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We have studied the quantum oscillations of the conductance for arrays ofconnected mesoscopic metallic rings, in the presence of an external magneticfield. Several geometries have been considered: a linear array of ringsconnected with short or long wires compared to the phase coherence length,square networks and hollow cylinders. Compared to the well-known case of theisolated ring, we show that for connected rings, the winding of the Browniantrajectories around the rings is modified, leading to a different harmonicscontent of the quantum oscillations. We relate this harmonics content to thedistribution of winding numbers. We consider the limits where coherence length$L_\varphi$ is small or large compared to the perimeter $L$ of each ringconstituting the network. In the latter case, the coherent diffusivetrajectories explore a region larger than $L$, whence a network dependentharmonics content. Our analysis is based on the calculation of the spectraldeterminant of the diffusion equation for which we have a simple expression onany network. It is also based on the hypothesis that the time dependence of thedephasing between diffusive trajectories can be described by an exponentialdecay with a single characteristic time $\tau_\varphi$ (model A) .  At low temperature, decoherence is limited by electron-electron interaction,and can be modelled in a one-electron picture by the fluctuating electric fieldcreated by other electrons (model B). It is described by a functional of thetrajectories and thus the dependence on geometry is crucial. Expressions forthe magnetoconductance oscillations are derived within this model and comparedto the results of model A. It is shown that they involve severaltemperature-dependent length scales.

Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders