We consider a 2D disordered composite material consisting of three constituents: a normal conductor, a super conductor and an insulator, and examine its macroscopic electrical response when subject to a strong magnetic field applied perpendicular to the plane of the composite. To this end, we exploit a discrete model of a random-resistor-network, and apply a Monte Carlo procedure for sampling ensembles of finite-size three-constituent networks of this kind.

Due to the immense Hall effect induced in the normal conductor in the strong field regime, a super conducting inclusion surrounded by a normally conducting neighborhood tends to expel the current, forcing it to flow mainly through the normal conductor. Namely, such a super conducting inclusion behaves almost like an insulating inclusion! In view of this surprising property, one expects that, in cases where the combination of normal and super conductors is above the percolation threshold, distinct behaviors of the macroscopic electrical response should appear, according to whether the normal conductor by itself is below or above the percolation threshold, leading to a critical point at the threshold itself. When the normal conductor is below the percolation threshold, the macroscopic magnetoresistance is expected to be much larger than when the normal conductor is above that threshold, because in the former case currents necessarily flow through super conducting inclusions embedded in normally conducting regions, i.e., through paths that are supposed to exhibit a large local magnetoresistance. Such an expectation has been also borne out by an asymptotic analysis of the self-consistent effective medium approximation carried out in the past.

When simulating this system, one encounters two problematic effects arising from network size and magnetic field strength. First, at a given small or moderate network size and a strong field, the sampled Ohmic conductances are found to range in a double-peaked distribution, the two parts of which differ by orders of magnitude. This is believed to be the result of boundary effects at the external equipotential boundaries. The effect is found to diminish as the network size is made larger. Second, large-scale current distortions, which occur due to local micro-structures and increase with the field, might extend over large spatial distances, or even over the entire network, significantly affecting its macroscopic response. Networks exhibiting such spatially extended flow distortions are inappropriate for representing the behavior of a real macroscopic material, where all length scales are much smaller than the typical size of the material. Reducing this undesirable effect compels simulating networks of a very large size (requiring large amounts of CPU time) at not too strong magnetic fields.

The results of the simulations support the expected critical behavior: Above the percolation threshold of the normal conductor, the macroscopic magnetoresistance is found to saturate, whereas below this threshold it increases as a function of the magnetic field. Furthermore, the increase with the field below threshold is found to be more rapid than the increase appearing at the threshold itself. This is an interesting instance of a critical phenomenon associated with both a geometrical percolation and the presence of a large Hall effect