  Broadly distributed random variables with a power-law distribution $f(m) \sim
m^{-(1+\alpha)}$ 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
$Y_k=\langle \sum_i m_i^k / (\sum_i m_i)^k\rangle$ ($k>1$), which takes a
finite value for $N \to \infty$ 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 $M
\sim N^{1+\delta}$ ($\delta >0$), hence interpolating between the unconstrained
condensation case (where the typical value of the total mass scales as $M\sim
N^{1/\alpha}$ for $\alpha<1$) and the extensive constrained mass. In particular
we show that for exponents $\alpha<1$ a condensate phase for values $\delta >
\delta_c=1/\alpha-1$ is separated from a homogeneous phase at $\delta <
\delta_c$ by a transition line, $\delta=\delta_c$, 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.
