  We use the random Green's matrix model to study the scaling properties of the
localization transition for scalar waves in a three-dimensional (3D) ensemble
of resonant point scatterers. We show that the probability density $p(g)$ of
normalized decay rates of quasi-modes $g$ is very broad at the transition and
in the localized regime and that it does not obey a single-parameter scaling
law for finite system sizes that we can access. The single-parameter scaling
law holds, however, for the small-$g$ part of $p(g)$ which we exploit to
estimate the critical exponent $\nu$ of the localization transition.
Finite-size scaling analysis of small-$q$ percentiles $g_q$ of $p(g)$ yields an
estimate $\nu \simeq 1.55 \pm 0.07$. This value is consistent with previous
results for Anderson transition in the 3D orthogonal universality class and
suggests that the localization transition under study belongs to the same
class.
