Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

Broadly distributed random variables with a power-law distribution f (m) ∼ m−(1+α) are known to generate condensation effects. This means that, when the exponent α 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 < α < 1) one finds unconstrained condensation, whereas for α > 1 constrained condensation takes places fixing the total mass to a large enough value M = ∑ i=1 mi > Mc. In both cases, a standard indicator of the condensation phenomenon is the participation ratio Yk = 〈∑i mk i /(∑i mi)〉 (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 ∼ N1+δ (δ > 0), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as M ∼ N1/α for α < 1) and the extensive constrained mass. In particular we show that for exponents α < 1 a condensate phase for values δ > δc = 1/α− 1 is separated from a homogeneous phase at δ < δc from a transition line, δ = δc, 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.