A search for the associations of distant radio-bright quasars with Abell   clusters: an effect of gravitational amplification bias ?
  The theory predicts that the effect of gravitational lensing by the matter associated with clusters of galaxies can magnify background sources, leading to an enhancement of source number density around foreground clusters of galaxies. We conduct a search for the associations of distant radio-bright quasars with Abell clusters using the 1-Jy and 2-Jy all-sky catalogs. Statistics turns to be very poor on the basis of the 1-Jy sample, which shows no correlations between the distant radio quasars with the foreground Abell clusters above $1\sigma$ level. However, an apparent association ($>1\sigma$) of the 2-Jy radio sources with the foreground Abell clusters has been detected on scale of $\sim20$ arcminutes. We point out that this enhancement is unlikely to be produced by the statistical gravitational lensing hypothesis utilizing the matter associated with a population of isolated clusters unless (1)their velocity dispersion is a few times larger than the presently adopted value and/or (2)the intrinsic counts of the radio-bright sources with flux have a steeper slope than the presently observed ones. This indicates that (1)the observed associations, if real, can be the integrated result of all the matter along the line of sight to the distant quasars, namely, the weak lensing effect of clusters of galaxies that trace large-scale structures of the universe, and/or (2)the number counts of the radio-bright sources have been seriously contaminated by lensing. 
Xiang-Ping Wu and Jinlin Han