# Condensation for random variables conditioned by the value of their sum

## TL;DR

This paper explores the phenomenon of condensation in sums of i.i.d. power-law distributed variables, analyzing the transition where a single variable dominates the sum, with a focus on comparing asymptotic and finite-size results.

## Contribution

It provides a clear, simplified analysis of condensation phenomena, reproduces known results, introduces new insights, and compares asymptotic estimates with exact finite-size calculations.

## Key findings

- Condensation transition occurs for large sums and many variables.
- Asymptotic estimates can be quantitatively compared to exact finite-size results.
- The role of dip and large deviation regimes is significant in the analysis.

## Abstract

We revisit the problem of condensation for independent, identically distributed random variables with a power-law tail, conditioned by the value of their sum. For large values of the sum, and for a large number of summands, a condensation transition occurs where the largest summand accommodates the excess difference between the value of the sum and its mean. This simple scenario of condensation underlies a number of studies in statistical physics, such as, e.g., in random allocation and urn models, random maps, zero-range processes and mass transport models. Much of the effort here is devoted to presenting the subject in simple terms, reproducing known results and adding some new ones. In particular we address the question of the quantitative comparison between asymptotic estimates and exact finite-size results. Simply stated, one would like to know how accurate are the asymptotic estimates of the observables of interest, compared to their exact finite-size counterparts, to the extent that they are known. This comparison, illustrated on the particular exemple of a distribution with L\'evy index equal to $3/2$, demonstrates the role of the contributions of the dip and large deviation regimes. Except for the last section devoted to a brief review of extremal statistics, the presentation is self-contained and uses simple analytical methods.