Finite-size localization scenarios in condensation transitions

We consider the phenomenon of condensation of a globally conserved quantity H=∑_{i=1}^{N}ε_{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}=〈ε_{i}^{2}〉/(Nh^{2}) on the size N of the system and on the control parameter δ=(h-h_{c}), 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(ε) 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 γ≥2 defined through the relation N^{*}≃δ^{-γ}, where N^{*} separates the delocalized region (N≪N^{*}, Y_{2} vanishes with increasing N) from the localized region (N≫N^{*}, Y_{2} is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.