Conductance in random inductance-capacitance networks.

Abstract

We consider the admittance Y of large square networks in which each link is randomly, and with equal probability, assigned a component of inductance L or capacitance C. When a potential sin(\ensuremath{\omega}t) is applied over the network, Y(\ensuremath{\omega}) is purely imaginary and alternates, as a function of \ensuremath{\omega}, between inductive and capacitive behavior, with resonances at frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{i}}$, which we find numerically. The corresponding frequency-density function is close to that of the logistic distribution in mathematical statistics. In the limit of large networks, the fraction 0.1964\ifmmode\pm\else\textpm\fi{}0.0007 of all ${\mathrm{\ensuremath{\omega}}}_{\mathit{i}}$ are 0 or \ensuremath{\infty}. That fraction is related to the number of L and C clusters in the network. We also discuss the amplitude of the potential at the nodes of the lattice, i.e., the eigenstates corresponding to the eigenvalues ${\mathrm{\ensuremath{\omega}}}_{\mathit{i}}$.