{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "GD3Q8G9GCV"
    }
  ],
  "body": "\n## Summary\n\nFree probability is a theory of noncommuting random variables in which the classical notion of independence is replaced by free independence.\nIt was introduced by Dan Voiculescu in the 1980s to study the von Neumann algebras of [[free groups|0KTW26C80J]].\nIt has analogues of the central limit theorem, of convolution and of cumulants.\nIn it the [[Wigner semicircle law|3QTT95DMFH]] plays the role of the normal distribution.\nIt also describes the limit of large random matrices, which become asymptotically free.\n\n## Explanations\n\n### L1 human\n\nIn ordinary probability, random numbers commute, and independence means that knowing one tells nothing about the other.\nMatrices and operators do not commute, and for them there is a different and very useful rule of independence, called freeness.\nIt is modelled on free groups, in which products of letters from different generators never cancel.\nThe sum of many free variables, suitably scaled, has a semicircle distribution rather than a bell curve.\nLarge random matrices chosen independently behave almost exactly like free variables, which makes the theory useful in physics and statistics.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46L54",
      "46L53"
    ],
    "name": "Free probability",
    "same_as": {
      "wikidata": "Q515893",
      "wikipedia": "Free probability"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Free_probability"
      }
    ],
    "tag": "11KYREAH1J"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/11KYREAH1J/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/11KYREAH1J.md",
  "msc": [
    "46L54",
    "46L53"
  ],
  "name": "Free probability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/11K/11KYREAH1J.md",
  "status": null,
  "tag": "11KYREAH1J",
  "uri": "m2://11KYREAH1J",
  "url": "https://math2.tamnd.com/wiki/free-probability/",
  "writing": {
    "tag": "11KYREAH1J",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}