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We have now all the needed ingredients for launching some explicit random matrix computations. Our goal will be that of computing the asymptotic moments, and then the asymptotic laws, with [math]N\to\infty[/math], for the main classes of large random matrices.

Let us begin by specifying the precise classes of matrices that we are interested in. First we have the complex Gaussian matrices, which are constructed as follows:

Definition

A complex Gaussian matrix is a random matrix of type

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

which has i.i.d. centered complex normal entries.

Here we use the notion of complex normal variable, introduced and studied in chapter 1. To be more precise, the complex Gaussian law of parameter [math]t \gt 0[/math] is by definition the following law, with [math]a,b[/math] being independent, each following the normal law [math]g_t[/math]:

[[math]] G_t=law\left(\frac{1}{\sqrt{2}}(a+ib)\right) [[/math]]

With this notion in hand, the assumption in the above definition is that all the matrix entries [math]Z_{ij}[/math] are independent, and follow this law [math]G_t[/math], for a fixed value of [math]t \gt 0[/math]. We will see that the above matrices have an interesting, and “central” combinatorics, among all kinds of random matrices, with the study of the other random matrices being usually obtained as a modification of the study of the Gaussian matrices.

As a somewhat surprising remark, using real normal variables in Definition 6.1, instead of the complex ones appearing there, leads nowhere. The correct real versions of the Gaussian matrices are the Wigner random matrices, constructed as follows:

Definition

A Wigner matrix is a random matrix of type

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

which has i.i.d. centered complex normal entries, up to the constraint [math]Z=Z^*[/math].

This definition is something a bit compacted, and to be more precise, a Wigner matrix is by definition a random matrix as follows, with the diagonal entries being real normal variables, [math]a_i\sim g_t[/math], for some [math]t \gt 0[/math], the upper diagonal entries being complex normal variables, [math]b_{ij}\sim G_t[/math], the lower diagonal entries being the conjugates of the upper diagonal entries, as indicated, and with all the variables [math]a_i,b_{ij}[/math] being independent:

[[math]] Z=\begin{pmatrix} a_1&b_{12}&\ldots&\ldots&b_{1N}\\ \bar{b}_{12}&a_2&\ddots&&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&a_{N-1}&b_{N-1,N}\\ \bar{b}_{1N}&\ldots&\ldots&\bar{b}_{N-1,N}&a_N \end{pmatrix} [[/math]]

As a comment here, for many concrete applications the Wigner matrices are in fact the central objects in random matrix theory, and in particular, they are often more important than the Gaussian matrices. In fact, these are the random matrices which were first considered and investigated, a long time ago, by Wigner himself .

However, as we will soon discover, the Gaussian matrices are somehow more fundamental than the Wigner matrices, at least from an abstract point of view, and this will be the point of view that we will follow here, with the Gaussian matrices coming first.

Finally, we will be interested as well in the complex Wishart matrices, which are the positive versions of the above random matrices, constructed as follows:

Definition

A complex Wishart matrix is a random matrix of type

[[math]] Z=YY^*\in M_N(L^\infty(X)) [[/math]]

with [math]Y[/math] being a complex Gaussian matrix.

As before with the Gaussian and Wigner matrices, there are many possible comments that can be made here, of technical or historical nature, as follows:

(1) First, using real Gaussian variables instead of complex Gaussian variables in the above definition leads to a less interesting combinatorics, and we will not do this.

(2) The complex Wishart matrices were introduced and studied by Marchenko and Pastur not long after Wigner, in , and so historically came second.

(3) Finally, in what regards their combinatorics and applications, the Wishart matrices quite often come first, before both the Gaussian and the Wigner ones.

So long for random matrix definitions and general talk about this, with all this being at this point quite subjective, but we will soon get to work, and prove results motivating all the above. Let us summarize this preliminary discussion in the following way: \begin{conclusion} There are three main types of random matrices, as follows:

• The Gaussian matrices, which can be thought of as being “complex”.
• The Wigner matrices, which can be thought of as being “real”.
• The Wishart matrices, which can be thought of as being “positive”.

