Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

Giacomo Gradenigo,Eric Bertin

doi:10.3390/e19100517

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 = ∑ 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 + 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.

Highlights

In the context of the sum of a large number of positive random variables, an interesting phenomenon occurs when a single variable carries a finite fraction of the sum [1]
To better grasp the nature of this difference, we study here the case where the total mass is fixed to a superextensive value M ∼ N 1+δ, with δ > 0, extending some of the results presented in [8]
The general motivation of this work was to better understand the connection between condensation in the unconstrained case and in the constrained case with extensive mass, because condensation occurs on opposite ranges of the exponent α, respectively α < 1 and α > 1

Summary

Introduction

In the context of the sum of a large number of positive random variables, an interesting phenomenon occurs when a single variable carries a finite fraction of the sum [1]. The situation is different, though, when one considers power-law distributed random variables with a fixed total sum M = ∑iN=1 mi , a case which we refer to as constraint-driven condensation, or constrained condensation. Such a phenomenon, which is related to the large of heavy-tailed sums (see, e.g., [26]), is found for instance in the stationary distribution of the discrete Zero Range Process and its continuous variables generalization [7,8,10]. Before dealing with the superextensive mass case, we will first recall in Section 2 how to compute the participation ratios in the case of unconstrained condensation, and present in Section 3 a simplified evaluation of the participation ratio in the case of constrained condensation with an extensive fixed mass

Unconstrained Condensation

Constrained Condensation

Preliminary Analysis of the Phase Diagram

Conclusions