Mean-square displacement of an isotropic planar random walk ($\mathbb{E}\Vert S_n{\Vert }^2=n$)

Independent planar unit steps with uniformly distributed direction have zero vector expected value. The squared Euclidean norm of their sum has expectation $n$, because each diagonal term is one and all cross terms vanish by independence. More generally, zero-mean independent steps with common second moment $s^2$ give mean-square displacement $n{s}^2$; nonzero drift adds a quadratic contribution.