An electron moving freely inside a planar metal with impurities can bedescribed by a Brownian trajectory weighted by its windings around some Poisson distributed points in the plane.

During this talk, I will try to present some of the aspect of Brownian windings in 2 dimensions and their rich structure, such as their Cauchy tail and their log-correlation.

Along the way, I'll also present some Monte--Carlo estimation for the Lévy area.