# Probability distributions with singularities

## TL;DR

This paper reviews probability distributions with singularities, exploring their mathematical mechanisms, connections to phase transitions, and implications for fluctuations and response behaviors in statistical mechanics.

## Contribution

It provides a comprehensive overview of singular probability distributions, linking their properties to phase transitions and fluctuation phenomena in statistical mechanics.

## Key findings

- Singular distributions are linked to phase transition mechanisms.
- Condensation of fluctuations is related to distribution singularities.
- Anomalous fluctuations cause giant responses in systems.

## Abstract

In this paper we review some general properties of probability distributions which exibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such paradigms, the underlying mathematical mechanism producing the singularity and other topics such as the condensation of fluctuations, the relationships with ordinary phase-transitions, the giant response associated to anomalous fluctuations, and the interplay with Fluctuation Relations.