The windings correlation for a closed random walk (Brownian, Feynmanian, or polymer loop)

This is a contribution to the theory of mathematical 'Brownian motion' (pure random walking: infinitely frequent, infinitesimal steps such that <Δr2>~Δt). As a correlation quantity concerning windings of closed walks, and therefore topology, it may have physical interest. Polymers can be such loops, or in a completely different context, spatially closed Feynman paths enact the quantum trace. Certainly for me it was physics – my winding path optics - that showed the correlation to be calculable, rather than trivial (zero or infinity), or intractable, like other statistics one might hope to calculate. Back to the mathematical specifics:
A random loop in a plane may or may not encircle the origin, possibly more than once. If it does so it may or may not also encircle another point fixed at some chosen radius from the origin. My calculation supplies the average product of these two winding numbers for Brownian loops. This 'windings correlation' is a function only of the radius chosen (in suitable units).