5c. Random matrices

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Our main results so far on the von Neumann algebras concern the finite dimensional case, where the algebra is of the form [math]A=\oplus_iM_{n_i}(\mathbb C)[/math], and the commutative case, where the algebra is of the form [math]A=L^\infty(X)[/math]. In order to advance, we must solve: \begin{question} What are the next simplest von Neumann algebras, generalizing at the same time the finite dimensional ones, [math]A=\oplus_iM_{n_i}(\mathbb C)[/math], and the commutative ones, [math]A=L^\infty(X)[/math], that we can use as input for our study? \end{question} In this formulation, our question is a no-brainer, the answer to it being that of looking at the direct integrals of matrix algebras, over an arbitrary measured space [math]X[/math]:

[[math]] A=\int_XM_{n_x}(\mathbb C)dx [[/math]]

However, when thinking a bit, all this looks quite tricky, with most likely lots of technical functional analysis and measure theory involved. So, we will leave the investigation of such algebras, which are indeed quite basic, and called of type I, for later.

Nevermind. Let us replace Question 5.21 with something more modest, as follows: \begin{question}[update] What are the next simplest von Neumann algebras, generalizing at the same time the usual matrix algebras, [math]A=M_N(\mathbb C)[/math], and the commutative ones, [math]A=L^\infty(X)[/math], that we can use as input for our study? \end{question} But here, what we have is again a no-brainer, because in relation to what has been said above, we just have to restrict the attention to the “isotypic” case, where all fibers are isomorphic. And in this case our algebra is a random matrix algebra:

[[math]] A=\int_XM_N(\mathbb C)dx [[/math]]

Which looks quite nice, and so good news, we have our algebras. In practice now, although there is some functional analysis to be done with these algebras, the main questions regard the individual operators [math]T\in A[/math], called random matrices. Thus, we are basically back to good old operator theory. Let us begin our discussion with:

Definition

A random matrix algebra is a von Neumann algebra of the following type, with [math]X[/math] being a probability space, and with [math]N\in\mathbb N[/math] being an integer:

[[math]] A=M_N(L^\infty(X)) [[/math]]

In other words, [math]A[/math] appears as a tensor product, as follows,

[[math]] A=M_N(\mathbb C)\otimes L^\infty(X) [[/math]]

of a matrix algebra and a commutative von Neumann algebra.

As a first observation, our algebra can be written as well as follows, with this latter convention being quite standard in the probability literature:

[[math]] A=L^\infty(X,M_N(\mathbb C)) [[/math]]

In connection with the tensor product notation, which is often the most useful one for computations, we have as well the following possible writing, also used in probability:

[[math]] A=L^\infty(X)\otimes M_N(\mathbb C) [[/math]]

Importantly now, each random matrix algebra [math]A[/math] is naturally endowed with a canonical von Neumann algebra trace [math]tr:A\to\mathbb C[/math], which appears as follows:

Proposition

Given a random matrix algebra [math]A=M_N(L^\infty(X))[/math], consider the linear form [math]tr:A\to\mathbb C[/math] given by:

[[math]] tr(T)=\frac{1}{N}\sum_{i=1}^N\int_X T_{ii}^xdx [[/math]]

In tensor product notation, [math]A=M_N(\mathbb C)\otimes L^\infty(X)[/math], we have then the formula

[[math]] tr=\frac{1}{N}\,Tr\otimes\int_X [[/math]]

and this functional [math]tr:A\to\mathbb C[/math] is a faithful positive unital trace.

Show Proof

The first assertion, regarding the tensor product writing of [math]tr[/math], is clear from definitions. As for the second assertion, regarding the various properties of [math]tr[/math], this follows from this, because these properties are stable under taking tensor products.

■

As before, there is a discussion here in connection with the other possible writings of [math]A[/math]. With the probabilistic notation [math]A=L^\infty(X,M_N(\mathbb C))[/math], the trace appears as:

[[math]] tr(T)=\int_X\frac{1}{N}\,Tr(T^x)\,dx [[/math]]

Also, with the probabilistic tensor notation [math]A=L^\infty(X)\otimes M_N(\mathbb C)[/math], the trace appears exactly as in the second part of Proposition 5.24, with the order inverted:

[[math]] tr=\int_X\otimes\,\,\frac{1}{N}\,Tr [[/math]]

To summarize, the random matrix algebras appear to be very basic objects, and the only difficulty, in the beginning, lies in getting familiar with the 4 possible notations for them. Or perhaps 5 possible notations, because we have [math]A=\int_XM_N(\mathbb C)dx[/math] as well.

