---
tag: 89ZY5YRETE
kind: concept
name: Tracy–Widom distribution
msc: [60B20, 15B52, "33E17"]
specializes: [0DVDCMA2DK]
sources:
  - url: https://en.wikipedia.org/wiki/Tracy–Widom_distribution
same_as:
  wikidata: Q7831889
  wikipedia: Tracy–Widom distribution
---

## Summary

The Tracy–Widom distributions are the limit laws of the largest eigenvalue of large random Hermitian or symmetric matrices, after centring and scaling.
There are three of them, $F_1$, $F_2$ and $F_4$, for the orthogonal, unitary and symplectic Gaussian ensembles.
They are given by a solution of the Painlevé II equation, or by a Fredholm determinant with the Airy kernel.
They also appear far from matrices, for example for the longest increasing subsequence of a random permutation and for random growth models.

## Explanations

### L1 human

The eigenvalues of a large random matrix fill an interval, by the [[semicircle law|3QTT95DMFH]].
The largest one sits near the right end of that interval, with small random fluctuations.
After zooming in by the right factor, these fluctuations follow a fixed law, the Tracy–Widom law.
It is not the normal law: it is asymmetric, with a left tail and a right tail that decay at different speeds.