Histogram of the total number of optical matches vs. separation radius for the 6287 XXL-S radio sources (a total of 6277 optical matches are within 3″). The blue curve represents the theoretical number of genuine matches as a function of radius (Eq. (1)), the red curve the theoretical number of spurious matches as a function of radius (Eq. (2)), and the green curve the total number of matches as a function of radius (the sum of Eqs. (1) and (2) for a given value of r).