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

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.