The Airy distribution: experiment, large deviations and additional statistics

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.

5 pages plus appendices

References in corpus (26)

• The large deviation approach to statistical mechanics
• Macroscopic fluctuation theory
• Nonequilibrium Markov processes conditioned on large deviations
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Introduction to dynamical large deviations of Markov processes
• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• From the area under the Bessel excursion to anomalous diffusion of cold atoms
• Exact statistical properties of the Burgers equation
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions
• Geometrical optics of constrained Brownian motion: three short stories
• An invariance principle to Ferrari-Spohn diffusions
• Spatial survival probability for one-dimensional fluctuating interfaces in the steady state
• Constraint Ornstein-Uhlenbeck bridges
• Average trajectory of returning walks
• Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
• Dyson Ferrari--Spohn diffusions and ordered walks under area tilts
• Low-temperature interfaces: Prewetting, layering, faceting and Ferrari-Spohn diffusions
• Large fluctuations of the area under a constrained Brownian excursion
• One-Dimensional Directed Sandpile Models and the Area under a Brownian Curve
• Mortal Brownian motion: three short stories
• Fluctuations of ring polymers

Cited by in corpus (23)

• Anomalous statistics of laser-cooled atoms in dissipative optical lattices
• Statistics of first-passage Brownian functionals
• Toward the full short-time statistics of an active Brownian particle on the plane
• Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting
• Nonequilibirum steady state for harmonically-confined active particles
• Slow dynamics and large deviations in classical stochastic Fredkin chains
• Statistics of the first passage area functional for an Ornstein-Uhlenbeck process
• Geometrical optics of large deviations of fractional Brownian motion
• Spectral density of individual trajectories of an active Brownian particle
• Geometrical optics of first-passage functionals of random acceleration
• Brownian flights over a circle
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Area fluctuations on a subinterval of Brownian excursion
• Geometrical optics of large deviations of Brownian motion in inhomogeneous media
• Nonequilibrium steady state of Brownian motion in an intermittent potential
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Short-time large deviation of constrained random acceleration process
• Full distribution of the ground-state energy of potentials with weak disorder
• The p-Airy distribution
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• The Tracy-Widom distribution at large Dyson index