Magnetoresistance of three-constituent composites: Percolation near a critical line

Scaling theory, duality symmetry, and numerical simulations of a random network model are used to study the magnetoresistance of a metal/insulator/perfect conductor composite with a disordered columnar microstructure. The phase diagram is found to have a critical line which separates regions of saturating and nonsaturating magnetoresistance. The percolation problem which describes this line is a generalization of anisotropic percolation. We locate the percolation threshold and determine the values of the critical exponents (formula presented) (formula presented) which are the same as in two-constituent two-dimensional isotropic percolation. We also determine the exponents which characterize the critical dependence on magnetic field, and confirm numerically that (formula presented) is independent of anisotropy. We propose and test a complete scaling description of the magnetoresistance in the vicinity of the critical line.