Two-dimensional disordered lattice networks with substrate

The paper deals with the development of a theory for describing two-dimensional (2D) random lattice networks of resistors with a particular topology. We consider a 2D anisotropic random lattice where each node of the network is connected to a reference node (substrate) through a given random resistor. This topology is of great interest both for theoretical and practical applications. Moreover, the theory is able to take into account the similar, but more interesting problem with a capacitive coupling with the substrate. The analytical results allow us to obtain the average behaviour of such networks, i.e. the electrical characterisation of the corresponding physical systems. This effective medium theory is developed starting from the properties of the lattice Green's function of the network and from an ad hoc mean field procedure. An accurate analytical study of the related lattice Green's functions has been conducted obtaining closed-form results expressed in terms of elliptic integrals. All the theoretical results have been verified by means of numerical Monte-Carlo simulations obtaining a remarkably good agreement between numerical and theoretical values, both in resistive and capacitive systems.