Getting to work now, as already said, the main questions about random matrix algebras regard the individual operators [math]T\in A[/math], called random matrices. To be more precise, we are interested in computing the laws of such matrices, constructed according to:

Theorem

Given an operator algebra [math]A\subset B(H)[/math] with a faithful trace [math]tr:A\to\mathbb C[/math], any normal element [math]T\in A[/math] has a law, namely a probability measure [math]\mu[/math] satisfying

[[math]] tr(T^k)=\int_\mathbb Cz^kd\mu(z) [[/math]]

with the powers being with respect to colored exponents [math]k=\circ\bullet\bullet\circ\ldots\,[/math], defined via

[[math]] a^\emptyset=1\quad,\quad a^\circ=a\quad,\quad a^\bullet=a^* [[/math]]

and multiplicativity. This law is unique, and is supported by the spectrum [math]\sigma(T)\subset\mathbb C[/math]. In the non-normal case, [math]TT^*\neq T^*T[/math], such a law does not exist.

Show Proof

We have two assertions here, the idea being as follows:

(1) In the normal case, [math]TT^*=T^*T[/math], we know from Theorem 5.2, based on the continuous functional calculus theorem, that we have:

[[math]] \lt T \gt =C(\sigma(T)) [[/math]]

Thus the functional [math]f(T)\to tr(f(T))[/math] can be regarded as an integration functional on the algebra [math]C(\sigma(T))[/math], and by the Riesz theorem, this latter functional must come from a probability measure [math]\mu[/math] on the spectrum [math]\sigma(T)[/math], in the sense that we must have:

[[math]] tr(f(T))=\int_{\sigma(T)}f(z)d\mu(z) [[/math]]

We are therefore led to the conclusions in the statement, with the uniqueness assertion coming from the fact that the operators [math]T^k[/math], taken as usual with respect to colored integer exponents, [math]k=\circ\bullet\bullet\circ\ldots[/math]\,, generate the whole operator algebra [math]C(\sigma(T))[/math].

(2) In the non-normal case now, [math]TT^*\neq T^*T[/math], we must show that such a law does not exist. For this purpose, we can use a positivity trick, as follows:

[[math]] \begin{eqnarray*} TT^*-T^*T\neq0 &\implies&(TT^*-T^*T)^2 \gt 0\\ &\implies&TT^*TT^*-TT^*T^*T-T^*TTT^*+T^*TT^*T \gt 0\\ &\implies&tr(TT^*TT^*-TT^*T^*T-T^*TTT^*+T^*TT^*T) \gt 0\\ &\implies&tr(TT^*TT^*+T^*TT^*T) \gt tr(TT^*T^*T+T^*TTT^*)\\ &\implies&tr(TT^*TT^*) \gt tr(TTT^*T^*) \end{eqnarray*} [[/math]]

Now assuming that [math]T[/math] has a law [math]\mu\in\mathcal P(\mathbb C)[/math], in the sense that the moment formula in the statement holds, the above two different numbers would have to both appear by integrating [math]|z|^2[/math] with respect to this law [math]\mu[/math], which is contradictory, as desired.

■

Back now to the random matrices, as a basic example, assume [math]X=\{.\}[/math], so that we are dealing with a usual scalar matrix, [math]T\in M_N(\mathbb C)[/math]. By changing the basis of [math]\mathbb C^N[/math], which won't affect our trace computations, we can assume that [math]T[/math] is diagonal:

[[math]] T\sim \begin{pmatrix} \lambda_1\\ &\ddots\\ &&\lambda_N \end{pmatrix} [[/math]]

But for such a diagonal matrix, we have the following formula:

[[math]] tr(T^k)=\frac{1}{N}(\lambda_1^k+\ldots+\lambda_N^k) [[/math]]

Thus, the law of [math]T[/math] is the average of the Dirac masses at the eigenvalues:

[[math]] \mu=\frac{1}{N}\left(\delta_{\lambda_1}+\ldots+\delta_{\lambda_N}\right) [[/math]]

As a second example now, assume [math]N=1[/math], and so [math]T\in L^\infty(X)[/math]. In this case we obtain the usual law of [math]T[/math], because the equation to be satisfied by [math]\mu[/math] is:

[[math]] \int_X\varphi(T)=\int_\mathbb C\varphi(x)d\mu(x) [[/math]]

At a more advanced level, the main problem regarding the random matrices is that of computing the law of various classes of such matrices, coming in series: \begin{question} What is the law of random matrices coming in series

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

in the [math]N \gt \gt 0[/math] regime? \end{question} The general strategy here, coming from physicists, is that of computing first the asymptotic law [math]\mu^0[/math], in the [math]N\to\infty[/math] limit, and then looking for the higher order terms as well, as to finally reach to a series in [math]N^{-1}[/math] giving the law of [math]T_N[/math], as follows:

[[math]] \mu_N=\mu^0+N^{-1}\mu^1+N^{-2}\mu^2+\ldots [[/math]]

As a basic example here, of particular interest are the random matrices having i.i.d. complex normal entries, under the constraint [math]T=T^*[/math]. Here the asymptotic law [math]\mu^0[/math] is the Wigner semicircle law on [math][-2,2][/math]. We will discuss this in chapter 6 below, and in the meantime we can only recommend some reading, from the original papers of Marchenko-Pastur , Voiculescu , Wigner , and from the books of Anderson-Guionnet-Zeitouni , Mehta , Nica-Speicher , Voiculescu-Dykema-Nica .

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].