---
tag: 11KYREAH1J
kind: concept
name: Free probability
msc: [46L54, 46L53]
sources:
  - url: https://en.wikipedia.org/wiki/Free_probability
same_as:
  wikidata: Q515893
  wikipedia: Free probability
---

## Summary

Free probability is a theory of noncommuting random variables in which the classical notion of independence is replaced by free independence.
It was introduced by Dan Voiculescu in the 1980s to study the von Neumann algebras of [[free groups|0KTW26C80J]].
It has analogues of the central limit theorem, of convolution and of cumulants.
In it the [[Wigner semicircle law|3QTT95DMFH]] plays the role of the normal distribution.
It also describes the limit of large random matrices, which become asymptotically free.

## Explanations

### L1 human

In ordinary probability, random numbers commute, and independence means that knowing one tells nothing about the other.
Matrices and operators do not commute, and for them there is a different and very useful rule of independence, called freeness.
It is modelled on free groups, in which products of letters from different generators never cancel.
The sum of many free variables, suitably scaled, has a semicircle distribution rather than a bell curve.
Large random matrices chosen independently behave almost exactly like free variables, which makes the theory useful in physics and statistics.