Probability distribution functions (PDFs) of the projected offsets between the BCG position and the SZ- or X-ray-determined cluster centers. Black and purple data points represent the offsets between the redMaPPer optical centers and the SZ (ACT DR5) or X-ray cluster centers, respectively. The solid blue and green lines show the best-fitting two-component 2D Gaussian models for the SZ and X-ray cases. The dashed orange and red lines show results from Ding et al. (2025): the orange line corresponds to all 186 optical–SZ cross-matched clusters, while the red line excludes clusters affected by astrophysical effects (e.g., ongoing mergers) or systematic biases in the HSC data and cluster-finding algorithm. The dashed gray line shows the X-ray–optical offset distribution from Oguri et al. (2018), based on HSC Wide S16A clusters. All offsets are shown in physical units of h⁻¹ kpc.