Four-terminal resistances in mesoscopic networks of metallic wires: Weak localisation and correlations

We consider the electronic transport in multi-terminal mesoscopic networks of weakly disordered metallic wires. After a brief description of the classical transport, we analyse the weak localisation (WL) correction to the four-terminal resistances, which involves an integration of the Cooperon over the wires with proper weights. We provide an interpretation of these weights in terms of classical transport properties. We illustrate the formalism on examples and show that weak localisation to four-terminal conductances may become large in some situations. In a second part, we study the correlations of four-terminal resistances and show that integration of Diffuson and Cooperon inside the network involves the same weights as the WL. The formulae are applied to multiconnected wire geometries.