Participation ratio for constraint-driven condensation with superextensive mass

Broadly distributed random variables with a power-law distribution f(m)∼ m^{1+\alpha} are known to generate condensation effects, in the sense that, when the exponent \alpha lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (0<\alpha<1) one finds unconstrained condensation, whereas for \alpha>1 constrained condensation takes places fixing the total mass to a large enough value M=\sum_{i=1}^N m_i > M_c. In both cases, a standard indicator of the condensation phenomenon is the participation ratio Y_k, which takes a finite value for N→∞ when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value M ~ N^{1+δ} (δ>0), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M∼N^{1/α} for α<1) and the extensive constrained mass. In particular we show that for exponents \alpha < 1 a condensate phase for values δ > δ_c= 1/α−1 is separated from a homogeneous phase at δ < δ_c by a transition line, δ = δ_c, where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass.