Participation ratio for constraint-driven condensation with superextensive mass

TL;DR

This paper investigates how the participation ratio signals condensation in sums of power-law distributed variables when the total mass scales superextensively with system size, revealing a phase transition between homogeneous and condensed phases.

Contribution

It introduces a unified framework for understanding condensation phenomena across unconstrained and constrained mass scenarios, focusing on superextensive total mass scaling.

Findings

Identification of a critical exponent  for phase transition

Demonstration of weak condensation at the transition line

Extension of participation ratio analysis to superextensive mass

Abstract

Broadly distributed random variables with a power-law distribution $f(m) \sim m^{-(1+\alpha)}$ are known to generate condensation effects, in the sense 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 $M=\sum_{i=1}^N m_i > M_c$. In both cases, a standard indicator of the condensation phenomenon is the participation ratio $Y_k=\langle \sum_i m_i^k / (\sum_i m_i)^k\rangle$ ($k>1$), which takes a finite value for $N \to \infty$ 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 $M \sim N^{1+\delta}$ ($\delta >0$), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as $M\sim N^{1/\alpha}$ for $\alpha<1$) and the extensive constrained mass. In particular we show that for exponents $\alpha<1$ a condensate phase for values $\delta > \delta_c=1/\alpha-1$ is separated from a homogeneous phase at $\delta < \delta_c$ by 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.