  We consider the phenomenon of condensation of a globally conserved quantity
$H=\sum_{i=1}^N \epsilon_i$ distributed on $N$ sites, occurring when the
density $h= H/N$ exceeds a critical density $h_c$. We numerically study the
dependence of the participation ratio $Y_2=\langle \epsilon_i^2\rangle/(Nh^2)$
on the size $N$ of the system and on the control parameter $\delta = (h-h_c)$,
for various models: (i)~a model with two conservation laws, derived from the
Discrete NonLinear Schr\"odinger equation; (ii)~the continuous version of the
Zero Range Process class, for different forms of the function $f(\epsilon)$
defining the factorized steady state. Our results show that various
localization scenarios may appear for finite $N$ and close to the transition
point. These scenarios are characterized by the presence or the absence of a
minimum of $Y_2$ when plotted against $N$ and by an exponent $\gamma\geq 2$
defined through the relation $N^* \simeq \delta^{-\gamma}$, where $N^*$
separates the delocalized region ($N\ll N^*$, $Y_2$ vanishes with increasing
$N$) from the localized region ($N\gg N^*$, $Y_2$ is approximately constant).
We finally compare our results with the structure of the condensate obtained
through the single-site marginal distribution.

Statistical Mechanics