6c. Semicircle laws

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In order to recapture the asymptotic measure of the Wigner matrices out of the moments, which are the Catalan numbers, there are several methods available, namely:

(1) Stieltjes inversion.

(2) Knowledge of [math]SU_2[/math].

(3) Cheating.

The first method, which is straightforward, without any trick, is based on the Stieltjes inversion formula, that we know from chapter 3. In fact, we have already applied in chapter 3 that formula to the Catalan numbers, with the following conclusion:

Proposition

The real measure having as even moments the Catalan numbers, [math]C_k=\frac{1}{k+1}\binom{2k}{k}[/math], and having all odd moments [math]0[/math] is the measure

[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

called Wigner semicircle law on [math][-2,2][/math].

Show Proof

This is something that we know, but since we will need the proof in what follows, in view of some generalizations, let us briefly recall it. The starting point is the formula in Theorem 6.13 for the generating series of the Catalan numbers, namely:

[[math]] \sum_{k=0}^\infty C_kz^k=\frac{1-\sqrt{1-4z}}{2z} [[/math]]

By using this formula with [math]z=\xi^{-2}[/math], we obtain the following formula, for the Cauchy transform of the real measure that we want to compute:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty C_k\xi^{-2k}\\ &=&\xi^{-1}\cdot\frac{1-\sqrt{1-4\xi^{-2}}}{2\xi^{-2}}\\ &=&\frac{\xi}{2}\left(1-\sqrt{1-4\xi^{-2}}\right)\\ &=&\frac{\xi}{2}-\frac{1}{2}\sqrt{\xi^2-4} \end{eqnarray*} [[/math]]

Now let us apply the Stieltjes inversion formula, from chapter 3, namely:

[[math]] d\mu (x)=\lim_{t\searrow 0}-\frac{1}{\pi}\,Im\left(G(x+it)\right)\cdot dx [[/math]]

The study of the limit on the right is then straightforward, going as follows:

(1) According to the general philosophy of the Stieltjes formula, the first term in the formula of [math]G(\xi)[/math], namely [math]\xi/2[/math], which is “trivial”, will not contribute to the density.

(2) As for the second term, which is something non-trivial, this will contribute to the density, the rule here being that the square root [math]\sqrt{\xi^2-4}[/math] will be replaced by the “dual” square root [math]\sqrt{4-x^2}\,dx[/math], and that we have to multiply everything by [math]-1/\pi[/math].

(3) As a conclusion, by Stieltjes inversion we obtain the following density:

[[math]] d\mu(x) =-\frac{1}{\pi}\cdot-\frac{1}{2}\sqrt{4-x^2}\,dx =\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

Thus, we have obtained the mesure in the statement, and we are done.

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More generally now, we have the following result:

Proposition

Given [math]t \gt 0[/math], the real measure having as even moments the numbers [math]M_{2k}=t^kC_k[/math] and having all odd moments [math]0[/math] is the measure

[[math]] \gamma_t=\frac{1}{2\pi t}\sqrt{4t-x^2}dx [[/math]]

called Wigner semicircle law on [math][-2\sqrt{t},2\sqrt{t}][/math].

Show Proof

This follows by redoing the above Stieltjes inversion computation, with a parameter [math]t \gt 0[/math] added. To be more precise, as before, the starting point is the formula from Theorem 6.13 for the generating series of the Catalan numbers, namely:

[[math]] \sum_{k=0}^\infty C_kz^k=\frac{1-\sqrt{1-4z}}{2z} [[/math]]

By using this formula with [math]z=t\xi^{-2}[/math], we obtain the following formula, for the Cauchy transform of the real measure that we want to compute:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty t^kC_k\xi^{-2k}\\ &=&\xi^{-1}\cdot\frac{1-\sqrt{1-4t\xi^{-2}}}{2t\xi^{-2}}\\ &=&\frac{\xi}{2t}\left(1-\sqrt{1-4t\xi^{-2}}\right)\\ &=&\frac{\xi}{2t}-\frac{1}{2t}\sqrt{\xi^2-4t} \end{eqnarray*} [[/math]]

