Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

Broadly distributed random variables with a power-law distribution f (m) similar to m(-() (1+alpha)) are known to generate condensation effects. This means 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 - Sigma(N) (i=1) m(i) > M-c. In both cases, a standard indicator of the condensation phenomenon is the participation ratio Y-k = ( k > 1), 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 similar to N1+delta (delta > 0), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M similar to N-1/(alpha) for alpha < 1) and the extensive constrained mass. In particular we show that for exponents a < 1 a condensate phase for values delta > delta(c) = 1/ alpha - 1 is separated from a homogeneous phase at delta < delta(c) from 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.