Free probability and asymptotic freeness

A tracial noncommutative probability space consists here of a unital star algebra with a positive normalized linear functional phi satisfying $\varphi(ab)=\varphi(ba)$. Think of phi as an abstract normalized trace. Unital subalgebras are free if every product of centered elements chosen from consecutively different subalgebras has expectation zero. Consecutively different does not mean that each subalgebra can occur only once: a first–second–first–second pattern is included. Centering applies to each entire selected element, which may itself be a polynomial.

For free centered unit-variance variables s and t, the alternating product $stst$ therefore has expectation zero. In contrast, the product $s^2t^2$ has expectation one, by writing each square as its mean plus a centered remainder and applying freeness. This does not conflict with the definition, because the squares are not centered as written. Freeness is a relation between generated algebras, not merely a pairwise covariance condition. Knowing the second mixed moment vanishes is far too little to establish it.

Take a fixed finite number of independent normalized GUE matrices of dimension N, each with off-diagonal squared-modulus mean $1/N$ and real diagonal variance $1/N$. All free upper coordinates are independent across the family. For every fixed noncommutative polynomial P, the expected normalized trace of P converges to its value under free standard semicircular variables. The expectation in the display is part of the assertion: it specifies the mode of moment convergence used here. Stronger concentration or almost-sure conclusions require additional arguments.

Gaussian Wick expansion pairs entries of matching matrices. After the normalized trace sums are counted, crossing pairings lose powers of dimension, while compatible noncrossing pairings survive. The same noncrossing combinatorics defines moments of free semicircular families. For one variable it produces the Catalan moments of the semicircle law; for several variables it also tracks the colors identifying which matrix supplied each entry. This is why separate semicircle marginals alone do not determine the joint limit.

The sum of two free standard semicircular variables is semicircular with variance two. Its support is therefore from minus twice the square root of two to twice the square root of two. One route is noncrossing pair counting: each matched pair in an even moment can carry either color, giving the variance-two Catalan sequence. Free additive convolution names the distribution of a sum of free self-adjoint variables, generalizing this example beyond semicircular inputs. It is defined through the joint freeness relation, not by adding independent scalar samples.

Classical independent commuting variables with the same semicircle marginals have a different fourth moment for their sum. Their alternating mixed product factors into the product of second moments instead of vanishing. Both independence notions add variances, so agreement at second order does not establish agreement of spectral laws. Likewise, the mixed-trace theorem does not automatically imply convergence of operator norms or exclude a vanishing proportion of outlying eigenvalues. Those observables require control beyond fixed polynomial moments, just as a global empirical law does.