Universal Randomness

TL;DR

This paper reviews the universality of the Tracy-Widom distribution across diverse random systems, illustrating its role in describing fluctuations in physical and mathematical models, especially directed polymers.

Contribution

It provides a comprehensive overview and a formal proof of the Tracy-Widom distribution's universality in describing free energy fluctuations in directed polymers.

Findings

Tracy-Widom distribution describes fluctuations in various random systems.

Universal behavior observed in both mathematical and physical models.

Formal proof provided for directed polymers in random media.

Abstract

During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by {\it the same} universal probability distribution function which is called the Tracy-Widom (TW) distribution. Among these systems we find both purely methematical problems, such as the longest increasing subsequences in random permutations, and quite physical ones, such as directed polymers in random media or polynuclear crystal growth. In the extensive Introduction we discuss in simple terms these various random systems and explain what the universal TW function is. Next, concentrating on the example of one-dimensional directed polymers in random potential we give the main lines of the formal proof that fluctuations of their free energy are described the universal TW distribution. The second part of the review consist of detailed appendices which provide necessary self-contained mathematical background for the first part.