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As an application of the semicircular and circular variable theory developed so far, and of free probability in general, let us go back now to the random matrices. Following Voiculescu's paper , we will prove now a number of key freeness results for them, complementing the basic random matrix theory developed in chapters 6-7. As a first result, completing our asymptotic law study for the Gaussian matrices, we have:

Theorem

Given a sequence of complex Gaussian matrices

[[math]] Z_N\in M_N(L^\infty(X)) [[/math]]

having independent [math]G_t[/math] variables as entries, with [math]t \gt 0[/math], we have

[[math]] \frac{Z_N}{\sqrt{N}}\sim\Gamma_t [[/math]]

in the [math]N\to\infty[/math] limit, with the limiting measure being Voiculescu's circular law.

Show Proof

We know from chapter 6, with this having been actually our very first moment computation for random matrices, in this book, that the asymptotic moments of the complex Gaussian matrices are given by the following formula:

[[math]] M_k\left(\frac{Z_N}{\sqrt{N}}\right)\simeq t^{|k|/2}|\mathcal{NC}_2(k)| [[/math]]

On the other hand, we also know from the above that an abstract noncommutative variable [math]a\in A[/math] is circular, following the law [math]\Gamma_t[/math], precisely when its moments are:

[[math]] M_k(a)=t^{|k|/2}|\mathcal{NC}_2(k)| [[/math]]

Thus, we are led to the conclusion in the statement.

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The above result is of course something quite theoretical, and having it formulated as such is certainly something nice. However, and here comes our point, it is actually possible to use free probability theory in order to go well beyond this, with this time some truly “new” results on the random matrices. We will explain this now, following Voiculescu's paper . Let us begin with the Wigner matrices. We have here:

Theorem

Given a family of sequences of Wigner matrices,

[[math]] Z^i_N\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

with pairwise independent entries, each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math], up to the constraint [math]Z_N^i=(Z_N^i)^*[/math], the rescaled sequences of matrices

[[math]] \frac{Z^i_N}{\sqrt{N}}\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

become with [math]N\to\infty[/math] semicircular, each following the Wigner law [math]\gamma_t[/math], and free.

Show Proof

This is something quite subtle, the idea being as follows:

(1) First of all, we know from chapter 6 that for any [math]i\in I[/math] the corresponding sequence of rescaled Wigner matrices becomes semicircular in the [math]N\to\infty[/math] limit:

[[math]] \frac{Z_N^i}{\sqrt{N}}\simeq\gamma_t [[/math]]

(2) Thus, what is new here, and that we have to prove, is the asymptotic freeness assertion. For this purpose we can assume that we are dealing with the case of 2 sequences of matrices, [math]|I|=2[/math]. So, assume that we have Wigner matrices as follows:

[[math]] Z_N,Z_N'\in M_N(L^\infty(X)) [[/math]]

We have to prove that these matrices become asymptotically free, with [math]N\to\infty[/math].

(3) But this something that can be proved directly, via various routine computations with partitions, which simplify as usual in the [math]N\to\infty[/math] limit, and bring freeness.

(4) However, we can prove this as well by using a trick, based on the result in Theorem 10.34. Consider indeed the following random matrix:

[[math]] Y_N=\frac{1}{\sqrt{2}}(Z_N+iZ_N') [[/math]]

This is then a complex Gaussian matrix, and so by using Theorem 10.34, we obtain that in the limit [math]N\to\infty[/math], we have:

[[math]] \frac{Y_N}{\sqrt{N}}\simeq\Gamma_t [[/math]]

Now recall that the circular law [math]\Gamma_t[/math] was by definition the law of the following variable, with [math]a,b[/math] being semicircular, each following the law [math]\gamma_t[/math], and free:

[[math]] c=\frac{1}{\sqrt{2}}(a+ib) [[/math]]

We are therefore in the situation where the variable [math](Z_N+iZ_N')/\sqrt{N}[/math], which has asymptotically semicircular real and imaginary parts, converges to the distribution of [math]a+ib[/math], equally having semicircular real and imaginary parts, but with these real and imaginary parts being free. Thus [math]Z_N,Z_N'[/math] become asymptotically free, as desired.

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Getting now to the complex case, we have a similar result here, as follows:

Theorem

Given a family of sequences of complex Gaussian matrices,

[[math]] Z^i_N\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

with pairwise independent entries, each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math], the rescaled sequences of matrices

[[math]] \frac{Z^i_N}{\sqrt{N}}\in M_N(L^\infty(X))\quad,\quad i\in I [[/math]]

become with [math]N\to\infty[/math] circular, each following the Voiculescu law [math]\Gamma_t[/math], and free.

Show Proof

This follows from Theorem 10.35, which applies to the real and imaginary parts of our complex Gaussian matrices, and gives the result.

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The above results are interesting for both free probability and random matrices. As an illustration here, we have the folowing application to free probability:

Theorem

Consider the polar decomposition of a circular variable in some von Neumann algebraic probability space with faithful normal state:

[[math]] x=vb [[/math]]

Then [math]v[/math] is Haar-unitary, [math]b[/math] is quarter-circular and [math](v,b)[/math] are free.

Show Proof

This is indeed easy to see in the Gaussian matrix model provided by Theorem 10.36 above, and for details here, we refer to Voiculescu's paper .

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There are many other applications along these lines, and conversely, free probability can be used as well for the detailed study of the Wigner and Gaussian matrices.

For further results on the topics discussed above, we recommend, besides Voiculescu's papers , , , , , and book with Dykema and Nica, , , , , , for general free probability, , , , , , , , for random matrix theory, and , , , , , for applications to operator algebras. But do not worry, we will come back to some of these topics, in what follows.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].