6b. Wigner matrices

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Moving ahead now, let us investigate the second class of random matrices that we are interested in, namely the Wigner matrices, which are by definition self-adjoint. Here our results will be far more complete than those for the Gaussian matrices.

Let us first recall from the above that a Wigner matrix is by definition a random matrix which has i.i.d. centered complex normal entries, up to the constraint [math]Z=Z^*[/math]. In practice, this means that our matrix is 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 starting point for the study of these matrices, we have the following simple fact, making the connection with the theory of Gaussian matrices developed above:

Proposition

Given a Gaussian matrix [math]Z[/math], with independent entries following the centered complex normal law [math]G_t[/math], with [math]t \gt 0[/math], if we write

[[math]] Z=\frac{1}{\sqrt{2}}(X+iY) [[/math]]

with [math]X,Y[/math] being self-adjoint, then both [math]X,Y[/math] are Wigner matrices, of parameter [math]t[/math].

Show Proof

This is something elementary, which can be done in two steps, as follows:

(1) As a first observation, the result holds at [math]N=1[/math]. Indeed, here our Gaussian matrix [math]Z[/math] is just a random variable, subject to the condition [math]Z\sim G_t[/math]. But recall that the law [math]G_t[/math] is by definition as follows, with [math]X,Y[/math] being independent, each following the law [math]g_t[/math]:

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

Thus in this case, [math]N=1[/math], the variables [math]X,Y[/math] that we obtain in the statement, as rescaled real and imaginary parts of [math]Z[/math], are subject to the condition [math]X,Y\sim g_t[/math], and so are Wigner matrices of size [math]N=1[/math] and parameter [math]t \gt 0[/math], as in Definition 6.2.

(2) In the general case now, [math]N\in\mathbb N[/math], the proof is similar, by using the basic behavior of the real and complex normal variables with respect to sums.

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The above result is quite interesting for us, because it shows that, in order to investigate the Wigner matrices, we are basically not in need of some new computations, starting from the Wick formula, and doing combinatorics afterwards, but just of some manipulations on the results that we already have, regarding the Gaussian matrices.

To be more precise, by using this method, we obtain the following result, coming by combining the observation in Proposition 6.11 with the formula in Theorem 6.8:

Theorem

Given a sequence of Wigner random 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], up to [math]Z_N=Z_N^*[/math], we have

[[math]] M_k\left(\frac{Z_N}{\sqrt{N}}\right)\simeq t^{k/2}|NC_2(k)| [[/math]]

for any integer [math]k\in\mathbb N[/math], in the [math]N\to\infty[/math] limit.

Show Proof

This can be deduced from a direct computation based on the Wick formula, similar to that from the proof of Theorem 6.8, but the best is to deduce this result from Theorem 6.8 itself. Indeed, we know from there that for Gaussian matrices [math]Y_N\in M_N(L^\infty(X))[/math] we have the following formula, valid for any colored integer [math]K=\circ\bullet\bullet\circ\ldots\,[/math], in the [math]N\to\infty[/math] limit, with [math]\mathcal{NC}_2[/math] standing for noncrossing matching pairings:

[[math]] M_K\left(\frac{Y_N}{\sqrt{N}}\right)\simeq t^{|K|/2}|\mathcal{NC}_2(K)| [[/math]]

By doing some combinatorics, we deduce from this that we have the following formula for the moments of the matrices [math]Re(Y_N)[/math], with respect to usual exponents, [math]k\in\mathbb N[/math]:

[[math]] \begin{eqnarray*} M_k\left(\frac{Re(Y_N)}{\sqrt{N}}\right) &=&2^{-k}\cdot M_k\left(\frac{Y_N}{\sqrt{N}}+\frac{Y_N^*}{\sqrt{N}}\right)\\ &=&2^{-k}\sum_{|K|=k}M_K\left(\frac{Y_N}{\sqrt{N}}\right)\\ &\simeq&2^{-k}\sum_{|K|=k}t^{k/2}|\mathcal{NC}_2(K)|\\ &=&2^{-k}\cdot t^{k/2}\cdot 2^{k/2}|\mathcal{NC}_2(k)|\\ &=&2^{-k/2}\cdot t^{k/2}|NC_2(k)| \end{eqnarray*} [[/math]]

Now since the matrices [math]Z_N=\sqrt{2}Re(Y_N)[/math] are of Wigner type, this gives the result.

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Summarizing, all this brings us into counting noncrossing pairings. But here, let us recall from chapter 3 that we have the following well-known result:

Theorem

The Catalan numbers [math]C_k=|NC_2(2k)|[/math] are as follows:

• They satisfy [math]C_{k+1}=\sum_{a+b=k}C_aC_b[/math].
• The series [math]f(z)=\sum_{k\geq0}C_kz^k[/math] satisfies [math]zf^2-f+1=0[/math].
• This series is given by [math]f(z)=\frac{1-\sqrt{1-4z}}{2z}[/math].
• We have the formula [math]C_k=\frac{1}{k+1}\binom{2k}{k}[/math].

Show Proof

This is something that we know well from chapter 3, with (1) coming from the definition of [math]C_k[/math], and with [math](1)\implies(2)\implies(3)\implies(4)[/math] being routine, using standard calculus. Alternatively, and also explained in chapter 3, the formula in (4) can be established as well via a bijective proof, by counting Dyck paths in the plane.

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Getting back now to the Wigner matrices, we can convert the main result that we have about them, Theorem 6.12, into something more concrete, as follows:

Theorem

Given a sequence of Wigner random 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], up to [math]Z_N=Z_N^*[/math], we have

[[math]] M_{2k}\left(\frac{Z_N}{\sqrt{N}}\right)\simeq t^kC_k [[/math]]

in the [math]N\to\infty[/math] limit. As for the asymptotic odd moments, these all vanish.

Show Proof

This follows from Theorem 6.12 and Theorem 6.13. Indeed, according to the results there, the asymptotic even moments are given by:

[[math]] M_{2k}\left(\frac{Z_N}{\sqrt{N}}\right)\simeq t^k|NC_2(2k)|=t^kC_k [[/math]]

As for the asymptotic odd moments, once again from Theorem 6.12, we know that these all vanish. Thus, we are led to the conclusion in the statement.

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Summarizing, we are done with the moment computations, and with the asymptotic study, for both the Gaussian and the Wigner matrices. It remains now to interpret the results that we have, with the computation of the corresponding laws. As explained before, for the Gaussian matrices this is something quite complicated, with the technology that we presently have, and this will have to wait a bit, until we do some free probability.

Regarding the Wigner matrices, however, the problems left here are very explicit, and quite elementary, and we will solve them next, in the remainder of this chapter.

General references

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