10a. Circular variables

[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.

We have seen so far that free probability theory leads to a remarkable free analogue of the CLT, with the limiting measure being the Wigner semicircle law. This is certainly something very interesting, theoretically speaking, and by reminding the fact that the Wigner laws appear in connection with many fundamental questions in mathematics, in relation with random walks on graphs, with Lie groups, and with random matrices as well, there are certainly many things to be done, as a continuation of this.

However, no hurry, and we will do this slowly. As a first objective, which is something quite straightforward, now that we have a free CLT, we would like to have as well a free analogue of the complex central limiting theorem (CCLT), adding to the classical CCLT, and providing us with free analogues [math]\Gamma_t[/math] of the complex Gaussian laws [math]G_t[/math].

This will be something quite technical, and in order to get started, let us begin by recalling the theory of the complex Gaussian laws [math]G_t[/math]. We first have:

Definition

The complex Gaussian law of parameter [math]t \gt 0[/math] is

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

where [math]a,b[/math] are independent, each following the law [math]g_t[/math].

There are many things that can be said about these laws, simply by adapting the known results from the real case, regarding the usual normal laws [math]g_t[/math]. As a first such result, the above measures form convolution semigroups:

Proposition

The complex Gaussian laws have the property

[[math]] G_s*G_t=G_{s+t} [[/math]]

for any [math]s,t \gt 0[/math], and so they form a convolution semigroup.

Show Proof

This is something that we know from chapter 1, coming from [math]g_s*g_t=g_{s+t}[/math], by taking the real and imaginary parts of all variables involved.

■

We have as well the following complex analogue of the CLT:

Theorem (CCLT)

Given complex variables [math]f_1,f_2,f_3,\ldots\in L^\infty(X)[/math] which are i.i.d., centered, and with variance [math]t \gt 0[/math], we have, with [math]n\to\infty[/math], in moments,

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^nf_i\sim G_t [[/math]]

where [math]G_t[/math] is the complex Gaussian law of parameter [math]t[/math].

Show Proof

This is something that we know too from chapter 1, which follows from the real CLT, by taking real and imaginary parts. Indeed, let us write:

[[math]] f_i=\frac{1}{\sqrt{2}}(x_i+iy_i) [[/math]]

The variables [math]x_i[/math] satisfy then the assumptions of the CLT, so their rescaled averages converge to a normal law [math]g_t[/math], and the same happens for the variables [math]y_i[/math]. The limiting laws that we obtain being independent, their rescaled sum is complex Gaussian, as desired.

■

Regarding now the moments, we have here the following result:

Proposition

The moments of the complex normal law are the numbers

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

where [math]\mathcal P_2(k)[/math] is the set of matching pairings of [math]\{1,\ldots,k\}[/math].

Show Proof

This is again something that we know well too, from chapter 1, the idea being as follows, with [math]c=\frac{1}{\sqrt{2}}(a+ib)[/math] being the variable in Definition 10.1:

(1) In the case where [math]k[/math] contains a different number of [math]\circ[/math] and [math]\bullet[/math] symbols, a rotation argument shows that the corresponding moment of [math]c[/math] vanishes. But in this case we also have [math]\mathcal P_2(k)=\emptyset[/math], so the formula in the statement holds indeed, as [math]0=0[/math].

(2) In the case left, where [math]k[/math] consists of [math]p[/math] copies of [math]\circ[/math] and [math]p[/math] copies of [math]\bullet[/math]\,, the corresponding moment is the [math]p[/math]-th moment of [math]|c|^2[/math], which by some calculus is [math]t^pp![/math]. But in this case we have as well [math]|\mathcal P_2(k)|=p![/math], so the formula in the statement holds indeed, as [math]t^pp!=t^pp![/math].

■

As a final basic result regarding the laws [math]G_t[/math], we have the Wick formula:

Theorem

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 Wick 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 from chapter 1 too, the idea being as follows:

(1) In the case where we have a single complex normal variable [math]X[/math], we have to compute 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 coincides with the one in Theorem 10.4, namely:

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

(2) In general now, when expanding [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), then making the product of the numbers that we found. But this amounts in counting the partitions in the statement.

