We present a study of the distribution of X-ray AGN in a representative sample of 26 massive clusters at 0.15<z<0.30, combining Chandra observations with highly complete spectroscopy of cluster members down to M_K*+2. In total we identify 48 X-ray AGN among the cluster members, with luminosities 2x10^41-1x10^44erg/s. In the stacked caustic diagram, the X-ray AGN appear to preferentially lie along the caustics, suggestive of an infalling population. They also appear to avoid the region with lowest cluster-centric radii and relative velocities (r_proj<0.4 r_500; |v-<v>|/sigma_v<0.8), which is dominated by the virialized population of galaxies accreted earliest into the clusters. Moreover the velocity dispersion of the 48 X-ray AGN is 1.51x that of the overall cluster population, which is consistent with the sqrt(2) ratio expected by simple energetic arguments when comparing infalling versus virialized populations. This kinematic segregation is significant at the 4.66-sigma level. When splitting the X-ray AGN sample into two according to X-ray or infrared (IR) luminosity, both X-ray bright and IR-bright sub-samples show higher velocity dispersions than their X-ray dim and IR-dim counterparts at >2sigma significance. This is consistent with the nuclear activity responsible for the X-ray and IR emission being slowly shut down as the host galaxies are accreted into the cluster. Overall our results provide the strongest observational evidence to date that X-ray AGN found in massive clusters are an infalling population, and that the cluster environment very effectively suppresses radiatively-efficient nuclear activity in its member galaxies. These results are consistent with the view that for galaxies to host an X-ray AGN they should be the central galaxy within their dark matter halo and have a ready supply of cold gas.
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