Is there a noncommutative Gaussian?

Question

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.

Answer 1

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 in Boolean convolution); monotone independence and monotone convolution (introduced by Muraki in Monotonic independence, monotonic central limit theorem and monotonic law of small numbers); and anti-monotone independence and anti-monotone convolution (the order-reversal of the previous notion). There are classification results of Speicher (On universal products) and Muraki (The five independences as natural products) 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 the recent thesis Evolution equations in non-commutative probability of David Jekel (and Chapter 5 of that thesis contains a more detailed history of the development of these notions of independence). For the classical and free independence concepts, at least, 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 (see Marcus, Spielman, and Srivastava - Finite free convolutions of polynomials), in which the analogue of the gaussian is the distribution of zeroes of Hermite polynomials.

Answer 2

If one does not insist on having a notion of "independence" in the background, but takes, as asked in the question, the Wick/Isserlis formula (which expresses general moments in terms of second moments via a nice combinatorial formula) as guiding principle, then there are also other nice non-commutative versions of multivariate Gaussian distributions. A prominent one is given by the q-Gaussian distribution, which interpolates between the classical Gaussian and the free Gaussian (i.e., semicircular) distribution.