  We consider the weak localization correction to the conductance of a ring
connected to a network. We analyze the harmonics content of the
Al'tshuler-Aronov-Spivak (AAS) oscillations and we show that the presence of
wires connected to the ring is responsible for a behaviour different from the
one predicted by AAS. The physical origin of this behaviour is the anomalous
diffusion of Brownian trajectories around the ring, due to the diffusion in the
wires. We show that this problem is related to the anomalous diffusion along
the skeleton of a comb. We study in detail the winding properties of Brownian
curves around a ring connected to an arbitrary network. Our analysis is based
on the spectral determinant and on the introduction of an effective perimeter
probing the different time scales. A general expression of this length is
derived for arbitrary networks. More specifically we consider the case of a
ring connected to wires, to a square network, and to a Bethe lattice.
