Finite-size localization scenarios in condensation transitions

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.