Large deviations, condensation, and giant response in a statistical system

TL;DR

This paper investigates the probability distribution of sums of non-identically distributed variables, revealing condensation phenomena akin to phase transitions, with general formulas and specific examples demonstrating their sensitivity and behavior.

Contribution

It derives a general expression for the distribution of sums of non-i.i.d. variables and analyzes the condensation phenomenon in analogy to phase transitions.

Findings

Condensation of fluctuations can occur in sums of non-i.i.d. variables.

The probability distribution is highly sensitive to the distribution of individual variables.

Analytical and numerical solutions confirm the theoretical predictions.

Abstract

We study the probability distribution $P$ of the sum of a large number of non-identically distributed random variables $n_m$. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted in analogy to phase-transitions in Statistical Mechanics. A general expression for $P$ is derived, and its sensitivity to the details of the distribution of a single $n_m$ is worked out. These general results are verified by the analytical and numerical solution of some specific examples.