Result of the cross-match of three synthetic catalogues with input values n_A = 40 000, n_B = 20 000n_C = 35 000n_AB = 6000, n_AC = 12 000, n_BC = 18 000 and n_ABC = 10 000. The error on catalogue A is a constant equal to 0.4′′. The circular error on catalogue B follows a linear distribution between 0.8 and 1.2′′. The circular error on catalogue C follows a Gaussian distribution of mean 0.75′′ and standard deviation of 0.1′′ between 0.5 and 1′′. The common surface area is a cone of radius 0.42°. Top left: histogram of all associations and theoretical curve from the input parameters. Top centre: histogram of real associations and theoretical curve from input parameters. Top right: histogram of “fully” spurious associations and theoretical curve from input paramameters. Bottom: histograms and theoretical curves of associations mixing a real association between two sources plus a spurious source.