Finite-size localization scenarios in condensation transitions

We consider the phenomenon of condensation of a globally conserved quantity H=-i=1N?i distributed on N sites, occurring when the density h=H/N exceeds a critical density hc. We numerically study the dependence of the participation ratio Y2=(?i2)/(Nh2) on the size N of the system and on the control parameter ?=(h-hc), for various models: (i) a model with two conservation laws, derived from the discrete nonlinear Schrödinger equation; (ii) the continuous version of the zero-range process class, for different forms of the function f(?) 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 Y2 when plotted against N and by an exponent ?>=2 defined through the relation N*??-?, where N* separates the delocalized region (N<>N*, Y2 is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.