■

Let us discuss now the free analogues of the above results. As in the classical case, there is actually not so much work to be done here, in order to get started, because we can obtain the free convolution and central limiting results, simply by taking the real and imaginary parts of our variables. Following Voiculescu , , we first have:

Definition

The Voiculescu circular law of parameter [math]t \gt 0[/math] is given by

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

where [math]a,b[/math] are free, each following the Wigner semicircle law [math]\gamma_t[/math].

In other words, the passage [math]\gamma_t\to\Gamma_t[/math] is by definition entirely similar to the passage [math]g_t\to G_t[/math] from the classical case, by taking real and imaginary parts. As before in other similar situations, the fact that [math]\Gamma_t[/math] is indeed well-defined is clear from definitions.

Let us start with a number of straightforward results, obtained by complexifying the free probability theory that we have. As a first result, we have, as announced above:

Proposition

The Voiculescu circular laws have the property

[[math]] \Gamma_s\boxplus\Gamma_t=\Gamma_{s+t} [[/math]]

so they form a [math]1[/math]-parameter semigroup with respect to free convolution.

Show Proof

This follows from our result feom chapter 9 stating that the Wigner laws [math]\gamma_t[/math] have the free semigroup convolution property, by taking real and imaginary parts.

■

Next in line, also as announced above, and also from , we have the following natural free analogue of the complex central limiting theorem (CCLT):

Theorem (Free CCLT)

Given random variables [math]x_1,x_2,x_3,\ldots[/math] which are f.i.d., centered, with variance [math]t \gt 0[/math], we have, with [math]n\to\infty[/math], in moments,

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^nx_i\sim\Gamma_t [[/math]]

where [math]\Gamma_t[/math] is the Voiculescu circular law of parameter [math]t[/math].

Show Proof

This follows indeed from the free CLT, established in chapter 9, by taking real and imaginary parts. Indeed, let us write:

[[math]] x_i=\frac{1}{\sqrt{2}}(y_i+iz_i) [[/math]]

The variables [math]y_i[/math] satisfy then the assumptions of the free CLT, and so their rescaled averages converge to a semicircle law [math]\gamma_t[/math], and the same happens for the variables [math]z_i[/math]:

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^ny_i\sim\gamma_t\quad,\quad \frac{1}{\sqrt{n}}\sum_{i=1}^nz_i\sim\gamma_t [[/math]]

Now since the two limiting semicircle laws that we obtain in this way are free, their rescaled sum is circular, in the sense of Definition 10.6, and this gives the result.

■

Summarizing, we have so far complex analogues of both the classical and free CLT, and the basic theory of the limiting measures, including their semigroup property. As a conclusion to all this, let us formulate the following statement:

Theorem

We have classical and free limiting theorems, as follows,

[[math]] \xymatrix@R=45pt@C=40pt{ FCLT\ar@{-}[r]\ar@{-}[d]&FCCLT\ar@{-}[d]\\ CLT\ar@{-}[r]&CCLT } [[/math]]

the limiting laws being the following measures,

[[math]] \xymatrix@R=45pt@C=55pt{ \gamma_t\ar@{-}[r]\ar@{-}[d]&\Gamma_t\ar@{-}[d]\\ g_t\ar@{-}[r]&G_t } [[/math]]

which form classical and free convolution semigroups.

Show Proof

This follows indeed from the various results established above. To be more precise, the results about the left edge of the square are from the previous chapter, and the results about the right edge are those discussed in the above.

■

Going ahead with more study of the Voiculescu circular variables, less trivial now is the computation of their moments. We will do this in what follows, among others in order to expand Theorem 10.9 into something much sharper, involving as well moments.

For our computations, we will need explicit models for the circular variables. Following , and the material in chapter 9, let us start with the following key result:

Proposition

Let [math]H[/math] be the complex Hilbert space having as basis the colored integers [math]k=\circ\bullet\bullet\circ\ldots[/math]\,, and consider the shift operators on this space:

[[math]] S:k\to\circ k\quad,\quad T:k\to\bullet k [[/math]]

We have then the following equalities of distributions,

[[math]] S+S^*\sim\gamma_1\quad,\quad S+T^*\sim\Gamma_1 [[/math]]

with respect to the state [math]\varphi(T)= \lt Te,e \gt [/math], where [math]e[/math] is the empty word.

