Condensation phenomena in fat-tailed distributions: a characterization by means of an order parameter

TL;DR

This paper investigates condensation phenomena in fat-tailed distributions, using the Density Functional Method to characterize phase transitions via an order parameter, revealing new insights into large deviation regimes and applications in finance.

Contribution

It introduces a novel approach using the Density Functional Method to analyze condensation transitions with an order parameter, overcoming limitations of Large Deviation Theory.

Findings

Identification of phase transitions in fat-tailed sums

Characterization of condensation via the Inverse Participation Ratio

Application to financial time-series data

Abstract

Condensation phenomena are ubiquitous in nature and are found in condensed matter, disordered systems, networks, finance, etc. In the present work we investigate one of the best frameworks in which condensation phenomena take place, namely, the sum of independent and fat-tailed distributed random variables. For large deviations of the sum, this system undergoes a phase transition and shifts from a democratic phase to a condensed phase, where a single variable (the condensate) carries a finite fraction of the sum. This phenomenon yields the failure of the standard results of the Large Deviation Theory. In this work we exploit the Density Functional Method to overcome the limitation of the Large Deviation Theory and characterize the condensation transition in terms of an order parameter, i.e. the Inverse Participation Ratio (IPR). This procedure leads us to investigate the system in the large-deviation regime where both the sum and the IPR are constrained, observing new phase transitions. As a sample application, the case of condensation phenomena in financial time-series is briefly discussed.