Abstract. We consider the weak localization correction to the conductance of a ring connected to a network. We analyse 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.

PACS numbers: 73.23.−b, 73.20.Fz, 72.15.Rn, 02.50.−r, 05.40.Jc

Print publication: Issue 15 (15 April 2005)
Received 9 November 2004, in final form 1 February 2005
Published 30 March 2005