we cannot otherwise specify. In order to address this problem, we determine which value of the scatter gives us an average luminosity as a function of mass that best fits the observed data. Since our original mass-luminosity relation was already quite a good fit to the data (see paper I), we must necessarily have only a small correction to it (i.e., a small value of $a$), which in turn implies a small value for the scatter, which is qualitatively in good agreement with observations (see further discussion in section 7).

Concentration model
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The other model for the CLF of BCGs we consider is based on the variation of the concentration of haloes with the same mass. The basic idea behind this is that the distribution of concentration in same mass haloes will lead to different mass in the inner region of the halo where the galaxy will be present; the luminosity of the hosted galaxy will then simply be proportional to this mass. In practice, we calculate the mass-to-light ratio of this inner region, for the average concentration and with an average luminosity given by equation (\[eq:mlmod\]), and then calculate the change in luminosity as the concentration changes by 