Is there a noncommutative Gaussian?

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.

• Probably you meant ,,noncommutative probability spaces'' instead of commutative Nov 27 '21 at 14:57
• You already picked the right tag - free-probability ... semicircle distribution, free convolution, etc. etc. Nov 27 '21 at 15:15

• @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. Nov 28 '21 at 19:08