\end{conclusion} We will study these three types of matrices in what follows, in the above precise order, with this order being the one that, technically, best fits us here. Let us also mention that there are many other interesting classes of random matrices, which are more specialized, usually appearing as modifications of the above. More on these later.

In order to compute the asymptotic laws of the Gaussian, Wigner and Wishart matrices, we use the moment method. Given a colored integer [math]k=\circ\bullet\bullet\circ\ldots\,[/math], we say that a pairing [math]\pi\in P_2(k)[/math] is matching when it pairs [math]\circ-\bullet[/math] symbols. With this convention, we have the following result, which will be our main tool for computing moments:

Theorem (Wick formula)

Given independent variables [math]X_i[/math], each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math] being a fixed parameter, we have the formula

[[math]] E\left(X_{i_1}^{k_1}\ldots X_{i_s}^{k_s}\right)=t^{s/2}\#\left\{\pi\in\mathcal P_2(k)\Big|\pi\leq\ker i\right\} [[/math]]

where [math]k=k_1\ldots k_s[/math] and [math]i=i_1\ldots i_s[/math], for the joint moments of these variables.

Show Proof

This is something that we know from chapter 1, the idea being as follows:

(1) In the case where we have a single complex normal variable [math]X[/math], which amounts in taking [math]X_i=X[/math] for any [math]i[/math] in the formula in the statement, what we have to compute are the moments of [math]X[/math], with respect to colored integer exponents [math]k=\circ\bullet\bullet\circ\ldots\,[/math], and the formula in the statement tells us that these moments must be:

[[math]] E(X^k)=t^{|k|/2}|\mathcal P_2(k)| [[/math]]

(2) But this is something that we know from chapter 1, the idea being that at [math]t=1[/math] this follows by doing some combinatorics and calculus, in analogy with the combinatorics and calculus from the real case, where the moment formula is identical, save for the matching pairings [math]\mathcal P_2[/math] being replaced by the usual pairings [math]P_2[/math], and then that the general case [math]t \gt 0[/math] follows from this, by rescaling. Thus, we are done with this case.

(3) In general now, with several variables as in the statement, when expanding the product [math]X_{i_1}^{k_1}\ldots X_{i_s}^{k_s}[/math] and rearranging the terms, we are left with doing a number of computations as in (1), and then making the product of the expectations that we found.

(4) But this amounts in counting the partitions in the statement, with the condition [math]\pi\leq\ker i[/math] there standing for the fact that we are doing the various type (1) computations independently, and then making the product. Thus, we obtain the result.

■

The above statement is one of the possible formulations of the Wick formula, and there are in fact many more formulations, which are all useful. Here is an alternative such formulation, which is quite popular, and that we will often use in what follows:

Theorem (Wick formula 2)

Given independent variables [math]f_i[/math], each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math] being a fixed parameter, we have the formula

[[math]] E\left(f_{i_1}\ldots f_{i_k}f_{j_1}^*\ldots f_{j_k}^*\right)=t^k\#\left\{\pi\in S_k\Big|i_{\pi(r)}=j_r,\forall r\right\} [[/math]]

for the non-vanishing joint moments of these variables.

Show Proof

This follows from the usual Wick formula, from Theorem 6.5. With some changes in the indices and notations, the formula there reads:

[[math]] E\left(f_{I_1}^{K_1}\ldots f_{I_s}^{K_s}\right)=t^{s/2}\#\left\{\sigma\in\mathcal P_2(K)\Big|\sigma\leq\ker I\right\} [[/math]]

Now observe that we have [math]\mathcal P_2(K)=\emptyset[/math], unless the colored integer [math]K=K_1\ldots K_s[/math] is uniform, in the sense that it contains the same number of [math]\circ[/math] and [math]\bullet[/math] symbols. Up to permutations, the non-trivial case, where the moment is non-vanishing, is the case where the colored integer [math]K=K_1\ldots K_s[/math] is of the following special form:

