Quasar-Cluster Associations and Gravitational Lensing by Large-Scale   Matter Clumps
  Motivated by the significant overdensity of background bright quasars recently detected behind the foreground clusters of galaxies on scale of $10$ arcminutes, we have investigated the possibility of attributing the quasar-cluster associations to gravitational lensing by large-scale matter inhomogeneities. Based on the conventional lensing models, we have shown that the reported quasar overdensity is unlikely to be generated by cluster matter alone. The situation does not change even if all the clusters of galaxies which follow their spatial two-point correlation function are taken into account, while matter clumps on scale of $>20$ Mpc are also found to be unable to provide the required mass surface density since their density contrast is strictly limited by the anisotropy measurements of the cosmic background radiation. Moreover, we have pointed out that the influence of a nonzero cosmological constant on the quasar-cluster associations is very minor. We conclude that either the observed quasar number counts have been seriously contaminated by the magnification bias of matter inhomogeneities of the universe or there should exists some intercluster matter on scale of less than $\sim20$ Mpc, e.g. from cluster-galaxy correlation, whose mean cosmic density is about an order of magnitude higher than that of clusters of galaxies. 
Xiang-Ping Wu and Li-Zhi Fang (University of Arizona)