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Condensation phenomena are ubiquitous in nature and are found in condensedmatter, disordered systems, networks, finance, etc. In the present work weinvestigate one of the best frameworks in which condensation phenomena takeplace, namely, the sum of independent and fat-tailed distributed randomvariables. For large deviations of the sum, this system undergoes a phasetransition and shifts from a democratic phase to a condensed phase, where asingle variable (the condensate) carries a finite fraction of the sum. Thisphenomenon yields the failure of the standard results of the Large DeviationTheory. In this work we exploit the Density Functional Method to overcome thelimitation of the Large Deviation Theory and characterize the condensationtransition in terms of an order parameter, i.e. the Inverse Participation Ratio(IPR). This procedure leads us to investigate the system in the large-deviationregime where both the sum and the IPR are constrained, observing new phasetransitions. As a sample application, the case of condensation phenomena infinancial time-series is briefly discussed.

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