The Airy distribution: experiment, large deviations and additional statistics

TL;DR

This paper experimentally measures the Airy distribution in a colloidal system, explores its theoretical large deviation properties, and uncovers phase transition phenomena in its behavior.

Contribution

It provides the first experimental measurement of the Airy distribution and advances its theoretical understanding through large deviation analysis and phase transition identification.

Findings

First experimental measurement of the Airy distribution.

Identification of third-order dynamical phase transitions.

Connection between small-area distribution and Ferrari-Spohn distribution.

Abstract

The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations - the Donsker-Varadhan formalism and the optimal fluctuation method - manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area, and measure its mean in the experiment. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, and we identify the reason for this coincidence.