| $$ (where $x_0$ is interpreted as $x_n$)? Or even more, what is the distribution of this maximum?

(As written, the expression $|x_{i}-x_{i-1}|$ represents distance in the Euclidean plane; I'd prefer to use the distance along the circumference of the circle, which isn't that different when $n$ is large.)

Trivially the maximum (using distance along the circumference of the circle) is at least $1/n$; I'm mostly interested in whether this is the correct order of magnitude (although the exact constant is an interesting question as well).