[[math]] K=\underbrace{\circ\circ\ldots\circ}_k\ \underbrace{\bullet\bullet\ldots\bullet}_k [[/math]]

So, let us focus on this case, which is the non-trivial one. Here we have [math]s=2k[/math], and we can write the multi-index [math]I=I_1\ldots I_s[/math] in the following way:

[[math]] I=i_1\ldots i_k\ j_1\ldots j_k [[/math]]

With these changes made, the above usual Wick formula reads:

[[math]] E\left(f_{i_1}\ldots f_{i_k}f_{j_1}^*\ldots f_{j_k}^*\right)=t^k\#\left\{\sigma\in\mathcal P_2(K)\Big|\sigma\leq\ker(ij)\right\} [[/math]]

The point now is that the matching pairings [math]\sigma\in\mathcal P_2(K)[/math], with [math]K=\circ\ldots\circ\bullet\ldots\bullet\,[/math], of length [math]2k[/math], as above, correspond to the permutations [math]\pi\in S_k[/math], in the obvious way. With this identification made, the above modified usual Wick formula becomes:

[[math]] E\left(f_{i_1}\ldots f_{i_k}f_{j_1}^*\ldots f_{j_k}^*\right)=t^k\#\left\{\pi\in S_k\Big|i_{\pi(r)}=j_r,\forall r\right\} [[/math]]

Thus, we have reached to the formula in the statement, and we are done.

■

Finally, here is one more formulation of the Wick formula, which is useful as well:

Theorem (Wick formula 3)

Given independent variables [math]f_i[/math], each following the complex normal law [math]G_t[/math], with [math]t \gt 0[/math] being a fixed parameter, we have the formula

[[math]] E\left(f_{i_1}f_{j_1}^*\ldots f_{i_k}f_{j_k}^*\right)=t^k\#\left\{\pi\in S_k\Big|i_{\pi(r)}=j_r,\forall r\right\} [[/math]]

for the non-vanishing joint moments of these variables.

Show Proof

This follows from our second Wick formula, from Theorem 6.6, simply by permuting the terms, as to have an alternating sequence of plain and conjugate variables. Alternatively, we can start with Theorem 6.5, and then perform the same manipulations as in the proof of Theorem 6.6, but with the exponent being this time as follows:

[[math]] K=\underbrace{\circ\bullet\circ\bullet\ldots\ldots\circ\bullet}_{2k} [[/math]]

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

■

Now by getting back to the Gaussian matrices, we have the following result:

Theorem

Given a sequence of Gaussian random matrices

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

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

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

for any colored integer [math]k=\circ\bullet\bullet\circ\ldots\,[/math], in the [math]N\to\infty[/math] limit.

Show Proof

This is something standard, which can be done as follows:

(1) We fix [math]N\in\mathbb N[/math], and we let [math]Z=Z_N[/math]. Let us first compute the trace of [math]Z^k[/math]. With [math]k=k_1\ldots k_s[/math], and with the convention [math](ij)^\circ=ij,(ij)^\bullet=ji[/math], we have:

[[math]] \begin{eqnarray*} Tr(Z^k) &=&Tr(Z^{k_1}\ldots Z^{k_s})\\ &=&\sum_{i_1=1}^N\ldots\sum_{i_s=1}^N(Z^{k_1})_{i_1i_2}(Z^{k_2})_{i_2i_3}\ldots(Z^{k_s})_{i_si_1}\\ &=&\sum_{i_1=1}^N\ldots\sum_{i_s=1}^N(Z_{(i_1i_2)^{k_1}})^{k_1}(Z_{(i_2i_3)^{k_2}})^{k_2}\ldots(Z_{(i_si_1)^{k_s}})^{k_s} \end{eqnarray*} [[/math]]

