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We study electronic transport through metallic multi-island Coulomb-blockadesystems. Based on a diagrammatic real-time approach, we develop a computeralgorithm that generates and calculates all transport contributions up tosecond order in the tunnel-coupling strengths for arbitrary multi-islandsystems. This comprises sequential and cotunneling, as well as termscorresponding to a renormalization of charging energies and tunnelingconductances. Multi-island cotunneling processes with energy transfer betweendifferent island are taken into account. To illustrate our approach we analyzethe current through an island in Coulomb blockade, that is electrostaticallycoupled to a second island through which a large current is flowing. In thisregime both cotunneling processes involving one island only as well asmulti-island processes are important. The latter can be understood asphoton-assisted sequential tunneling in the blockaded island, where the photonsare provided by potential fluctuations due to sequential tunneling in thesecond island. We compare results of our approach to a P(E)-theory forphoton-assisted tunneling in the weak coupling limit.Comment: 14 pages, 7 figures, published version; minor changes in Sec. IV

Transport in metallic multi-island Coulomb blockade systems: A systematic perturbative expansion in the junction transparency