Show Proof

This is standard free probability, the idea being as follows:

(1) The first formula, namely [math]S+S^*\sim\gamma_1[/math], is something that we already know, in a slightly different formulation, from chapter 9, when proving the CLT.

(2) As for the second formula, [math]S+T^*\sim\Gamma_1[/math], this follows from the first formula, by using the freeness results and the rotation tricks established in chapter 9.

■

At the combinatorial level now, we have the following result, which is in analogy with the moment theory of the Wigner semicircle law, developed above:

Theorem

A variable [math]a\in A[/math] follows the law [math]\Gamma_1[/math] precisely when its moments are

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

for any colored integer [math]k=\circ\bullet\bullet\circ\ldots[/math]

Show Proof

By using Proposition 10.10, it is enough to do the computation in the model there. To be more precise, we can use the following explicit formulae for [math]S,T[/math]:

[[math]] S:k\to\circ k\quad,\quad T:k\to\bullet k [[/math]]

With these formulae in hand, our claim is that we have the following formula:

[[math]] \lt (S+T^*)^ke,e \gt =|\mathcal{NC}_2(k)| [[/math]]

In order to prove this formula, we can proceed as for the semicircle laws, in chapter 9 above. Indeed, let us expand the quantity [math](S+T^*)^k[/math], and then apply the state [math]\varphi[/math].

With respect to the previous computation, from chapter 9, what happens is that the contributions will come this time via the following formulae, which must succesively apply, as to collapse the whole product of [math]S,S^*,T,T^*[/math] variables into a 1 quantity:

[[math]] S^*S=1\quad,\quad T^*T=1 [[/math]]

As before, in the proof for the semicircle laws, from chapter 9, these applications of the rules [math]S^*S=1[/math], [math]T^*T=1[/math] must appear in a noncrossing manner, but what happens now, in contrast with the computation from the proof in chapter 9 where [math]S+S^*[/math] was self-adjoint, is that at each point where the exponent [math]k[/math] has a [math]\circ[/math] entry we must use [math]T^*T=1[/math], and at each point where the exponent [math]k[/math] has a [math]\bullet[/math] entry we must use [math]S^*S=1[/math]. Thus the contributions, which are each worth 1, are parametrized by the partitions [math]\pi\in\mathcal{NC}_2(k)[/math]. Thus, we obtain the above moment formula, as desired.

■

More generally now, by rescaling, we have the following result:

Theorem

A variable [math]a\in A[/math] is circular, [math]a\sim\Gamma_t[/math], precisely when its moments are given by the formula

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

for any colored integer [math]k=\circ\bullet\bullet\circ\ldots[/math]

Show Proof

This follows indeed from Theorem 10.11, by rescaling. Alternatively, we can get this as well directly, by suitably modifying Proposition 10.10 first.

■

Even more generally now, we have the following free version of the Wick rule:

Theorem

Given free variables [math]a_i[/math], each following the Voiculescu circular law [math]\Gamma_t[/math], with [math]t \gt 0[/math] being a fixed parameter, we have the Wick type formula

[[math]] tr(a_{i_1}^{k_1}\ldots a_{i_s}^{k_s})=t^{s/2}\#\left\{\pi\in\mathcal{NC}_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, with the inequality [math]\pi\leq\ker i[/math] on the right being taken in a technical, appropriate sense.

Show Proof

This follows a bit as in the classical case, the idea being as follows:

(1) In the case where we have a single complex normal variable [math]a[/math], we have to compute the moments of [math]a[/math], with respect to colored integer exponents [math]k=\circ\bullet\bullet\circ\ldots\,[/math], and the formula in the statement coincides with the one in Theorem 10.12, namely:

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

(2) In general now, when expanding the product [math]a_{i_1}^{k_1}\ldots a_{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. But this amounts precisely in counting the partitions in the statement, with the condition [math]\pi\leq\ker i[/math] there standing precisely for the fact that we are doing the various type (1) computations independently.

■

All the above was a bit brief, based on Voiculescu's original paper , and on his foundational free probability book with Dykema and Nica . The combinatorics of the free families of circular variables, called “circular systems”, is something quite subtle, and there has been a lot of work developed in this direction. For a complement to the above material, with a systematic study using advanced tools from combinatorics, we refer to the more recent book by Nica and Speicher . We will be actually back to this, in this book too, namely in chapter 12 below, when talking about free cumulants.