(2) Next, we rescale our variable [math]Z[/math] by a [math]\sqrt{N}[/math] factor, as in the statement, and we also replace the usual trace by its normalized version, [math]tr=Tr/N[/math]. Our formula becomes:

[[math]] tr\left(\left(\frac{Z}{\sqrt{N}}\right)^k\right)=\frac{1}{N^{s/2+1}}\sum_{i_1=1}^N\ldots\sum_{i_s=1}^N(Z_{(i_1i_2)^{k_1}})^{k_1}(Z_{(i_2i_3)^{k_2}})^{k_2}\ldots(Z_{(i_si_1)^{k_s}})^{k_s} [[/math]]

Thus, the moment that we are interested in is given by:

[[math]] M_k\left(\frac{Z}{\sqrt{N}}\right)=\frac{1}{N^{s/2+1}}\sum_{i_1=1}^N\ldots\sum_{i_s=1}^N\int_X(Z_{(i_1i_2)^{k_1}})^{k_1}(Z_{(i_2i_3)^{k_2}})^{k_2}\ldots(Z_{(i_si_1)^{k_s}})^{k_s} [[/math]]

(3) Let us apply now the Wick formula, from Theorem 6.5. We conclude that the moment that we are interested in is given by:

[[math]] \begin{eqnarray*} &&M_k\left(\frac{Z}{\sqrt{N}}\right)\\ &=&\frac{t^{s/2}}{N^{s/2+1}}\sum_{i_1=1}^N\ldots\sum_{i_s=1}^N\#\left\{\pi\in\mathcal P_2(k)\Big|\pi\leq\ker\left((i_1i_2)^{k_1},(i_2i_3)^{k_2},\ldots,(i_si_1)^{k_s}\right)\right\}\\ &=&t^{s/2}\sum_{\pi\in\mathcal P_2(k)}\frac{1}{N^{s/2+1}}\#\left\{i\in\{1,\ldots,N\}^s\Big|\pi\leq\ker\left((i_1i_2)^{k_1},(i_2i_3)^{k_2},\ldots,(i_si_1)^{k_s}\right)\right\} \end{eqnarray*} [[/math]]

(4) Our claim now is that in the [math]N\to\infty[/math] limit the combinatorics of the above sum simplifies, with only the noncrossing partitions contributing to the sum, and with each of them contributing precisely with a 1 factor, so that we will have, as desired:

[[math]] \begin{eqnarray*} M_k\left(\frac{Z}{\sqrt{N}}\right) &=&t^{s/2}\sum_{\pi\in\mathcal P_2(k)}\Big(\delta_{\pi\in NC_2(k)}+O(N^{-1})\Big)\\ &\simeq&t^{s/2}\sum_{\pi\in\mathcal P_2(k)}\delta_{\pi\in NC_2(k)}\\ &=&t^{s/2}|\mathcal{NC}_2(k)| \end{eqnarray*} [[/math]]

(5) In order to prove this, the first observation is that when [math]k[/math] is not uniform, in the sense that it contains a different number of [math]\circ[/math], [math]\bullet[/math] symbols, we have [math]\mathcal P_2(k)=\emptyset[/math], and so:

[[math]] M_k\left(\frac{Z}{\sqrt{N}}\right)=t^{s/2}|\mathcal{NC}_2(k)|=0 [[/math]]

(6) Thus, we are left with the case where [math]k[/math] is uniform. Let us examine first the case where [math]k[/math] consists of an alternating sequence of [math]\circ[/math] and [math]\bullet[/math] symbols, as follows:

[[math]] k=\underbrace{\circ\bullet\circ\bullet\ldots\ldots\circ\bullet}_{2p} [[/math]]

In this case it is convenient to relabel our multi-index [math]i=(i_1,\ldots,i_s)[/math], with [math]s=2p[/math], in the form [math](j_1,l_1,j_2,l_2,\ldots,j_p,l_p)[/math]. With this done, our moment formula becomes:

