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In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not know how to make this precise, but non-precisely what I mean is that 1. Gaussian shows up everywhere, and 2. it is universal/canonical/... in some sense, e.g. as in the central limit theorem.

My question is whether for many noncommutative probability spaces (an algebra $A$ over $\mathbf{C}$ and a map $E:A\to\mathbf{C}$, with conditons), there also exists a "nicest quadratic" random variable $X\in A$, satisfying analogous properties to the Gaussian.

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    $\begingroup$ Probably you meant ,,noncommutative probability spaces'' instead of commutative $\endgroup$
    – truebaran
    Nov 27 '21 at 14:57
  • $\begingroup$ You already picked the right tag - free-probability ... semicircle distribution, free convolution, etc. etc. $\endgroup$ Nov 27 '21 at 15:15
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The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of free independence and free convolution (introduced by Voiculescu), but there is also boolean independence and boolean convolution (introduced by Speicher and Woroudi); monotone independence and monotone convolution (introduced by Muraki); and anti-monotone independence and anti-monotone convolution (the order-reversal of the previous notion). There are classification results of Speicher and Muraki that show that these are the only notions of independence (or convolution) that obey some natural set of axioms. (Speicher's classification assumed that convolution is commutative, so omitted the monotone and anti-monotone cases that were later discovered by Muraki.)

For each such concept of independence, there is a central limit theorem. Classically, the limiting distribution is the gaussian; in free probability it is the semicircular law; in the boolean case it is the Bernoulli distribution; and in the monotone and anti-monotone cases it is the arcsine law. See Section 9.2.1 of this recent thesis of David Jekel (and Chapter 5 of that thesis contains a more detailed history of the development of these notions of independence). For each of these independence concepts there is also an associated notion of entropy, and these distributions extremise the entropy amongst all distributions of a fixed mean and variance; again, Jekel's thesis has further information. (For the free case, of course, pretty much any introduction to free probability will contain these facts.)

EDIT: There is also finite free convolution, in which the analogue of the gaussian is the distribution of zeroes of Hermite polynomials.

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    $\begingroup$ It's interesting that they're all bounded (although I guess that the Gaussian is, for most practical purposes, bounded.) $\endgroup$
    – Will Sawin
    Nov 27 '21 at 17:17
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    $\begingroup$ @WillSawin In noncommutative settings the $2k^{th}$ moment of a sum of $2k$ independent mean zero variables (for any of the non-commutative notions of independence) contains only exponentially many non-trivial terms, as opposed to factorially many in the classical case. (compare for instance the number of non-crossing perfect matchings in $\{1,\dots,2k\}$ against the number of unrestricted perfect matchings). This already largely explains the boundedness phenomenon. $\endgroup$
    – Terry Tao
    Nov 28 '21 at 19:08

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