On the same topic, let us mention as well that various technical extensions and generalizations of the above results can be found, hidden as technical lemmas, throughout the random matrix and operator algebra literature, in connection with free probability, with the notable users of the circular systems including, besides Voiculescu himself, Dykema , Mingo, Nica, Speicher , , , , , and Shlyakhtenko .

Getting back now to the case of the single variables, from Theorem 10.12, the formula there has the following more conceptual interpretation:

Theorem

The moments of the Voiculescu laws are the numbers

[[math]] M_k(\Gamma_t)=\sum_{\pi\in\mathcal{NC}_2(k)}t^{|\pi|} [[/math]]

with “[math]\mathcal{NC}_2[/math]” standing for the noncrossing matching pairings.

Show Proof

This follows from the formula in Theorem 10.12. Indeed, we know from there that a variable [math]a\in A[/math] is circular, of parameter [math]t \gt 0[/math], precisely when we have the following formula, for any colored integer [math]k=\circ\bullet\bullet\circ\ldots\,[/math]:

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

Now since the number of blocks of a pairing [math]\pi\in\mathcal{NC}_2(k)[/math] is given by [math]|\pi|=|k|/2[/math], this formula can be written in the following alternative way:

[[math]] tr(a^k)=\sum_{\pi\in\mathcal{NC}_2(k)}t^{|\pi|} [[/math]]

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

■

All this is quite nice, when compared with the similar results from the classical case, regarding the complex Gaussian laws, that we established above, and with other results of the same type as well. As a conclusion to these considerations, we can now formulate a global result regarding the classical and free complex Gaussian laws, as follows:

Theorem

The complex Gaussian laws [math]G_t[/math] and the circular Voiculescu laws [math]\Gamma_t[/math], given by the formulae

[[math]] G_t=law\left(\frac{1}{\sqrt{2}}(a+ib)\right)\quad,\quad \Gamma_t=law\left(\frac{1}{\sqrt{2}}(\alpha+i\beta)\right) [[/math]]

where [math]a,b/\alpha,\beta[/math] are independent/free, following [math]g_t/\gamma_t[/math], have the following properties:

• They appear via the complex CLT, and the free complex CLT.
• They form semigroups with respect to the operations [math]*[/math] and [math]\boxplus[/math].
• Their moments are [math]M_k=\sum_{\pi\in D(k)}t^{|\pi|}[/math], with [math]D=\mathcal P_2,\mathcal{NC}_2[/math].

Show Proof

This is a summary of results that we know, the idea being as follows:

(1) This is something quite straightforward, by using the linearization results provided by the logarithm of the Fourier transform, and by the [math]R[/math]-transform.

(2) This is quite straightforward, too, once again by using the linearization results provided by the logarithm of the Fourier transform, and by the [math]R[/math]-transform.

(3) This comes by doing some combinatorics and calculus in the classical case, and some combinatorics and operator theory in the free case, as explained above.

■

More generally now, we can put everything together, with some previous results included as well, and we have the following result at the level of the moments of the asymptotic laws that we found so far, in classical and free probability:

Theorem

The moments of the various central limiting measures, namely

[[math]] \xymatrix@R=48pt@C=53pt{ \gamma_t\ar@{-}[r]\ar@{-}[d]&\Gamma_t\ar@{-}[d]\\ g_t\ar@{-}[r]&G_t } [[/math]]

are always given by the same formula, involving partitions, namely

[[math]] M_k=\sum_{\pi\in D(k)}t^{|\pi|} [[/math]]

where the sets of partitions [math]D(k)[/math] in question are respectively

[[math]] \xymatrix@R=50pt@C=45pt{ NC_2\ar[d]&\mathcal{NC}_2\ar[l]\ar[d]\\ P_2&\mathcal P_2\ar[l]} [[/math]]

and where [math]|.|[/math] is the number of blocks.

Show Proof

This follows by putting together the various moment results that we have, from the previous chapter, and from Theorem 10.15.

■

Summarizing, we are done with the combinatorial program outlined in the beginning of the present chapter. We will be back to this in the next chapter, by adding some new laws to the picture, coming from the classical and free PLT and CPLT, and then in the chapter afterwards, 12 below, with full conceptual explanations for all this.

General references

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