Airy processes and variational problems

TL;DR

This paper reviews Airy processes, their mathematical formulation, and their role in describing large-scale fluctuations in one-dimensional growth models, including formulas for probabilities and applications to variational problems.

Contribution

It provides a comprehensive overview of Airy processes, their boundary value formulas, and their applications in physical variational problems and local behavior analysis.

Findings

Formulas expressing probabilities as Fredholm determinants.

Connections between Airy processes and physical variational problems.

Insights into local behavior of Airy processes.

Abstract

We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they lie below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.