Entropy, Vol. 19, Pages 517: Participation Ratio for Constraint-Driven Condensation with Superextensive Mass (Entropy)

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