Thus, by Stieltjes inversion we obtain the following density, as claimed:

[[math]] d\mu(x)=\frac{1}{2\pi t}\sqrt{4t-x^2}\,dx [[/math]]

But simplest is in fact, perhaps a bit by cheating, simply using the result at [math]t=1[/math], from Proposition 6.15, along with a change of variables. Indeed, by using Proposition 6.15, the even moments of the measure in the statement are given by:

[[math]] \begin{eqnarray*} M_{2k} &=&\frac{1}{2\pi t}\int_{-2\sqrt{t}}^{2\sqrt{t}}\sqrt{4t-x^2}\,x^{2k}\,dx\\ &=&\frac{1}{2\pi t}\int_{-1}^1\sqrt{4t-ty^2}\,(\sqrt{t}y)^{2k}\,\sqrt{t}\,dy\\ &=&\frac{t^k}{2\pi}\int_{-1}^1\sqrt{4-y^2}\,y^{2k}\,dy\\ &=&t^kC_k \end{eqnarray*} [[/math]]

As for the odd moments, these all vanish, because the density of [math]\gamma_t[/math] is an even function. Thus, one way or another, we are led to the conclusion in the statement.

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Talking cheating, another way of recovering Proposition 6.15, this time without using the Stieltjes inversion formula, but by knowing instead the answer to the question, namely the semicircle law, in advance, which is of course cheating, is as follows:

Proposition

The Catalan numbers are the even moments of

[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

called Wigner semicircle law. As for the odd moments of [math]\gamma_1[/math], these all vanish.

Show Proof

The even moments of the Wigner law can be computed with the change of variable [math]x=2\cos t[/math], and we are led to the following formula:

[[math]] \begin{eqnarray*} M_{2k} &=&\frac{1}{\pi}\int_0^2\sqrt{4-x^2}x^{2k}\,dx\\ &=&\frac{1}{\pi}\int_0^{\pi/2}\sqrt{4-4\cos^2t}\,(2\cos t)^{2k}2\sin t\,dt\\ &=&\frac{4^{k+1}}{\pi}\int_0^{\pi/2}\cos^{2k}t\sin^2t\,dt\\ &=&\frac{4^{k+1}}{\pi}\cdot\frac{\pi}{2}\cdot\frac{(2k)!!2!!}{(2k+3)!!}\\ &=&2\cdot 4^k\cdot\frac{(2k)!/2^kk!}{2^{k+1}(k+1)!}\\ &=&C_k \end{eqnarray*} [[/math]]

As for the odd moments, these all vanish, because the density of [math]\gamma_1[/math] is an even function. Thus, we are led to the conclusion in the statement.

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More generally, we have the following result, involving a parameter [math]t \gt 0[/math]:

Proposition

The numbers [math]t^kC_k[/math] are the even moments of

[[math]] \gamma_t=\frac{1}{2\pi t}\sqrt{4t-x^2}dx [[/math]]

called semicircle law on [math][-2\sqrt{t},2\sqrt{t}][/math]. As for the odd moments of [math]\gamma_t[/math], these all vanish.

Show Proof

This follows indeed from what we have in Proposition 6.17, via a quick change of variables, as explained at the end of the proof of Proposition 6.16.

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In any case, one way or another, we have our semicircle measures, and by putting now everything together, we obtain the Wigner theorem, 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]] \frac{Z_N}{\sqrt{N}}\sim\frac{1}{2\pi t}\sqrt{4t-x^2}dx [[/math]]

in the [math]N\to\infty[/math] limit, with the limiting measure being the Wigner semicircle law [math]\gamma_t[/math].

Show Proof

This follows indeed by combining Theorem 6.14 either with Proposition 6.16, and doing here an honest job, or with Proposition 6.18.

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There are many other things that can be said about the Wigner matrices, which appear as variations of the above, and we refer here to the standard random matrix books , , , . We will be back to them later on in this book, in chapter 10 below.

General references

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