[[math]] M_k\left(\frac{Z}{\sqrt{N}}\right) =t^p\sum_{\pi\in\mathcal P_2(k)}\frac{1}{N^{p+1}}\#\left\{j,l\in\{1,\ldots,N\}^p\Big|\pi\leq\ker\left(j_1l_1,j_2l_1,j_2l_2,\ldots,j_1l_p\right)\right\} [[/math]]

Now observe that, with [math]k[/math] being as above, we have an identification [math]\mathcal P_2(k)\simeq S_p[/math], obtained in the obvious way. With this done too, our moment formula becomes:

[[math]] M_k\left(\frac{Z}{\sqrt{N}}\right) =t^p\sum_{\pi\in S_p}\frac{1}{N^{p+1}}\#\left\{j,l\in\{1,\ldots,N\}^p\Big|j_r=j_{\pi(r)+1},l_r=l_{\pi(r)},\forall r\right\} [[/math]]

(7) We are now ready to do our asymptotic study, and prove the claim in (4). Let indeed [math]\gamma\in S_p[/math] be the full cycle, which is by definition the following permutation:

[[math]] \gamma=(1 \, 2 \, \ldots \, p) [[/math]]

In terms of [math]\gamma[/math], the conditions [math]j_r=j_{\pi(r)+1}[/math] and [math]l_r=l_{\pi(r)}[/math] found above read:

[[math]] \gamma\pi\leq\ker j\quad,\quad \pi\leq\ker l [[/math]]

Counting the number of free parameters in our moment formula, we obtain:

[[math]] \begin{eqnarray*} M_k\left(\frac{Z}{\sqrt{N}}\right) &=&\frac{t^p}{N^{p+1}}\sum_{\pi\in S_p}N^{|\pi|+|\gamma\pi|}\\ &=&t^p\sum_{\pi\in S_p}N^{|\pi|+|\gamma\pi|-p-1} \end{eqnarray*} [[/math]]

(8) The point now is that the last exponent is well-known to be [math]\leq 0[/math], with equality precisely when the permutation [math]\pi\in S_p[/math] is geodesic, which in practice means that [math]\pi[/math] must come from a noncrossing partition. Thus we obtain, in the [math]N\to\infty[/math] limit, as desired:

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

This finishes the proof in the case of the exponents [math]k[/math] which are alternating, and the case where [math]k[/math] is an arbitrary uniform exponent is similar, by permuting everything.

■

This was for the computation, but in what regards now the interpretation of what we found, things are more complicated. The precise question is as follows: \begin{question} What is the abstract asymptotic distribution that we found, having as moments the numbers

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

for any colored integer [math]k=\circ\bullet\bullet\circ\ldots[/math]? \end{question} As a first observation, the above moment formula is very similar to the one for the usual complex Gaussian variables [math]G_t[/math], from chapter 1, which was as follows:

[[math]] N_k=t^{|k|/2}|\mathcal P_2(k)| [[/math]]

It is possible to make many speculations here, for instance in relation with the combinatorics from chapters 3-4, but we will do this later, once we will know more. Let us record however our observation as a partial answer to Question 6.9, as follows: \begin{answer} The abstract asymptotic distribution that we found appears as some sort of “free analogue” of the usual complex normal law [math]G_t[/math], with the underlying matching pairings being now replaced by underlying matching noncrossing pairings. \end{answer} Obviously, some interesting things are going on here. We will see in a moment, after doing some more combinatorics, this time in connection with the Wigner matrices, that there are some good reasons for calling this mysterious law “circular”.

Thus, for ending with our present study with a nice conclusion, we can say that the Gaussian matrices become “asymptotically circular”, with this meaning by definition that the [math]N\to\infty[/math] moments are those computed above. This is of course something quite vague, and we will be back to it in chapters 9-12 below, when doing free probability.

General references

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