Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

Abstract

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 = ∑ N i=1 m i > M c . In both cases, a standard indicator of the condensation phenomenon is the participation ratio, 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 α < 1 a condensate phase for values δ > δ c = 1/α − 1 is separated from a homogeneous phase at δ < δ c from 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.

Condensation transition and ensemble inequivalence in the discrete nonlinear Schrödinger equation

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Probability Distributions with Singularities

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Transport in disordered systems: The single big jump approach

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